# Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends

код для вставкиСкачатьUNIVERSITY OF CALIFORNIA, IRVINE Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Electrical and Computer Engineering by Javier Rodriguez De Luis Dissertation Committee: Professor Franco De Flaviis, Chair Associate Professor Pai H. Chou Assistant Professor Filippo Capolino 2011 UMI Number: 3456974 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3456974 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Chapter 2 © 2010 IEEE Portion of Chapter 3 © 2009 IEEE. Portion of Chapters 4 to 7 © 2011 IEEE All other materials © 2011 Javier R. De Luis Dedication To my parents ii TABLE OF CONTENTS Page LIST OF FIGURES vii LIST OF TABLES xvii ACKNOWLEDGMENTS xix CURRICULUM VITAE xx ABSTRACT OF THE DISSERTATION xxi 1. INTRODUCTION 1 1.1 Interest in Reconfigurable antennas and microwave circuits. 1 1.2 Document organization 3 2. A FREQUENCY AGILE SWTICHED BEAM ANTENNA ARRAY SYSTEM 6 2.1 Introduction 6 2.2 Design of the single antenna element 9 2.2.1 Gap Size Effect on the resonant frequencies. 11 2.2.2 Optimum number of diodes analysis. 12 2.2.3 Radiation Efficiency analysis 16 The dual frequency phase shifter 17 2.3.1 The Dual Frequency phase delay section. 20 2.3.2 The Dual Frequency DC Bias. 21 2.4 Simulations and measurement results 22 2.5 Conclusions 28 2.3 References 28 3. BANDWIDTH ENHANCEMENT OF HIGH ISOLATION ISO-FREQUENCY REPEATERS USING MEMS-RECONFIGURABLE LOADED PARASITICS 32 3.1 Introduction 32 3.2 Description of the repeater system 34 3.3 Design Of In-plane (IPPs) and Out-of-plane (OUPPs) parasitics 36 iii 3.4 Combined isolation and matching optimization using the genetic algorithm optimizer. 37 3.5 Isolation and impedance bandwidth enhancement. 40 3.6 Additional Information. 41 43 References 4. AN ANALYTICAL ALGORITHM FOR PI-NETWORK IMPEDANCE TUNERS. 45 4.1 Introduction 45 4.2 Matching parameters and tunable pi-network configuration 47 4.3 Explanation of composite components frequently used in the algorithm 49 4.3.1 Equivalent Series Inductance & Its Characteristics 50 4.3.2 Composite Load and Source Impedance 51 Formulas and algorithm for perfect match 53 4.4.1 Perfect Match Solutions 53 4.4.2 Algorithm for Perfect Match 54 Formulas and algorithm for best match 55 4.5.1 Formulas for Best Match 55 4.5.2 Algorithm for Best Match 59 4.5.3 Algorithm for Branch 3 60 4.6 Comparison of algorithm and optimizer results 62 4.7 Algorithm extension for different Pi-network topologies. 67 4.4 4.5 4.7.1 Shunt Tunable Capacitor Having a Fixed Inductor Connected in Parallel. 68 4.8 4.9 4.7.2 Series portion of a pi-network consisting of series LC combination 68 Application examples. 69 4.8.1 Replacing Optimizer to Analyze Matching Performance 69 4.8.2 Predetermining the Match Tuning Setting for Known Antenna Load Impedance 70 4.8.3 Control Algorithm for Closed loop Impedance Matching 71 Conclusions. 73 Appendix A. Derivation of formulas (4.14)-(4.17) iv 73 Appendix B. Derivation of input reflection coefficient. 76 Appendix C. Parameters of Pi-Network tuner. 77 References 78 5. TUNABLE MEMS NOTCH FILTER AND ITS FREQUENCY CONTROL LOOP FOR WIRELESS TRANSCEIVERS 81 5.1 Introduction 81 5.2 Tunable Notch filter design 83 5.3 Notch filter frequency control loop based on reflected reference signal 85 5.3.1 System Description. 85 5.3.2 Mathematical Model and Formulation 89 5.3.3 Calculations and Simulations of Notch Filter Frequency Control Loop Performance 92 Implementations and measurements 96 5.4.1 Standalone Tunable Filter 96 5.4.2 Notch Filter Control Loop 100 Conclusions 104 5.4 5.5 Appendix A. Derivation of the transfer function of the notch filter. 104 Appendix B. Derivation of the frequency control loop differential equation. 105 References 107 6. APPROXIMATE CLOSE FORM DESIGN EXPRESSIONS FOR CAPACITIVELY LOADED PLANAR INVERTED-F ANTENNA 109 6.1 Introduction 109 6.2 Closed form expressions for capacitively loaded PIFA design 111 6.2.1 Transmission line model for a capacitively loaded PIFA 111 6.2.2 Input impedance. 113 6.2.3 Determining the patch length for a given operation frequency and loading capacitance. 115 6.2.4 Required loading capacitance for a given length and operating frequency. 118 6.2.5 Radiation efficiency and quality factor. 119 v 6.2.6 Impedance bandwidth. 126 6.2.7 Design Guidelines. 128 6.3 Measurement results 130 6.4 Conclusions 133 134 References 7. TUNABLE DUPLEXING ANTENNA SYSTEM FOR WIRELESS TRANSCEIVERS 137 7.1 Introduction 137 7.2 Duplexing Antenna System 139 7.2.1 Advantages from a system perspective 139 7.2.2 Single Tunable Element Design 141 7.2.3 Antenna Pair Design 148 7.2.4 Single Antenna Radiated Test 152 Conclusions 153 7.3 156 References vi LIST OF FIGURES Page Fig. 1.1 Conventional non tunable cell phone system architecture from Wispry Inc (www.wispry.com). ........................................................................................................... 2 Fig. 1.2 Novel tunable cell phone system architecture from Wispry Inc (www.wispry.com). ...... 2 Fig. 2.1 (a) Ominidirectional antenna system with single frequency operation showing interference between aircrafts. (b) switched beam dual frequency antenna system proposed in this chapter. .................................................................................................... 7 Fig. 2.2 Complete system diagram formed by a 4x1 dual frequency phased array and two different types of switched line phase shifters (Type 1, Type 2) ...................................... 8 Fig. 2.3 The antenna element consists of a dual microstrip patch topology connected through PIN diodes, printed over RT duroid 5880 (εr=2.2,tanδ=0.0009, 35µm copper thickness , 1.575mm dielectric thickness) with dimensions: Lhf=11.1mm, Whf=12.15mm, Llf=19.5mm, Wlf=21mm, g=0.5mm, gd=0.3mm, iip=2mm, iop=0.34mm, Wdc=0.2mm. Diode length=0.7mm, width=0.3mm and pad gap=0.3mm ....................... 9 Fig. 2.4 Diode circuit model for the ‘On’ and ‘Off’ states. Rs=1.52Ω, Ls=0.25nH and Coff=47fF, Rp=10KΩ. Maximum insertion loss @4-8GHz 0.4dB. Minimum isolation @4-8GHz 11dB. ............................................................................................................................... 10 Fig. 2.5 Scalar current distribution over the antenna at 4.7GHz for the cases: (a) Ideal ‘On’ with metallic connections (b) Ideal ‘Off’ with no connections, (c) Real ‘On’ with diode connections and (d) Real ‘Off’ with diode connections. Center of figure, standard rectangular patch resonating at 4.7GHz. ......................................................................... 14 vii Fig. 2.6 Resonant frequency change with increasing gap between IP and OP for cases: (a) Ideal ‘On’: Metal connection between Patches (low frequency) (b) Ideal ‘Off’: No connection between patches (high frequency), (c) Real ‘On’: Diodes between patches in ‘On’ state (low frequency) and (d) Real ‘Off’: Diodes between patches in ‘Off’ state (high frequency). ....................................................................................................................... 14 Fig. 2.7 Scalar current distribution over antenna different number of diodes in ‘On state: 1 diode (a) first resonance, (b) second resonance; 2 diodes (c) first resonance, (d) second resonance; 3 diodes (e) first resonance, (f) second resonance and 5 diodes (j) first resonance, (k) second resonance ..................................................................................... 15 Fig. 2.8 Imaginary part of the antenna edge impedance versus frequency showing first and second resonances for the cases of (a) 1 diode ‘On’, (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’ ................................................................................................................................. 15 Fig. 2.9 Normalized radiation pattern (dB) corresponding to the second resonance of cases (a) 1 diode ‘On’ (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’ ........................................................ 15 Fig. 2.10 Radiation Efficiency as a function of IP and OP slit lengths. The dynamic range between worst and best results is found to be 35%. ........................................................ 17 Fig. 2.11 Complete Type 1 Phase Shifter with phase delays from top to bottom branch of {0,45,90,135} degrees ..................................................................................................... 19 Fig. 2.12 Zoom into the phase shifter input region that contains the DF matching network, input SP4T distribution section and equal length launches. The Output SP4T distribution section is also included in the figure to show the direction of the diodes. ...................... 19 Fig. 2.13 Dual frequency phase delay section with dual stub configuration and showing optimization variables. .................................................................................................... 20 viii Fig. 2.14 Simulation results from a zero degree phase delay section.Input/Output magnitude of the reflection coefficient in dB (left axis) and Input/Output phase delay in degrees (right axis). Both frequencies remain matched while providing the desired zero degrees phase delays. .............................................................................................................................. 21 Fig. 2.15 Dual frequency DC bias formed by two quarter wavelength stubs, a dual frequency quarterwave section and a DC input pad to bias the diodes. ........................................... 22 Fig. 2.16 Fabricated Single element antenna and Zoom into the diode region. The devices are attached using silver epoxy to avoid overheating ........................................................... 24 Fig. 2.17 Simulated versus measured antenna magnitude of the reflection coefficient. For simplicity, the cases ‘On’ and ‘Off’ corresponding to operation at 4.7GHz and 7.5GHz are superimposed in the same graph. ............................................................................... 25 Fig. 2.18 Measured radiation patterns for the single element antenna operating in the (a-b) ‘On’ and (c-d) ‘Off’ cases. (a) and (c) show the E-plane while (b) and (d) show the H-plane. Solid and dashed line represents co-pol and cross-pol components respectively. The measured efficiency is 79% and 68% with associated gains of 5.4dB and 5dB at 7.5GHz and 4.7GHz, respectively. ............................................................................................... 25 Fig. 2.19 Fabricated 4x1 switched beam antenna array and phase shifters. ................................. 26 Fig. 2.20 Simulated with IE3D infinite ground (solid) and measured (dashed) normalized E-total radiation patterns for 4.7GHz (lf) and 7.5GHz (hf) corresponding to different progressive phases between antennas (a) lf,α=-135 sim-gain=10.8dB (b)lf, α=-45, simgain=12.5dB (c) lf,α=+135,sim-gain=10.5dB (d) lf,α=+45 sim-gain=13.7dB; (e) hf, α=135 sim- gain=14.8dB (f) hf, α=-45 sim-gain=14.7dB (g) hf, α=+135 sim-gain=12.7dB (h) hf, α=+45 sim-gain=15.2dB. ..................................................................................... 27 ix Fig. 3.1 Block Diagram of an Iso-frequency repeater. ................................................................. 33 Fig. 3.2 Proposed Iso-frequency repeater using RF-MEMS loaded parasitics. ............................ 35 Fig. 3.3 UCI’s RF MEMS switch picture. This specific device was built over a quartz substrate and is intended to be used as depicted in Fig. 3.2 ........................................................... 35 Fig. 3.4 A) OUPPs detail with parameters wo =0.5mm,= so =5mm,= do =20mm No =8 (each side) B) IPPs details. wi=0.5mm,= si =0.75mm Ni=46 (each side). ......................................... 37 Fig. 3.5 Network cascaded connection representing the loading conditions for each of the parasitic ports. The loading block is represented by a set of reflection coefficients being +1 for the open circuit and -1 for the short circuit conditions. ........................................ 39 Fig. 3.6 Results after optimization. (a) case 1, no obstacle. (b) case 2, obstacle at 1λx0.5λ. ....... 42 Fig. 3.7 Results after optimization of the isolation bandwidth. Case 2 and different fitness functions. ......................................................................................................................... 42 Fig. 3.8 a) Radiation patterns after optimization for case 1 and 2. b) Aperture coupled microstrip antenna with optimized impedance bandwidth and high isolation.................................. 42 Fig. 4.1 (a) ideal tunable low-pass pi-network and (b) equivalent circuit of practical pi-network tuner with fixed inductor and tunable capacitors. ........................................................... 48 Fig. 4.2 Circuit topology of pi-network tuner with driving source and load to be matched......... 49 Fig. 4.3 Equivalent or composite inductance Le as a function of frequency for different values of C3 ranging (a) 0.5 pF to 1.5 pF and (b) 2 pF to 5 pF. ...................................................... 51 Fig. 4.4 Tuning algorithm flow chart for Branch 1 and 2 ............................................................. 58 Fig. 4.5 Algorithm flow chart for Branch 3 .................................................................................. 61 Fig. 4.6 Input VSWR contour plots for 0.05 ≤ |ΓL| ≤ 0.95 obtained with (a) the algorithm and (b) the optimizer simulation at 700 MHz ............................................................................. 62 x Fig. 4.7 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm and (b) optimizer at 2170 MHz for the case of lossless network. ... 64 Fig. 4.8 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm and (b) optimizer at 700 MHz. ......................................................... 67 Fig. 4.9 Pi-network shunt C2 having a fixed inductor Ls connected in parallel........................... 68 Fig. 4.10 Pi-network with a series LC circuit in its series path. ................................................... 69 Fig. 4.11 VSWR versus frequency of a given antenna impedance with and without using the Pinetwork tuner. The tuner has components L= 6.8 nH, C1,min = C2,min = 0.8 pF, C1,max = C2, max = 5 pF, C3,min = 0.25 pF and C3,max = 4 pF and the values of the capacitor settings have been found using the presented tuning algorithm. ..................... 71 Fig. 4.12 Conceptual block diagram of closed loop impedance matching control system. The proposed algorithm is used here to adjust the tunable capacitor settings once the antenna complex impedance is detected. ...................................................................................... 72 Fig. 4.13 Equivalent circuit for (4.14) and (4.15) derivations. ..................................................... 74 Fig. 4.14 Equivalent circuit for (4.16) and (4.17) derivations. ..................................................... 75 Fig. 4.15 Equivalent circuit for Γin derivation ............................................................................. 76 Fig. 4.16 Equivalent circuit of pi-network tuner........................................................................... 77 Fig. 5.1 Block Diagram of an Iso-frequency repeater. ................................................................. 82 Fig. 5.2 Circuit schematic of the SPSZ tunable filter design. ....................................................... 84 Fig. 5.3 S-parameter response of notch filter without the nearby co-exisisting pass band........... 86 Fig. 5.4 Block diagram of tunable notch filter automatic frequency control loop utilizing reflected reference signal................................................................................................. 87 xi Fig. 5.5 Mathematical model of notch filter frequency automatic control loop utilizing the reflection signal. .............................................................................................................. 88 Fig. 5.6 Comparison of Closed form expressions and ADS simulations for the second order filter response with In this experiment, 20, ∆ 15 1000 and 0 10 ............................................................................................................. 93 Fig. 5.7 Simulated filter frequency control loop transient response for different values with =20KHz, A=0.12 and ∆ =15MHz, τ=0us. .............................................................. 94 Fig. 5.8 Simulated filter frequency control loop transient response for different values with =80, A=0.12 and ∆ =15MHz, =0us. ..................................................................... 95 Fig. 5.9 Simulated filter frequency control loop transient response for different A values with =20KHz, =80 and ∆ =15MHz, =0us ............................................................... 95 Fig. 5.10 Simulated filter frequency control loop transient response for different values with =80, =20KHz and ∆ =15MHz , =0.03............................................................. 96 Fig. 5.11 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell; b) 3D image of capacitance bits ............................................. 97 Fig. 5.12 Detailed TDC layout. Red cells and green cells are used for the resonator and insertion loss blocks, respectively .................................................................................................. 98 Fig. 5.13 Fabricated SPSZ tunable filter prototype board and filter structure detail. ................... 98 Fig. 5.14 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when tuning resonator block capacitor 2................................................. 99 Fig. 5.15 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when tuning insertion loss block capacitors 3 and 4. ......................... 100 Fig. 5.16 Discrete component implementation of the proposed notch filter control loop. ......... 101 xii Fig. 5.17 Measured filter frequency control loop transient response for different A=0.3 and ∆ =49MHz. ................................................................................................................ 102 Fig. 5.18 Measured filter frequency control loop transient response for different A=3 and ∆ =50MHz. ................................................................................................................ 103 Fig. 5.19 Simple notch filter circuit topology ............................................................................. 105 Fig. 5.20 Low pass loop filter and integrator .............................................................................. 106 Fig. 6.1 Capacitively loaded tunable PIFA ................................................................................. 111 Fig. 6.2 Transmission line equivalent model for the capacitively loaded PIFA ......................... 111 Fig. 6.3 (a) Equivalent transmission line model of a capacitively loaded PIFA, (b) the capacitor has been replaced by an open circuited section of transmission line ............................ 113 Fig. 6.4 Magnitude of the reflection coefficient using the transmission line model and method of moments simulations (IE3D) for a PIFA over air with 2, 4, ! 34.5, ! 1. Error refers to the absolute value of the difference in frequency between simulations and closed form expressions. ....................................................... 116 Fig. 6.5 Magnitude of the reflection coefficient using the transmission line model and method of moments simulations (IE3D) for a PIFA over air with 4, 4, ! 33.5, ! 1. Error refers to the absolute value of the difference in frequency between simulations and closed form expressions. ....................................................... 116 Fig. 6.6 Simulations and closed form (CF) expressions results for the total patch length (LT) required to make the PIFA resonating at 2GHz versus capacitve loading for different height values and 4 ...................................................................................... 118 xiii Fig. 6.7 Simulations and closed form (CF) expressions results for the resonance frequency versus capacitive loading for different height values. The patch total length was chosen to resonate at #$ 2% in absence of capacitive loading ( 0&'........................... 119 Fig. 6.8 (a) Surface current distribution in the fundamental mode opeation for a regular half wavelength microstrip antenna, (b) current distribution in a capacitively loaded PIFA antenna. The shadowed region indicates the current integration area. .......................... 122 Fig. 6.9 (a) Radiation efficiency versus capacitive loading obtained by MoM simulations (Zeland IE3D) and obtained closed form (CF) expressions. The antenna has parameters 4, #$ 2% ()* 0&', +$ 1, ,1, - 6/ (b) Radiation efficiency versus resonant frequency with ,0&, - 2&/. ................................... 123 Fig. 6.10 Efficiency reduction versus capacitive loading for different values of with 2. ............................................................................................................................. 125 Fig. 6.11 Efficiency reduction versus capacitive loading for different values of with 4, ............................................................................................................................. 125 Fig. 6.12 a) Impedance bandwidth reduction versus capacitive loading for different values of with 0 4. (b) Impedance bandwidth reduction versus frequency when loading capacitance is changed from C=0pF to 2pF. The patch total length was chosen to resonate at #$ 2% in absence of capacitive loading ( 0&' for each case. .... 127 Fig. 6.13 Relative impedance bandwidth reduction versus capacitive loading for different values of with 2 ................................................................................................... 128 Fig. 6.14 Relative impedance bandwidth reduction versus capacitive loading for different values of with 4. .................................................................................................. 129 xiv Fig. 6.15 Fabricated PIFA prototype over large ground plane. The ceramic capacitor is loaded at the antenna radiating edge. ............................................................................................ 131 Fig. 6.16 Measured magnitude of the PIFA reflection coefficient for different capacitive loadings. ........................................................................................................................ 132 Fig. 6.17 Fabricated Wheeler Cap covering the PIFA antenna for radiation efficiency measurement. ................................................................................................................. 132 Fig. 7.1 Duplexing antenna tunable front end concept depicting tunable narrowband antennas in combination with tunable notch filter, (b) Conventional non tunable front end comprising of a multiband broadband antenna with diplexer and external multiple duplexer modules. ......................................................................................................... 140 Fig. 7.2 Tunable PIFA antenna concept with shunt capacitor loading on antenna open edge. .. 141 Fig. 7.3 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell; b) 3D image of capacitance bits. .......................................... 142 Fig. 7.4 Tunable PIFA with two shunt tunable capacitors loading the antenna radiating edge and a series external chip capacitor in series with the tunable capacitors. .......................... 144 Fig. 7.5 Designed tunable single element and PCB dimensions................................................. 144 Fig. 7.6 Top and bottom views of the fabricated prototype. The top view shows the duplexing antenna structure, tunable capacitor dies and RF ground plane. The bottom side contains the control circuitry and interfacing multi-pin connector.. ........................................... 145 Fig. 7.7 Picture of the duplexing antenna built prototype........................................................... 146 Fig. 7.8 Simulated (HFSS) and Measured reflection coefficient of the single element antenna for different values of capacitive loading. .......................................................................... 147 xv Fig. 7.9 Measured reflection coefficient of the single element antenna for different values of capacitive loading. ......................................................................................................... 147 Fig. 7.10 Different configurations considered in the duplexing antenna isolation study. The distance ‘d’ is varied in discrete intervals and the S21 parameter is recorded for each variation at high and low frequency pair situations. ..................................................... 149 Fig. 7.11 Simulated S-parameters resulting from configuration of Fig. 7.10(a) when the two antennas are separated a distance (a)11.8mm and (b)36.8mm. Antenna with port 1 and port 2 are operating at 1850MHz (transmit antenna) 1930MHz (receive antenna), respectively. ................................................................................................................... 151 Fig. 7.12 Simulated parametric study of isolation (S21) between the two antennas operating at high (1980MHz, 2170MHz) and low (1850MHz, 1930MHz) frequency pairs for the configurations depicted in Fig. 7.10. ............................................................................. 154 Fig. 7.13 Measured reflection coefficients and S21(dB) between the two antennas operating at (a) high (Tx-Rx offset=190MHz) and (b) low (Tx-Rx offset=80MHz) frequency pairs. ....................................................................................................................................... 154 Fig. 7.14 Simulated (blue solid) and measured (red dashed) normalized radiation patterns for three orthogonal planes at two different loading conditions: (a) C=Cmax and (b) C=Cmin.. ....................................................................................................................... 155 xvi LIST OF TABLES Page Table 2.1 Phase delays for each antenna phase shifter ................................................................. 19 Table 2.2 Simulated and measured antenna center frequencies ................................................... 26 Table 2.3 Simulated and measured phase delays at 4.7GHz ........................................................ 26 Table 2.4 Simulated and measured phase delays at 7.5GHz ........................................................ 26 Table 3.1 Reflection coefficient (S11) for several combinations of OUPPs and optimized reflection coefficient using best configuration of IPPs. Bit ‘0’ meaning open circuited parasitic and bit ‘1’ meaning short circuited ................................................................ 37 Table 4.1 comparison of average VSWR resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9 .................................................................................................. 63 Table 4.2 comparison of average ∆GT resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9 .................................................................................................. 64 Table 4.3 Comparison of ∆GT resulting from algorithm and optimizer at different frequencies for low loss network case. .................................................................................................. 67 Table 4.4 Matching performance and tunable capacitor settings considering matching network with continuous capacitance. ........................................................................................ 71 Table 4.5 Matching performance and tunable capacitor settings considering matching network with discrete capacitance steps. .................................................................................... 71 Table 5.1 Final Frequency Error versus Value ........................................................................ 94 Table 5.2 Summary of Suppression and insertion losses within 5MHz Bandwidth ................... 100 Table 6.1 antenna resonant frequency versus capacitive loading .............................................. 131 Table 6.2 antenna impedance bandwidth versus capacitive loading .......................................... 132 xvii Table 6.3 antenna radiation efficiency versus capacitive loading .............................................. 133 Table 7.1 Wireless standards frequency specifications .............................................................. 138 Table 7.2 Requirement for cell phone antenna bandwidths ........................................................ 138 Table 7.3 antenna radiation efficiency versus capacitive loading ............................................. 153 xviii ACKNOWLEDGMENTS I would like to begin this dissertation expressing my most sincere appreciation to Prof. Franco De Flaviis whose valuable advice, guidance and trust have been the light of this journey. I would also like to thank Prof. Lluis Jofre, Prof. Roger Rangel and Mr. Pete Balsells for giving me the opportunity of becoming part of this family and enabling the dreams of others. I would not be at this point without them. Special thanks to the members of my committee Prof. Filippo Capolino and Prof. Pai Chou for their willingness, commitment and encouragement. To my friends and colleagues, Dr. Alfred Grau, Salvatore Campione, Anna Papio, Andrea Massenz, Nick Chopra and Ali Hosseini for their friendship and enjoyable presence during these years. Special thanks to Dr. Qizheng Gu and Dr. Art Morris for giving me the opportunity to be part of what they do and helping me to become a better engineer. To my friends back at my second home, Jose A. Parras, Miquel Cardona, Pere Salvatella, David Navarrete, Roger Piqueras, Raul Macule and Mika Pedros for making me laugh and sharing the best experiences of our life together. To the most loving people of my life, my parents. For making their unconditional support the role model of my life. I will never find enough words to express my gratitude. To my brother Roberto and my sister Carolina, for the incredible gift of growing up with them. To Evonne, for making my days happier, transforming me into a better person and always being with me no matter what. This work has been funded by the California Catalonia Innovation Program (2007-2008 and 2010-2011) and the Balsells fellowship for graduate studies (2006-2007 and 2009-2010). My deepest appreciation to everyone involved on these programs. xix CURRICULUM VITAE Javier R. De Luis 2000-06 B.S. in Telecommunications Engineering Universitat Politecnica Catalunya, Barcelona, Spain 2006-08 M.S in Electrical Engineering and Computer Science University of California, Irvine, USA 2008-09 Teaching Assistant, Henry Samueli School of Engineering University of California, Irvine 2006-11 Ph.D in Electrical and Computer Engineering University of California, Irvine, USA 2009-11 Intern Antenna/RF Engineer Wispry Inc, Irvine, USA FIELD OF STUDY Reconfigurable Antennas and microwave circuits, Electrical and Computer Engineering. SELECTED PUBLICATIONS De Luis, J.R.; de Flaviis, F.; , "Frequency Agile Switched Beam Antenna Array System," Antennas and propagation, IEEE Transactions on , vol.58, no.10, pp.3196-3204, Oct. 2010 De Luis, J. R.; Morris III, A.; Gu, Q.; De Flaviis, F.;,” A Tunable Asymmetric Notch Filter using RFMEMS” Microwaves theory and Techniques Society International Symposium, 2010. MTT-S 2010 De Luis, J.R.; Gu, Q.; Morris III, A.;De Flaviis, F, “Tunable MEMS Notch Filter and its Frequency Control Loop for Wireless Transceivers”, Microwave Theory and Techniques, IEEE Transactions on, (Submitted, under review). Vallechi, A.; De Luis, J.R; Capolino, F.; De Flaviis, F., “A Low Profile Folded Dipole Antenna on a Reactive High Impedance Substrate”, Antennas and propagation, IEEE Transactions on. (submitted, under review). De Luis, J.R.; Capdevila, S. Gu, Q.; Morris III, A.;De Flaviis, F, “Closed Form Expressions for Capacitively Loaded Planar Inverted-F Antenna Design”. Antennas and propagation, IEEE Transactions on. (submitted, under review) U.S Patent provisional app. “Tuning methods for tunable matching networks”. Q. Gu and J.R De Luis U.S Patent provisional app. “MEMS tunable notch filter frequency automatic control loop” Q. Gu and J.R De Luis. xx ABSTRACT OF THE DISSERTATION Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends By Javier Rodriguez De Luis Doctor of Philosophy in Electrical and Computer Engineering University of California, Irvine, 2011 Professor Franco De Flaviis, Chair Reconfigurable antennas and microwave circuits have attracted much attention in recent years due to their advantages compared to conventional designs. It is recognized that systems exploiting a certain extent of tunability potentially benefit from multi-functionality and performance enhancement, which ultimately translates into lower component count, size and overall cost of the transceiver. In this work, a collection of tunable/reconfigurable designs comprising of antennas and passive microwave circuits that spans to different application areas is presented in a comprehensive manner. In chapter 2 a dual frequency reconfigurable antenna array using PIN diodes and its associated phase shifter are presented. The system is capable of operation at two independent frequencies (4.7GHz and 7.5GHz) while switching between four different radiation patterns types for each frequency. Chapter 3 explores the feasibility of a novel of iso-frequency repeater system based on reconfigurable parasitic elements to maximize the isolation between the transmitting and receiving antennas. In chapter 4, an analytical tuning algorithm for a reconfigurable impedance matching pi-network based on tunable capacitors is presented. The algorithm is able to determine all tunable network component values for matching any given load impedance. xxi A single pole single zero notch filter using RF MEMS tunable capacitors and its associated frequency control loop system for automatic frequency tracking are presented in chapter 5. The high Q value of the tunable capacitors enables this filter to achieve 22dB rejection with less than 0.8dB insertion loss over a 5MHz bandwidth in the International Mobile Telecommunications band (IMT, 2.1 GHz). In chapter 6, a set of approximate closed form expressions for the input impedance, resonant frequency, radiation efficiency, quality factor and impedance bandwidth of capacitively-tunable loaded planar inverted-F antennas (PIFAs) used for cell phone applications are presented. Finally, chapter 7 presents a tunable antenna pair for wireless transceivers using RF MEMS tunable capacitors. This design utilizes antenna frequency agility to cover a frequency tuning range from 1850MHz to 2170MHz (16%) corresponding to the cellular wireless bands IMT-I and PCS and is designed to provide built-in filtering between antennas. xxii 1. INTRODUCTION 1.1 Interest in reconfigurable antennas and microwave circuits. Reconfigurable antennas and microwave circuits have received much attention in the past years playing an important role in the design of smart and adaptive systems. The evolution of such systems has been enormously benefited from recent improvements in the performance of the existing switching/tunable solid state or RF MEMS technologies. In contrast to conventional microwave designs, reconfigurable systems are able to adapt one or more of its operational characteristics in order to enhance performance or provide additional functionalities to the system. This multi-functional advantage most often enables a reduction in the number of required components compared to non tunable solutions, which ultimately impacts the overall cost of the front end. As an illustrative example of the potential advantages allowed by tunability let’s consider a classic cell phone system architecture as the one shown in Fig. 1.1. The conventional approach uses multiple multiband antennas followed by arrays of switches or duplexer modules in order to select the desired transmission branch specific of each standard. Additionally, each transmission path contains its own filtering and amplifiers stages. Recognizing the fact that the cell phone market evolves towards the increase in number of frequencies and standards, the classic problem approach leads to a complex front end and elevated number of components, which profoundly affects the design cycle time and cost of the transceiver. A solution based on a novel tunable front end is shown as an alternative in Fig. 1.2 where the different antennas have been substituted by a multiband antenna with a tunable matching network (or a tunable antenna), followed by a lower number of switching sections, tunable 1 filters, amplifiers and duplexers. The comparison of Fig. 1.1 and Fig. 1.2 makes apparent the flexibility that allows the reduction on the transceiver component number and provides a sensible argument for the need of research in the reconfigurable RF field. Fig. 1.1 Conventional non tunable cell phone system architecture from Wispry Inc (www.wispry.com). Fig. 1.2 Novel tunable cell phone system architecture from Wispry Inc (www.wispry.com). 2 Although a wide debate and extensive comparison can be found in literature regarding the most suited tunable device technology, the reality is that RF MEMS and solid state components offer advantages and disadvantages that must be clear prior to the design stage. The objective of this thesis is not to discuss the several technology options but to present different design concepts based on reconfigurable antennas and microwave circuits that cover different application areas from telemetry to cellular communications and use both RF MEMS and PIN diode devices. 1.2 Document organization In chapter 2 a dual frequency reconfigurable antenna array and its associated phase shifter are presented. The system is capable of operation at two independent frequencies (4.7GHz and 7.5GHz) while switching between four different radiation patterns pointing at different spatial locations for each frequency. The reconfigurability is achieved by using high performance PIN diodes acting as microwave switches. Chapter 3 explores the capacity and feasibility of a new concept of iso-frequency repeater system based on reconfigurable parasitic elements that is able to maximize the isolation between the transmitting and receiving antennas. This is done with the objective to allow a higher the power amplifier gain which results in an increased signal coverage area. In chapter 4, an analytical tuning algorithm for a reconfigurable pi-network impedance matching network is presented. The pi-network consists of tunable capacitors with finite tuning range, and a fixed value inductor. The algorithm is able to determine all tunable network component values for matching any given load impedance. The resulting matching performance measured either in terms of the input VSWR or transducer gain, is equivalent to that obtained from a commercial iterative optimization methods, but the algorithm runs significantly faster than the optimizer simulation. The proposed algorithm can be extended to a network tuner topology comprised of 3 four or less tunable components as long as it can be transformed into an equivalent pi-network topology. It is suitable for the design of fixed as well as tunable matching networks. A single pole single zero notch filter using RF MEMS tunable capacitors and its associated frequency control loop system are presented in chapter 5. The high Q value of the tunable capacitors enables this filter to achieve 22dB rejection with less than 0.8dB insertion loss over a 5MHz bandwidth in the International Mobile Telecommunications band (IMT, 2.1 GHz). The proposed design allows for independent tuning of the rejection and pass bands. A filter frequency control loop based on sensing the reflection phase change of the notch filter is developed to achieve automatic channel frequency tracking. The filter frequency control loop uses a reference signal (in this case, the transmitter carrier) to keep the tunable filter tracking the transmitter operation frequency. In chapter 6, a set of approximate closed form expressions for the input impedance, resonant frequency, radiation efficiency, quality factor and impedance bandwidth of capacitively-tunable loaded planar inverted-F antennas (PIFAs) used for cell phone applications are presented. These expressions yield insight into the effects of the most important design parameters over antenna performance and are useful to establish fundamental tradeoffs in early stages of the design. Experimental results by means of measurements and full wave simulations are presented to support the validity of the proposed closed form expressions showing good agreement between them. Finally, chapter 7 presents a tunable antenna pair for wireless transceivers using RF MEMS tunable capacitors. This design utilizes antenna frequency agility to cover a frequency tuning range from 1850MHz to 2170MHz (16%) corresponding to the cellular wireless bands IMT-I and PCS, although the same concept can be extended to additional bands. Each antenna in the 4 pair is designed to operate at the transmit and receive channels of the band, respectively, while the narrow impedance bandwidth characteristics of the single element provides built-in filtering between antennas. The level of isolation between elements is studied for several antenna arrangements depending on their relative orientation within the handset and can be exploited from a system perspective towards relaxing the duplexer specifications or allowing its substitution by narrowband RF MEMS tunable notch filters. A fabricated prototype is presented and measured to proof the concept and validate the design performance. Chapters 2 and 3: Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Chapters 4 to 7: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. 5 2. A FREQUENCY AGILE SWTICHED BEAM ANTENNA ARRAY SYSTEM 2.1 Introduction Increasing the angular resolution of antenna systems enhances the performance of the wireless mobile communication link. A phased array comprising of several elements and a control algorithm provides virtually unlimited control over azimuth and elevation scan angles at high speeds. This capability, however, comes with a high complexity and cost associated with a large number of elements, each having a digital phasing network with switching semiconductor or electromechanical elements. The high complexity and cost become more severe if a wideband (or multiple band) phased array is needed. Thus, as of today the use of phased array antennas is nearly limited to sophisticated military and space systems. However, many applications do not require a full scan capability involving the complexity of a phased array. In these cases, a simpler system, such as a reconfigurable switched beam antenna array, can be used. Reconfigurable antennas have received much attention in the past years playing an important role in the design of smart and adaptive systems [1]. Recent improvements in the performance of switching technologies such as RFMEMS and solid state switches, integrated in antennas and microwave circuits, have proved to be useful for a wide range of applications [2]-[10], including switched beam arrays. 3D structures, based on circular array configurations having a single active antenna surrounded by several loaded parasitics have been proposed to provide endfire beam switching for single [11] and dual frequency applications [12]. A multi-layered three beam system using triangular microstrip antennas has been presented in [13]. Other multi-beam approaches using 6 Butler Matrix [14] or phase switched solutions [15] can be found in the literature. However, the systems in [13]-[15] are limited to operation at a single frequency band. In this chapter, a novel dual frequency reconfigurable microstrip antenna array for next generation telemetry application with beam switching capability at 4.7GHz and 7.5GHz is presented. In contrast to [12], only one of the two operating bands remains active at each time, rejecting the non-active frequency without the need of an external diplexer. Furthermore, the system is a true planar implementation printed in a two layer PCB, reducing the complexity, space and total cost of the overall system. The dual frequency switching capability will provide a better usage of the spectrum resources and will also allow several nearby systems to communicate at high speed with the ground station simultaneously. A system comprising of ominidirectional antennas and single frequency operation as shown in Fig. 2.1(a) may create harmful interferences between aircraft. Fig. 2.1 (a) Ominidirectional antenna system with single frequency operation showing interference between aircrafts. (b) switched beam dual frequency antenna system proposed in this chapter. However, a system deployed as shown in Fig. 2.1(b), with two different frequencies and switched beam capability, minimizes the jamming risk and enables a fully functional system. Based on 7 this concept, the system proposed in this chapter is composed of a dual frequency (DF) reconfigurable antenna and a single feed dual frequency reconfigurable switched line phase shifter. Two different features, frequency and beam maximum position, can be selected by introducing PIN diodes in the system. Progressive phases between antennas of {+45,+135,-45,135} can be chosen to generate a switched beam with beam maxima pointing to {15,45,-15,45} and {13,30,-13,-30} degrees at 4.7GHz and 7.5GHz respectively. The proposed design is shown in Fig. 2.2 and can be divided in two blocks: a DF reconfigurable microstrip antenna and DF switched line phase shifter explained in sections 2.2 and 2.3, respectively. Section 2.4 presents system simulations and measurement results. The input and output of each block is matched to a system impedance of 50Ω. The modular design approach of each sub-block allows for a simple design process flow that can be reproduced for any desired frequency ratio. Fig. 2.2 Complete system diagram formed by a 4x1 dual frequency phased array and two different types of switched line phase shifters (Type 1, Type 2) 8 Fig. 2.3 The antenna element consists of a dual microstrip patch topology connected through PIN diodes, printed over RT duroid 5880 (εr=2.2,tanδ=0.0009, 35µm copper thickness , 1.575mm dielectric thickness) with dimensions: Lhf=11.1mm, Whf=12.15mm, Llf=19.5mm, Wlf=21mm, g=0.5mm, gd=0.3mm, iip=2mm, iop=0.34mm, Wdc=0.2mm. Diode length=0.7mm, width=0.3mm and pad gap=0.3mm 2.2 Design of the single antenna element The antenna element used in the 4x1 linear array of Fig. 2.2 is shown in detail in Fig. 2.3. The design consists of a printed patch over a 1.575mm thick Duroid RT-5880 substrate with εr=2.2, tanδ=0.0009 at f=10GHz and 34µm copper thickness, backed with a metallic ground plane. A small inner rectangular patch (IP) is partially surrounded by a larger U-shaped outer patch (OP). Both patches are connected through three high frequency GaAs PIN diodes (model MA4AGCFCP910, vendor M/A-COM) acting as microwave switches with equivalent circuit model shown in Fig. 2.4. The element values of the equivalent lumped circuit model were found in order to match a single diode measurement mounted in a microstrip line test fixture over the frequency range from 4GHz to 8GHz. The device was characterized and the test fixture was deembedded from the measurement, using a TRL calibration procedure. The antenna is fed by a microstrip line connected to the middle point of the IP left edge. In the fundamental mode, the left and right edges of the antenna are responsible for the radiation 9 phenomena and the RF current resonates along the x-axis (Fig. 2.3). The diodes are placed on the right edge of the inner patch to allow the currents to flow to the OP when required. Fig. 2.4 Diode circuit model for the ‘On’ and ‘Off’ states. Rs=1.52Ω, Ls=0.25nH and Coff=47fF, Rp=10KΩ. Maximum insertion loss @4-8GHz 0.4dB. Minimum isolation @4-8GHz 11dB. The patches are separated by a distance of 0.5mm, which is sufficient to leave space for the diode placement and to provide physical isolation. When the diodes are in the 'Off’ state (or reverse bias), the RF signal flows mainly in the inner patch while the outer patch acts as a parasitic element and the antenna operates in the high frequency mode (7.5GHz). Similarly, when the diodes are switched to the 'On' state (forward bias), the current flows in both patches resulting in an increase of the effective area of the antenna and the system operates in the low frequency mode (4.7GHz). The sizes of both patches have been optimized simultaneously for resonance at both frequency bands, as explained in the next subsections. The DC current path returning to ground necessary for the diode biasing, is designed using a high impedance 4.7GHz quarter wave microstrip line ending in a pad connected to the antenna ground plane. The short circuit at the pad is transformed into an open circuit seen from the OP edge, making the ground path section transparent to the RF signal. It was observed that the effect of the quarterwave biasing line on the 7.5GHz resonance, when the diodes are in the ‘Off’ state, was negligible due to the small RF current flowing in the OP as shown in Fig. 2.5. When the diodes are in the ‘On’ state, the effect of the biasing line on the radiation pattern at 4.7GHz produced a decrease of 0.4dB on the antenna gain. 10 Due to the different topology for both operation states, the edge resonance resistance seen from the feeding point has different values depending on the operative frequency band. Therefore, a dual frequency matching network based on a dual stub configuration was designed. 2.2.1 Gap Size Effect on the resonant frequencies. As a first step in the design of any dual-frequency dual-patch antenna, it is important to understand the effects of the gap size between the IP and OP on the high and low resonant frequencies. In order to analyze this effect, two stand alone rectangular patches were designed to resonate at 4.7GHz (L=20mm 0.46λeff at 4.7GHz, W=22mm) and 7.5GHz (L=12.2mm 0.45λeff at 7.5GHz, W=13.25mm) respectively according to [16]-[18]. Both independent patches were then combined together into the same space leaving a 0.5mm gap between them as shown in Fig. 2.5(ad). Depending on the interconnection and operating state, four different cases of the current distribution in the antenna were plotted. The antenna current distribution gives a qualitative idea of the level of IP/OP isolation in the diode ‘Off’ state and the current ability to flow from IP to OP in the ‘On’ state. Fig. 2.5 shows the low frequency operation mode with three metallic connections (ideal ‘On’ state) and their absence (ideal ‘Off’ state), respectively. On the other hand, Fig. 2.5 shows the high frequency operation using the forward biased diode S-parameters (real ‘On’ state) and reverse biased diode parameters (real ‘Off’ state) respectively. It is observed that the current distributions of the antenna at low frequency for the ‘ideal’ and ‘real’ ‘On’ states are very similar to the standard reference patch in the center of the figure. Fig. 2.5(b) gives an idea on how much current is induced in the OP only due to the proximity coupling effect when both patches are physically isolated. In addition to this coupling, Fig. 2.5(d) produces an additional current leakage near the diode connections due to the imperfect isolation of the diode in ‘Off’ state. 11 Intuitively, it is expected that the biggest deviation in frequency value from the stand alone patch case would be produced for the case of Fig. 2.5(d) because the IP current distribution presents more differences with respect to the stand alone patch. To verify this statement and to study the gap size effect on the antenna resonant frequency, a numerical experiment using full wave analysis was performed. The gap size was increased from 0.05mm to 2mm in 0.05mm steps, while the associated frequency change was monitored as shown in Fig. 2.6. The difference, between ‘ideal’ (a) and ‘real’ (c) ‘On’ states curves for low frequency is clearly less significant than the difference between (b) and (d) ‘Off’ state (high frequency) cases, which matches with the current distribution comparison. The average low frequency in Fig. 2.6 is 4.6GHz which is only 1% lower than the stand alone patch case. This decrease is due to the slightly longer current path forced by the connections between both patches. When the gap distance gets larger, curve (b) shows no difference in frequency with respect to the 7.5GHz stand alone patch due to the negligible proximity coupling between IP and OP. However, this effect is not seen on curve (d) due to existing signal leakage produced by the diode in the ‘Off’ state. Similarly, when the gap distance is small, both curves (b) and (d) show a decrease in the resonant frequency produced by the effective increase on antenna size created by the IP/OP mutual coupling. 2.2.2 Optimum number of diodes analysis. In this study we found that the number of diodes integrated in the antenna affects the radiation efficiency and system losses. In addition, a higher number of devices will require more DC power consumption and will have higher noise generation. Therefore, a minimum number of diodes for good operation at both frequencies must be found. One to five diodes were integrated in the antenna as shown in Fig. 2.7 and considered in the following analysis. Fig. 2.8 shows the 12 behavior of the imaginary part of the input impedance with frequency when one, two and three diodes are switched ‘On’ forcing the antenna to operate at low frequency. The fundamental resonant modes for these are observed at 4.48GHz, 4.59GHz and 4.6GHz respectively. When a smaller number of diodes is used, frequency is lowered due to the increased current path enforced by the topology. Even more important is the higher frequency resonances observed at 8.5GHz, 8.73GHz and 9.2GHz for the one, two and three diode cases respectively. In order to study all the mentioned resonances, the average current distribution over the patch operating in ‘On’ state is shown in Fig. 2.7. The low frequency fundamental modes and higher frequency resonances are shown in the top and bottom rows, respectively. An additional case of 5 diodes is also introduced (Fig. 2.7(j-k)) to show that further increase in diode number does not create a significant difference in the patch current distribution. This is because the two diodes placed horizontally in the non radiating edges do not contribute to the vertical current flow in the patch. The fundamental mode current flow and the value of the associated resonant frequency are clearly closer to the stand alone patch case of Fig. 2.5 when three diodes are used. The radiation patterns corresponding to the higher frequency resonances for all considered cases are shown in Fig. 2.9. The pattern for the second resonance (at 8.5GHz), corresponding to 1-diode case has a radiation behavior typical of a fundamental mode. This is due to the fact that the current is very confined to the diode region when crossing to the OP, thus the IP remains resonant resulting in a dual frequency antenna (instead of a switchable one). One can actually perceive that behavior on the current plot of Fig. 2.7(b). However, when three diodes are used, the second resonance appears considerably higher at 9.2GHz (around 2flow) and the radiation pattern shows a null in the normal direction typical of a higher order mode of the radiating OP (instead 13 of a fundamental one). In this case, the IP is not contributing to radiation and switching becomes effective. The two diode pattern case of Fig. 2.9(b) is in between the two previous cases. If only one diode was chosen for the design, its 8.5Ghz resonance could create an undesirable spurious radiating mode close to the target high frequency of 7.5GHz. This could potentially harm the receiver if filtering circuitry is not implemented. Therefore, three diodes will be used for the design to avoid requiring an external diplexer while guaranteeing a switchable behavior with good operation at 4.7GHz. Fig. 2.5 Scalar current distribution over the antenna at 4.7GHz for the cases: (a) Ideal ‘On’ with metallic connections (b) Ideal ‘Off’ with no connections, (c) Real ‘On’ with diode connections and (d) Real ‘Off’ with diode connections. Center of figure, standard rectangular patch resonating at 4.7GHz. 7.75 7.50 7.25 7.00 Frequency (GHz) 6.75 (a) Metal Connection between patches (b) No connection between patches (c) Diodes between pacthes in ON state (d) Diodes between patches in OFF state 6.50 6.25 6.00 5.75 Gap=2mm Gap=0.05mm 5.50 5.25 5.00 4.75 4.50 4.25 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 1.95 Gap (mm) Fig. 2.6 Resonant frequency change with increasing gap between IP and OP for cases: (a) Ideal ‘On’: Metal connection between Patches (low frequency) (b) Ideal ‘Off’: No connection between patches (high frequency), (c) Real ‘On’: Diodes between patches in ‘On’ state (low frequency) and (d) Real ‘Off’: Diodes between patches in ‘Off’ state (high frequency). 14 Fig. 2.7 Scalar current distribution over antenna different number of diodes in ‘On state: 1 diode (a) first resonance, (b) second resonance; 2 diodes (c) first resonance, (d) second resonance; 3 diodes (e) first resonance, (f) second resonance and 5 diodes (j) first resonance, (k) second resonance 200 (a) 1 Diode (b) 2 Diodes (c) 3 Diodes 150 100 Im(Zin) 50 0 -50 -100 -150 -200 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Frequency(GHz) Fig. 2.8 Imaginary part of the antenna edge impedance versus frequency showing first and second resonances for the cases of (a) 1 diode ‘On’, (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’ Fig. 2.9 Normalized radiation pattern (dB) corresponding to the second resonance of cases (a) 1 diode ‘On’ (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’ 15 2.2.3 Radiation Efficiency analysis Each diode can be considered like a passive device that introduces losses into the system. The loss sources are mainly ohmic loss (insertion loss) due to the device itself and mismatch loss due to the diode placement within the antenna. The transducer gain expression [19] takes into account both loss sources and can be expressed as: 2 GT = 1 − Γs Pload = Pavailable 1 − Γs Γin 2 S 21 2 1 − ΓL 2 1 − S 22 ΓL 2 (2.1) Where Pload and Pavailable in this cases, refer the power delivered to the outer patch and inner patch respectively. Γs is the reflection coefficient looking towards the IP from the diode positive terminal. Γin is the reflection coefficient looking to the positive terminal of the diode from the IP . ΓL is the reflection coefficient looking to the OP from the diode negative terminal, S22 is the reflection coefficient looking towards the IP from the negative terminal of the diode and S21 represents the transmission coefficient in 50Ω environment. Dealing with passive devices, the maximum achievable transducer gain is 0dB. Assuming ohmic losses, the optimum transducer gain for a specific device can be achieved satisfying simultaneous matching conditions (SMC) imposed by Γs = Γin* and ΓL = S22*, which minimize the mismatch loss. This is an important condition as long and it can be related to the radiation efficiency of the antenna. The input impedance seen at the edge of the IP and OP where the terminals of the diode are connected can be considerably different. In order to find impedances that satisfy the SMC condition, slits are introduced in the IP and OP sections as shown in Fig. 2.3. The length of both slits is changed independently from 0mm (no slit) to 5.3mm (half of the IP length). The associated radiation efficiency of the antenna is evaluated using Full Wave (Zeland IE3D MoM base software) simulations for each different pair of slit length for a total of 2500 combinations. The radiation 16 efficiency versus slit length information is then extracted from the simulation data and plotted in Fig. 2.10. Efficiency varies from 40% to 74% between the two extreme cases. The efficiency improvement due to the proper slit length has an associated gain increase from 5.88dB to 6.48dB. This result demonstrates the importance of finding SMC for any diode placed within the antenna. In a second study, the slits for the center and corner diodes were optimized independently, but no improvement above the mentioned 74% efficiency was observed by using this approach. Fig. 2.10 Radiation Efficiency as a function of IP and OP slit lengths. The dynamic range between worst and best results is found to be 35%. 2.3 The dual frequency phase shifter A microwave phase shifter is a two port network that provides a specific phase delay to a signal travelling from its input to output ports at a given frequency. Different design approaches and topologies are widely found in the literature such us loaded line phase shifters [20], quadrature reflection phase shifters [21] or Schiffman phase shifter [22]. However, when identical phase delays are required at two different frequencies, a solution based on a non-tunable passive circuit 17 is not available or its complexity increases [23] allowing only a very specific combination of phase delays. In general, switchable solutions are required for dual frequency operation, but the complexity and fabrication challenges may play an important role as in some reported solutions [24]. In this chapter, a novel reconfigurable switched line phase shifter using commercial PIN diodes is presented following a simple design procedure. In a switched line phase shifter, the RF input signal can travel between different transmission paths before arriving to the output resulting in different phase delays. In this system, four paths or branches can be selected as shown in Fig. 2.11. The phase shifters (PS) for antennas #1 and #4 (Type 1 PS) shown in Fig. 2.2 are identical, and can provide a switchable phase delay of {0,45,90,135} degrees at 4.7GHz and 7.5GHz simultaneously. Similarly, the PS connected to antennas #2 and #3 (Type 2 PS) can provide a delay of {270,45,90,135} degrees at both frequencies. The different combinations of states in PS #1 to #4 shown in Table I can produce four different progressive phases between antennas of {-135,-45,+45,+135}, that have associated radiation beam maxima at {+15,+45,-15,-45} degrees for 4.7GHz and {+13,+30,-13,-30} degrees for 7.5GHz. The design consists of four passive sub-circuits that are designed separately and cascaded to form the complete system. Each of these will be explained in the following sections: the input/output matching networks, the SP4T distribution blocks, the dual frequency phase delay section and the dual frequency DC bias network. The design shown in Fig. 2.11 is fully planar using microstrip transmission lines printed on a RT Duroid 6006 substrate (εr=6.15, tanδ=0.0027 at f=10GHz, 35µm copper thickness and 0.635mm board thickness). The same PIN diode model used for the antenna element was also used for the phase shifter design. 18 Fig. 2.11 Complete Type 1 Phase Shifter with phase delays from top to bottom branch of {0,45,90,135} degrees Desired Progressive Phase (deg) -135 -45 45 135 Table 2.1 Phase delays for each antenna phase shifter Phase Phase Phase PS # 1 PS # 2 PS # 3 (deg) (deg) (deg) 45 270 135 135 90 45 0 45 90 0 135 270 Phase PS # 4 (deg) 0 0 135 45 Fig. 2.12 Zoom into the phase shifter input region that contains the DF matching network, input SP4T distribution section and equal length launches. The Output SP4T distribution section is also included in the figure to show the direction of the diodes. In reality, due to the non perfect isolation of the diode, the non active ‘Off’ branches always contribute with some degree of shunt parasitic reactance, deteriorating return loss conditions in the input and output ports of the phase shifter. Therefore, dual frequency input and output 19 matching networks based on stepped impedance transmission lines are connected to the input and output SP4T ports (Fig. 2.12), to provide good reflection coefficient characteristics (VSWR<2) for all eight possible combinations (four phases and two frequencies). Both, matching networks and distribution sections must be identical for both types of phase shifter to avoid any phase perturbation. 2.3.1 The Dual Frequency phase delay section. Fig. 2.13 Dual frequency phase delay section with dual stub configuration and showing optimization variables. The DF phase delay section of Fig. 2.13 is the fundamental building block of the phase shifter. It consists of a transmission line loaded with two shunt opened stubs that are able to provide the same phase delay at two different frequencies simultaneously. In [25], a dual frequency delay network is presented and the closed form expressions are obtained for the particular case of 90 degrees phase delay. However, phases different from 90 degrees make the mathematical approach unfeasible so an optimization tool is required. The length (Lstub), width (Wstub) and separation (L_line) between stubs are set as optimization variables in a microwave circuit simulator (Applied Wave Research, AWR) to achieve each specific phase delay goal. IE3D was used afterwards to perform full wave analysis and for fine tuning of the phase shifter purposes. The simulation result for the case of zero degree phase delay at 4.7GHz and 7.5 GHz with 20 magnitude of the reflection coefficient below -20dB at both frequencies is shown in Fig. 2.14. The same approach is used for the rest of the targeted phases with similar results. 0 200 100 50 -20 0 -30 -50 Phase (Deg) Mag. Reflection Coefficient (dB) 150 -10 -100 -40 (a) Input/Output Return Loss (b) Input/Output Phase Difference -150 -50 -200 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Frequency (GHz) Fig. 2.14 Simulation results from a zero degree phase delay section.Input/Output magnitude of the reflection coefficient in dB (left axis) and Input/Output phase delay in degrees (right axis). Both frequencies remain matched while providing the desired zero degrees phase delays. 2.3.2 The Dual Frequency DC Bias. Biasing current is required to control the diode states for switching the circuit ‘On’ and ‘Off’. Surface mount components may be used for this purpose. However, when operating at high frequencies, catalog rated values may change and issues may arise with component SRF. In these cases a printed approach is preferred such as the DC Bias circuit shown in Fig. 2.15, which does not require surface mount components. Two quarter wavelength radial open stubs are connected in shunt to a microstrip line. Each of them transforms the open circuit in point 'A' into a short circuit seen from point 'B' for each frequency. The network after point 'B' is a dual frequency quarterwave transformer that transforms the short circuit in ‘B’ to open circuit in ‘C’ for 4.7GHz and 7.5GHz. Thus, the network seen from point ‘C’ becomes transparent to the RF signal flow. Point ‘D’ is used as a DC-current input pad that allows the diodes to be biased in the circuit. Anything surrounding the DC pad should not affect RF performance if both radial stubs are 21 properly designed. Each diode requires a current of 10mA in forward mode (‘On’), while zero volts are used in reverse mode (‘Off’). If the RF signal needs to travel through the upper branch of Fig. 2.11, 0V are applied to the pads DC#1,3,4,5,6 while a voltage that provides 10mA would be applied to DC pad #2. Fig. 2.15 Dual frequency DC bias formed by two quarter wavelength stubs, a dual frequency quarterwave section and a DC input pad to bias the diodes. 2.4 Simulations and measurement results The single element antenna, with optimized slit lengths and patch sizes was fabricated as shown in Fig. 2.16. The three PIN diodes were placed by hand using conductive epoxy to avoid overheating the device. For testing purposes, the bias current was introduced through an external coaxial bias T. However, a printed dual frequency bias could be designed for the same purpose. The magnitude of the reflection coefficient comparison between simulated (IE3D) and measured results is in very good agreement as shown in Fig. 2.17, where both frequency bands present good matching (<-15dB). For convenience, the ‘On’ and ‘Off’ cases are superimposed in the same graph. Table II shows the target versus simulated and measured frequencies. In all cases, return loss below 10dB was achieved for the target frequency. 22 The measured radiation patterns in both principal planes for the single antenna element at 4.7GHz and 7.5GHz are shown in Fig. 2.18(a-b) and Fig. 2.18(c-d), respectively. Peak gains of 5dB and 5.4dB are obtained for 4.7GHz and 7.5GHz respectively. Efficiency values of 68% for the low band and 79% for the high band were obtained. Four identical antenna elements were used to form the 4x1 linear array. Deciding the distance between elements in a dual frequency array is not a simple task and requires a tradeoff analysis of array gain and side lobe level (e.g. in order to obtain such gain at 4.7GHz, an increase in SLL must be accepted at 7.5GHz). For this specific application a high system gain greater than 10dB was required. Therefore a separation distance between antennas of 30mm (0.47λ at 4.7GHz, 0.75λ at 7.5GHz) was chosen. With this distance, the simulated array gains are greater than 10dB at both frequencies, despite of an increased side lobe level at 7.5GHz. In addition, the distance avoids input impedance distortion due to mutual coupling, keeping good return loss conditions in the antennas. Phase shifter Types 1 and 2 were fabricated and arranged resembling Fig. 2.2, and all the possible phase combinations were measured to evaluate their performance. Tables III and IV show the simulated and measured progressive phase values between antennas compared to the ideal target values for 4.7GHz and 7.5GHz respectively. All measured values remain close to the simulations and target phases with an average phase error with respect to target values of 4.43 degrees at 4.7GHz and 4.63 degrees at 7.5GHz. Magnitude of the reflection coefficient conditions below 10dB were measured for the input and output phase shifter ports when all eight paths were activated at both frequencies. Average insertion loss of 2.8dB including connectors and diode losses was measured. 23 The final system is shown in Fig. 2.19. The four input ports of the phase shifters shown in Fig. 2.19 were connected to a 4-way SMA power divider to converge to a single input port. The DC control lines for the PIN diodes were externally controlled by a Labjack U3 device [26] connected through a USB interface to a PC, were custom software was designed. The system radiation patterns were measured in the UCI far field anechoic chamber. The simulated (solid line) and measured (dashed line) normalized radiation patterns for the array system with each one of the progressive phases between antennas provided by the phase shifter is shown in Fig. 2.20. A total of four switchable beams at each frequency can be obtained by using the proposed phase shifter, with the measured patterns resulting reasonably close to the simulated ones. The simulated patterns were obtained using Zeland IE3D with infinite ground plane, which explains the absence of the back radiation that is present in measurements. On the other hand, in order to simulate all different patterns the antenna array without phase shifter was fed directly with the theoretical phases. Therefore, the simulated absolute peak gain is on average 3.1dB higher than the measured one, which corresponds to the phase shifter and inter-stage connector average losses. Fig. 2.16 Fabricated Single element antenna and Zoom into the diode region. The devices are attached using silver epoxy to avoid overheating 24 0 Mag. Reflection Coefficient (dB) -5 -10 -15 -20 Sim. 4.7GHz (Diode ON) Sim. 7.5GHz (Diode OFF) Meas. 7.5GHz (Diode OFF) Meas. 4.7GHz (Diode ON) -25 -30 -35 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Frequency (GHz) Fig. 2.17 Simulated versus measured antenna magnitude of the reflection coefficient. For simplicity, the cases ‘On’ and ‘Off’ corresponding to operation at 4.7GHz and 7.5GHz are superimposed in the same graph. Fig. 2.18 Measured radiation patterns for the single element antenna operating in the (a-b) ‘On’ and (c-d) ‘Off’ cases. (a) and (c) show the E-plane while (b) and (d) show the H-plane. Solid and dashed line represents co-pol and cross-pol components respectively. The measured efficiency is 79% and 68% with associated gains of 5.4dB and 5dB at 7.5GHz and 4.7GHz, respectively. 25 Target Frequencies (GHz) 4.7 7.5 Target Values At 4.7GHz -135 -45 45 135 Target Values At 7.5GHz -135 -45 45 135 Table 2.2 Simulated and measured antenna center frequencies Simulated Measured Return Loss<10dB at Frequency Frequency target frequency (GHz) (GHz) 4.72 4.7 YES 7.53 7.55 YES Table 2.3 Simulated and measured phase delays at 4.7GHz Sim./Meas. Sim. /Meas. Sim. /Meas. Ant#1~Ant#2 Ant#2~Ant#3 Ant#3~Ant#4 (deg) (deg) (deg) -131.5/-128.73 -137.5/-136.37 -134.5/-139.76 -48.50/-50.00 -45.20/-44.55 -45.50/-49.31 45.50/45.77 45.20/42.37 48.50/50.00 134.50/139.76 137.50/136.37 131.50/131.77 Table 2.4 Simulated and measured phase delays at 7.5GHz Sim./Meas. Sim. /Meas. Sim. /Meas. Ant#1~Ant#2 Ant#2~Ant#3 Ant#3~Ant#4 (deg) (deg) (deg) -134.50/-130.79 -131.00/-138.96 -138.50/-130.14 -43.50/-40.60 -49.00/-49.05 -46.00/-44.53 46.00/46.09 49.00/50.00 43.50/40.60 138.5/130.14 131.00/138.96 134.50/134.47 Fig. 2.19 Fabricated 4x1 switched beam antenna array and phase shifters. 26 120° 135° 105° 90° 75° 120° 135° 60° 45° 150° 0° -30° 105° 90° 75° 30° 165° 0 -10 -20 -30 ± 180° 0° -30° 120° 135° -30° 0° -15° 45° 30° 15° 0° -165° -15° -30° -135° -45° -120° -60° -105° -90° -75° -135° -45° -120° -60° -105° -90° -75° 60° 120° 135° 45° 30° 105° 90° 75° 150° 15° 60° 45° 30° 165° 0 -10 -20 -30 ± 180° 0° -165° 60° -150° -30° 165° 0 -10 -20 -30 ± 180° 105° 90° 75° 165° 0 -10 -20 -30 ± 180° 15° -165° -15° -150° 0° 150° 30° 150° 15° -150° 45° 105° 90° 75° 30° -135° -45° -120° -60° -105° -90° -75° 165° 0 -10 -20 -30 ± 180° 120° 135° 45° -15° 60° -150° 60° -165° -135° -45° -120° -60° -105° -90° -75° 150° 105° 90° 75° 165° 0 -10 -20 -30 ± 180° -15° 105° 90° 75° -30° 150° 15° -165° 120° 135° -15° 120° 135° 45° -150° 0° -135° -45° -120° -60° -105° -90° -75° 60° 150° 15° -150° -135° -45° -120° -60° -105° -90° -75° 120° 135° 30° -165° -15° -150° 45° 165° 0 -10 -20 -30 ± 180° 15° -165° 60° 150° 30° 165° 0 -10 -20 -30 ± 180° 105° 90° 75° -30° 0° -165° -15° -150° -135° -45° -120° -60° -105° -90° -75° 15° -30° -135° -45° -120° -60° -105° -90° -75° Fig. 2.20 Simulated with IE3D infinite ground (solid) and measured (dashed) normalized E-total radiation patterns for 4.7GHz (lf) and 7.5GHz (hf) corresponding to different progressive phases between antennas (a) lf,α=-135 simgain=10.8dB (b)lf, α=-45, sim-gain=12.5dB (c) lf,α=+135,sim-gain=10.5dB (d) lf,α=+45 sim-gain=13.7dB; (e) hf, α=-135 sim- gain=14.8dB (f) hf, α=-45 sim-gain=14.7dB (g) hf, α=+135 sim-gain=12.7dB (h) hf, α=+45 simgain=15.2dB. 27 2.5 Conclusions In this chapter, a dual frequency reconfigurable switched beam antenna array with phase shifter using PIN diodes for telemetry applications was presented. A dual patch approach was used to design the single antenna element, while a switched line topology was chosen for the phase shifter. The system is capable of switching the beam to four selectable space positions at two different frequencies with 1.6:1 ratio. References [1] C. A. Balanis and P. I. Ioannides. Introduction to Smart Antennas. Seattle, WA: Morgan and Claypool. Synthesis Lectures on Antennas #5, 2007. [2] S. Liu, M. Lee, C. W. Jung, G. P. Li, and F. De Flaviis. “A frequency-reconfigurable circularly polarized patch antenna by integrating MEMS switches,” in Proc. IEEE Antennas and Propagation Society Int. Symposium, 2005, pp. 413-416. [3] J. R. De Luis and F. De Flaviis. “A reconfigurable dual frequency switched beam antenna array and phase shifter using PIN diodes,” in Proc. IEEE Antennas and Propagation Society Int. Symposium, 2009, pp. 1-4. [4] C. W. Jung and F. De Flaviis. “Reconfigurable multi-beam spiral antenna with RF-MEMS capacitive series switches fabricated on rigid substrates,” in Proc. IEEE Antennas and Propagation Society Int. Symposium, 2005, pp. 421-424. [5] S. V. Shynu, C. K. Gijo Augustin, P. Aanandan, P. Mohanan and K. Vasudevan. “A reconfigurable dual-frequency slot-loaded microstrip antenna controlled by PIN diodes.” Microwave and Optical Technology Letters, vol. 44, No. 4, pp. 374-376, Feb. 2005. 28 [6] A. Daryoush, K. Bontzos and P. Herczfeld. “Optically tuned patch antenna for phased array applications,” in Proc. IEEE Antennas and Propagation Society Int. Symposium, 1986, pp. 361-364. [7] F. Yang and Y. Rahmat-Samii. "Patch Antenna with switchable slot (PASS): dual frequency operation." Microwave and optical Technology Letters, vol. 31, No. 3, pp. 165168, Sep. 2001. [8] F. Yang and Y. Rahmat-Samii. "Switchable dual-band circularly polarized patch antenna with single feed." Electronics Letters, vol. 37, No. 16, pp. 1002-1003, Aug. 2001. [9] J. Costantine, C. G. Christodoulou, C. T. Abdallah and S. E. Barbin. “Optimization and complexity reduction of switch-reconfigured antennas using graph models.” IEEE Antennas and Wireless Propagation Letters, vol. 8, pp. 1072-1075, Sep. 2009. [10] L. M. Feldner, C. D. Nordquist and C. G. Christodoulou. “RF MEMS reconfigurable triangular patch antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symposium, 2005, pp. 388-391. [11] M. D. Migliore, D. Pinchera and F. Schettino. “A simple and robust adaptive parasitic antenna.” IEEE Transactions on Antennas and Propagation, vol. 53, No. 10, pp. 3262-3272, Oct. 2005. [12] R. W. Schlub. “Practical realization of switched and adaptive parasitic monopole radiating structures.” Ph.D. Thesis, Griffith University, Brisbane, Queensland, Australia, 2004. [13] Basari, M. F. Purnomo, K. Saito, M. Takahashi and K. Ito. “Simple switched-beam array antenna system for mobile satellite communications.” Communications, vol. E92-B, No. 12 pp. 3861-3868, Dec. 2009. 29 IEICE Transactions on [14] C. H. Tseng, C. J. Chen, and T. H. Chu. “A Low-Cost 60-GHz Switched-Beam Patch Antenna.” IEEE Antennas and Wireless Propagation Letters, vol. 7, pp. 432-435, Dec. 2008. [15] M. Barba, J. E. Page, and J. A. Encinar, “Planar C-Band Antenna with Electronically Controllable Switched Beams,” International Journal of Antennas and Propagation, vol. 2009, pp. 1-7, Oct. 2008. [16] D. R. Jackson and N. G. Alexopoulos. “Simple approximate formulas for input resistance, bandwidth, and efficiency of a resonant rectangular patch.” IEEE Transactions on Antennas and Propagation, vol. 39, No. 3, pp. 407-410, Mar. 1991. [17] R. Garg, P. Bhartia, I. Bahl and A. Ittipiboon. Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001, pp 265-269. [18] C. A. Balanis. Antenna Theory: Analysis and Design, 3rd Edition. New York, NY: Wiley, 2005, pp 819-820. [19] G. Gonzalez. Microwave Transistor Amplifiers Analysis and Design 2nd Edition. Upper Saddle River, NJ: Prentice-Hall, 1997, pp.183-184. [20] H. A. Atwater. “Circuit design of the loaded-line phase shifter.” IEEE Transactions on Microwave Theory and Techniques, vol. 33, pp. 626-634, Jul. 1985. [21] D. M. Klymyshyn, S. Kumar and A. Mohammadi. “Linear reflection phase shifter with optimised varactor gamma.” Electronics Letters, vol. 33, No. 12, pp. 1054-1055, Jun. 1997. [22] B. M. Schiffman. “A new class of broad-band microwave 90-degree phase shifters.” IRE Transactions on Microwave Theory and Techniques, vol. 6, No. 2, pp. 232-237, Apr. 1958. [23] C. Collado, A. Grau, and F. De Flaviis. “Dual-band butler matrix for WLAN systems,” in Proc. IEEE Microwave Theory and Techniques society Int. Symposium, 2005, pp. 22472250. 30 [24] K, Tang, Y. Wu, Q. Wu, H. Wang, H. Zhu and L. Li. “A Novel Dual-Frequency RF MEMS Phase Shifter,” in Proc. Asia-Pacific Symposium on Electromagnetic compatibility APEMC, 2008, pp. 750-753. [25] K. K. M. Cheng, and F.L. Wong. “A novel approach to the design and implementation of a dual-band compact planar 90 branch-line coupler.” IEEE Transactions on Microwave Theory and Techniques, vol. 52, No. 11, pp. 2458-2463, Nov 2004. [26] “Labjack: Measurements and Authomation Simplified.” Internet: http://labjack.com/, [Nov 20, 2009]. 31 3. BANDWIDTH ENHANCEMENT OF HIGH ISOLATION ISOFREQUENCY REPEATERS USING MEMS-RECONFIGURABLE LOADED PARASITICS 3.1 Introduction One of the main problems in wireless communications systems is the lack of signal coverage in some specific areas shadowed by obstacles. The use of signal repeaters becomes a cost-effective solution to extend the coverage to areas of low signal strength. To avoid frequency conversions, an iso-frequency repeater (gap-filler) can be used. This device acts as a directional amplifier that receives signal from a donor base station and retransmits it with the same frequency after an amplification stage. A block diagram of a general repeater is shown in Fig. 3.1. On the other hand, compact gap-filler solutions result in designs with a small physical separation between transmit and receive antennas and its radio-frequency sections. As a consequence a strong undesired electromagnetic coupling between the transmitting and receiving stages may arise, thus bringing unacceptable isolation levels. This lack of isolation between transmitter and receiver may severely degrade the performance of the system in the following ways: • Magnitude and phase errors in the repeated signal. • Magnification of high undesired spurious signals • Instability and oscillation in the amplifier • Mismatch between the antennas and the amplifier. In addition, the nature of this coupling do not only depends on the system architecture but also on the characteristics of the time-variant surrounding environment in which the gap-filler is working. As a consequence of poor isolation, the active gain of the gap-filler must be reduced, 32 and the maximum coverage range reduces drastically. To improve isolation in iso-frequency repeaters, a sheltering panel between the transmitter and the receiver can be used, or the physical separation between the transmit and receive antennas can be increased [1]. However, those are not practical solutions for compact systems. Some studies have focused in the design of an adaptive feedback cancellation system [2] or store-and forward techniques [3, 4], but the computational complexity and power consumptions in these solutions increase their cost drastically. In [5], we presented a preliminary concept of a narrowband back-to-back iso-frequency repeater antenna system which used MEMS-reconfigurable parasitics to improve signal isolation. In this work, we present a bandwidth enhanced solution based on a different and more flexible repeater architecture which instead uses MEMS-reconfigurable loaded parasitics and an aperture coupling feeding mechanism. The proposed gap-filler can be smartly reconfigured in real-time to optimize its performance (matching and isolation level and bandwidth) in the presence of a changing surrounding environment. Fig. 3.1 Block Diagram of an Iso-frequency repeater. 33 3.2 Description of the repeater system The proposed MEMS-reconfigurable iso-frequency repeater shown in Fig. 3.2 is composed of two active microstrip patch antennas placed back-to-back over a 2x2 in quartz substrate (εr=3.78 and δ≈0) with a thickness of 1.587 mm. The feeding mechanism is done through coaxial probe in an optimum antenna location to ensure impedance matching at the operating frequency of 5.2 GHz. The ground planes of the microstrip patches (1.5λx1.5λ) also act as sheltering panels between the transmitting and receiving side. At a distance of 0.13λ, two sets of out-of-plane parasitics (OUPPs) are placed symmetrically with respect to the sheltering planes. These parasitic elements are loaded with MEMS switches or PIN diodes, which allow them to be reconfigured into a short- or open-circuit state. These parasitics act as directors of the active patches when they are short-circuited while they will become transparent to the system in the open–circuit state. The OUPPs are used to improve the isolation capabilities of the gap-filler over a certain frequency band. On the other hand, to recover from mismatches of the antenna due to a specific environment configuration (such as close obstacles, etc.), two sets of in-plane parasitics (IPPs) are placed next to each microstrip antenna and loaded as well with MEMS. Therefore, these IPPs can also be set into an open- or short-circuit state. The capacitive couplings produced from the main patch to these elements in combination with the capability to select the state of their respective switches, allows one to effectively elongate the antenna width and thus to incorporate an additional degree of freedom to reconfigurable the impedance matching over a certain frequency band. Also, the IPPs can contribute to minimize the coupling between the antennas. Both sets of parasitics (92 in total) can be driven to dynamically enhance the isolation and matching conditions and thus to optimize the performance of the system under a wide variety of working environment. 34 Fig. 3.2 Proposed Iso-frequency repeater using RF-MEMS loaded parasitics. Fig. 3.3 UCI’s RF MEMS switch picture. This specific device was built over a quartz substrate and is intended to be used as depicted in Fig. 3.2 35 3.3 Design Of In-plane (IPPs) and Out-of-plane (OUPPs) parasitics The width (wo), separation (so), distance (do) and Number (No) of OUPPs (Fig. 3.4) was optimized first in order to obtain the maximum single tone isolation level for all cases with obstacles up to 1.5x1.5 λ distance and 45 degrees tilt. This study concluded with an optimum number of eight OUPPs that can be switched in two different states, open circuit (bit ‘0’ or non active) and short circuited (bit ‘1’ or active). When the antenna is placed near a metallic object, the OUPPs will activate or de-activate to find the optimum condition that satisfies maximum isolation. However, the input impedance of the antenna will change, resulting on reflection coefficient increase and thus mismatch with the amplifier stage. When this happens, the amplified signal may not be radiated efficiently into the space. So this is one major problem that needs to be addressed. It is known that the input impedance and bandwidth of a microstrip patch antenna can be modified by controlling its width. The idea behind the In-plane parasitic concept is being able to dynamically change this dimension to recover a good antenna reflection coefficient condition (<10dB) in all case scenarios. In plane parasitics (IPPs) are capacitively coupled to the microstrip patch, increasing its effective width. These parasitics will also change the radiation pattern characteristics, contributing to maximize the isolation between radiating elements. The width (wi), separation (si) and Number (Ni) of IPPs (Fig. 3.4) was optimized in order to recover the minimum antenna reflection coefficient conditions for six standard loading cases for the OUPPs. 36 Table 3.1, shows the antenna S11 for several combinations of OUPPs without the effect of the IPPs and how an optimal S11 condition can be achieved with and optimal configuration of the IPPs. Table 3.1 Reflection coefficient (S11) for several combinations of OUPPs and optimized reflection coefficient using best configuration of IPPs. Bit ‘0’ meaning open circuited parasitic and bit ‘1’ meaning short circuited Binary Combination for OUPPs S11(dB) S11 optimized (dB) [1 0 1 0 0 1 0 1] [1 1 1 1 1 1 1 1] [1 1 1 0 1 1 1 1] [1 0 1 0 1 0 1 0] -4.2966 -14.1311 -11.3433 -4.6566 -18.212 (+14) -16.9427 (+2) -58.8151 (+47) -40.1912 (+36) Fig. 3.4 A) OUPPs detail with parameters wo =0.5mm,= so =5mm,= do =20mm No =8 (each side) B) IPPs details. wi=0.5mm,= si =0.75mm Ni=46 (each side). 3.4 Combined isolation and matching optimization using the genetic algorithm optimizer. In order to optimize the matching and isolation in the proposed reconfigurable gap-filler, we use a genetic optimization algorithm (GOA) to select the optimal state of the MEMS switches for each of the parasitic elements. This evolutionary technique has been successfully used in several electromagnetic problems in the last years [6].Using the GOA, a binary chromosome (array) composed of 92 bits is evaluated as a potential solution candidate. This binary array contains the 37 state information of each MEMS switch. MEMS in the OFF state (open-circuit) are assigned a bit value of ‘1’, while MEMS in the ON state (short-circuit) are assigned a bit value of ‘0’. Each chromosome candidate receives a score according to a specific fitness function. Greater scores are assigned to solutions that maximize the isolation between active antennas (S12) at 5.2 GHz while maintaining a minimum reflection coefficient level (S11,S22) of 15 dB. The GOA maximizes the following fitness function: abs( S11 ) if abs( S11 ) < 15dB fitness = abs( S12 @ 5.2 GHz ) − f ( S11@ 5.2GHz ) where f ( S11 ) = 0 otherwise (3.1) By using this approach, only one full wave simulation need to be performed which makes the optimization process feasible. In this case the structure is drawn in HFSS and each of the radiating antenna elements and parasitics are assigned a lumped port. The structure is then EM simulated resulting in a S-parameter matrix of N by M dimensions SNxM, where N is the number of radiating antennas (in this case N=2) and M is the number of parasitic elements (in this case M=92). This 184 element matrix can be divided into four different sub-matrices named: • S11NxN: A NxN square matrix containing the reflection coefficient information from the radiating elements • S21MxN,S12NxM: A MxN or NxM rectangular matrix containing the transmission coefficient information between the radiating and parasitic elements and viceversa. • S22MxM: A MxM square matrix containing the reflection coefficient information from the parasitic elements Once these sub-matrices have been defined, we can mathematically load the M parasitic elements with ideal open or short circuits as depicted in Fig. 3.5, where the cascaded ΓR vector represents the reflection coefficients from the open and short circuiting loads taking values of +1 38 or -1, respectively. The resulting network after the parasitic loading now consists of a NxN matrix containing the S-parameter information (isolation and matching) of just the radiating elements. With proper manipulation we can find this matrix (Sin) by using expression (3.2). Fig. 3.5 Network cascaded connection representing the loading conditions for each of the parasitic ports. The loading block is represented by a set of reflection coefficients being +1 for the open circuit and -1 for the short circuit conditions. ( NxN NxM MxM MxM S in = S11 + S12 I − ΓR S 22 ( NxN NxM MxM MxM S in = S11 − S12 I + S 22 ) −1 ) −1 MxN ΓR S 21 → ΓR( i ,i ) = ±1 short _ circuit ( − ) open _ circuit ( + ) (3.2) MxN S 21 After a unique full wave multiport simulation is performed, the parasitic elements are loaded using a combination decided by the genetic algorithm (assigning bit ‘1’ to ΓR,i=1 and bit ‘0’ to ΓR,i=-1). The genetic algorithm is then in charge of evaluating the fitness of each solution candidate and conclude with a global optimum genome. In order to verify the reconfigurable capabilities of the proposed gap-filler, we consider for now a single-tone analysis (at 5.2 GHz) for the following three cases. Case 1 corresponds to a standalone reconfigurable gap-filler without any scatterers surrounding the antennas. In case 2, a 1λx1λ metallic plate is placed around the proposed repeater at a distance of 1λx0.5λ and forming 39 a 45º angle with respect to the antennas ground plane. Finally, in order to compare the results to those of a conventional (non-reconfigurable) repeater, a reference case (Case ref.) is also considered which basically consists of two back-to-back microstrip patches. Fig. 3.6(a-b) shows the simulation results after optimization for cases 1 and 2, respectively, using the proposed gapfiller system. A maximum level of 85 dB of isolation is achieved for case 1 and 125 dB for the case 2, at the operating frequency 5.2 GHz. The dashed line shows the reference case corresponding to the isolation of the conventional repeater, which is approximately 30 dB at 5.2 GHz. Therefore, a 55 dB and 95 dB isolation improvement is achieved in cases 1 and 2, respectively. The impedance bandwidth at -10dB is around 40 MHz (0.77%), while the isolation bandwidth at -50 dB is around 0.6%, in both cases. Fig. 3.8(a) shows the resultant 3D radiation patterns after optimization for case 1 and 2. Peak gains of approximately 8 dBi are achieved in both cases. Notice that the reconfigurable repeater, in the adaptive process, tries to place a radiation null in the direction of the metallic obstacle to avoid the undesired back-scattered radiation. 3.5 Isolation and impedance bandwidth enhancement. We now investigate the capabilities of the proposed reconfigurable gap-filler to improve its bandwidth for a particular isolation level (-50 dB) and matching level (-10 dB). In this occasion, for the configuration shown in case 2 of the previous section, we let the GOA maximize any of following two fitness functions: 5 fitness1 = ∑ abs ( S12@ f k GHz ) − f ( S11@ 5.2 GHz ), k =1 fitness2 = min(abs ( S12@ f k GHz )) − f ( S11@ 5.2 GHz ) (3.3) where fk corresponds to frequencies points ([5.18, 5.19, 5.2, 5.21, 5.22] GHz) within the considered band. The results of the optimization are shown in Fig. 3.7(a) and Fig. 3.7(b), when 40 using the fitness function 1 and 2, respectively. Values of 3.3% and 2.5% isolations bandwidths at -50 dB are achieved using the fitness functions 1 and 2, respectively. This represents a significant improvement in the isolation bandwidth with respect to the 0.6% bandwidth achieved with the single-tone analysis (case 2 in . The impedance bandwidth is 0.77% and 1.15% for the fitness functions 1 and 2, respectively. On the other hand, in order to further improve the inherent narrowband frequency response of the used microstrip antennas, we now use a thicker substrate (4.762 mm) in combination with an aperture coupling feeding technique [7]. Applying this idea, the proposed repeater can achieve higher impedance and isolation bandwidths. Results after optimization with this approach are shown in Fig. 3.8, where one can observe that the impedance bandwidth has now been increased to 17% while the isolation bandwidth is 1.5%. 3.6 Additional Information. For additional information on this topic the reader is referred to a recent publication where a prototype of a version of this repeater using PIN diodes has been built and successfully tested [8, 9]. Additionally, such repeater system includes some extent of beam steering capabilities besides isolation enhancement and good matching capabilities. 41 Fig. 3.6 Results after optimization. (a) case 1, no obstacle. (b) case 2, obstacle at 1λx0.5λ. Fig. 3.7 Results after optimization of the isolation bandwidth. Case 2 and different fitness functions. Fig. 3.8 a) Radiation patterns after optimization for case 1 and 2. b) Aperture coupled microstrip antenna with optimized impedance bandwidth and high isolation. 42 References [1] W. T. Slingsby and J. P. McGeehan, "Antenna isolation measurements for on-frequency radio repeaters," in Ninth International Conference on Antennas and Propagation (Conf. Publ. No.407), Eindhoven, Netherlands, 1995, pp. 239-43 vol.1. [2] S. J. Kim, et al., "Adaptive feedback interference cancellation system (AF-ICS)," in Microwave Symposium Digest, 2003 IEEE MTT-S International, 2003, pp. 627-630 vol.1. [3] J. J. Nehez, "Multiple Channel Same Frequency Repeater flight test," in Proceedings of the IEEE 1974 National Aerospace and Electronics Conference, Dayton, OH,, 1974, pp. 333340. [4] T. Gluszczak, et al., "Wideband Digital UHF Same Frequency Repeater," MAL-TR-73- 22, 1973. [5] A. Grau, et al., "Back-to-back high-isolation iso-frequency repeater antenna using MEMS-Reconfigurable-Parasitics," in Antennas and Propagation Society International Symposium, 2007 IEEE, 2007, pp. 497-500. [6] Y. Rahmat-Samii, "Genetic algorithm (GA) and particle swarm optimization (PSO) in engineering electromagnetics," in Conference Proceedings ICECom 2003. 17th International Conference on Applied Electromagnetics and Communications, Dubrovnik, Croatia, 2003, pp. 15. [7] P. B. R.Garg, I. Bahl, A. Ittipiboon, Microstrip Antenna Design Handbook: Artech House antennas and propagation, 2001. [8] E. Diaz, et al., "Pixeled-Dipole Based Isofrequency Reconfigurable RF Repeater," presented at the European Conference on Antennas and Propagation, EUCAP, Rome, 2011. 43 [9] E. Diaz, et al., "Isofrequency Reconfigurable 8-bit RF Repeater," IEEE, Transactions on Antennas and Propagation, (Submitted) 2011. 44 4. AN ANALYTICAL ALGORITHM FOR PI-NETWORK IMPEDANCE TUNERS. 4.1 Introduction Tunable matching networks are expected to play an important role in the realization of adaptive and reconfigurable radio front-end architectures. One particular example is the compensation of handset antenna impedance mismatch loss caused by user proximity effects using tunable antenna impedance matching networks [1-5]. Different matching network topologies have been reported in the literature. Basic L-type networks are able to achieve conjugate matching over a limited Smith chart region [6-8]. On the other hand, pi-networks [1, 3, 5, 9, 10] similar to (Fig. 4.1(a)) provide an extra degree of freedom that enables conjugate matching over a substantially wider impedance range. In the ideal case where the component values range is unbounded [0,∞] pi-networks can provide complete Smith Chart coverage and the component values for perfect conjugation matching can be calculated by using the approaches given in [10-12]. The finite nature of the component tunable range is due to practical implementation limitations such as parasitic influences and component properties. For a matching network with finite component tuning ranges the perfect conjugate match can be achieved only if the load impedance lays within the matching domain [11, 13]. In reality, the network component available range may be predetermined and the unknown load impedance may often be located outside of the matching domain. Therefore, in practical impedance tuners with finite component tuning range, where a perfect conjugation match may not exist, optimization techniques have been commonly used to minimize the reflected signal (minimize VSWR) [5, 14-18]. Different optimization approaches, 45 such as simplex and single step [5], genetic algorithm [15, 16, 18], or simulated annealing [17, 18] have been used to minimize the network input reflection coefficient as much as possible or at least down to an acceptable level. These optimization methods search for the right component tuning setting through an iterative process, consuming a considerable amount of time to reach the tuning goal. In addition, depending on the optimizer choice and its initial settings, there is a risk of converging into local minima. Thus it is desirable to develop a deterministic approach to directly compute the final component tuning setting for the impedance match in order to reduce the tuning time and avoid the intermediate tuning states. This chapter presents an analytical algorithm to determine the required component values for a tunable pi-network having finite tuning range components matching any load impedance on the Smith chart at a given frequency. The analytical tuning algorithm is based on closed-form formulas and a direct calculation procedure. For a given load impedance, the algorithm first attempts to find a perfect conjugation match solution, and in case it does not exist, the algorithm then calculates the setting providing the maximum power transfer to the load. It is acknowledged that in the case of a matching network without loss, tuning for achieving conjugation match or minimizing the reflection coefficient means maximizing the power transfer to the load. However in reality, the matching network has a certain amount of loss and the above statements are no longer equivalent. Thus any impedance matching approach or algorithm based on (or partially based on) minimizing the input reflection coefficient, only has good accuracy for lossless and low loss matching networks or tuners where the final matching goal is maximizing the power delivered to the load. The scope of application of this algorithm is not exclusive to 46 antenna impedance tuning control; it can be used for tunable or non-tunable matching network design or performance analysis to replace an optimization tool. Sections 4.4 to 4.5 of this chapter present the algorithm for perfect and optimal matching conditions based on VSWR or maximum power transfer criteria. Section 4.6 compares the experimental results from this algorithm with those obtained by means of the robust and simplex optimization methods [19]. The results are shown to be in very good agreement, but the algorithm using significantly less time than the iterative optimizer to achieve the same results. The application of the algorithm is not constrained to a specific pi-network topology, but may also be extended to other pi-network topologies as described in Section 4.7. Section 4.8 provides additional application examples to illustrate use of the algorithm. 4.2 Matching parameters and tunable pi-network configuration In this chapter, we evaluate the performance of a pi-network tuner algorithm based on the input voltage standing wave ratio (VSWR) and transducer gain and/or relative transducer gain. The VSWR at the input port of the pi-network tuner is related to its input reflection coefficient ( Γin ) as VSWRin = 1 + Γin 1 − Γin (4.1) The impedance match is achieved by minimizing the magnitude of reflection coefficient or equivalently the VSWR. In the case of perfect match, the magnitude of the reflection coefficient is zero and VSWR is equal to 1. On the other hand, the transducer gain (GT) of the pi-network tuner is defined as the ratio of the power delivered to the load to the available power from the source, and can be expressed as [20] 47 2 GT = ( 2 S 21 1 − ΓL 1 − S 22 ΓL 2 ) (4.2) where |S21|2 and S22 are the insertion loss and output reflection coefficient, respectively, of the pi-network tuner. ΓL is the load reflection coefficient. The relative transducer gain is the ratio of the GT to the transmission loss (1-|ΓL|2) caused by the mismatched load impedance. The relative transducer gain (∆GT) is the GT improvement achieved by introducing the pi-network tuner and it can be expressed as: ∆ GT = S 21 2 1 − S 22 ΓL 2 (4.3) Maximizing the GT or ∆GT for a given load impedance, i.e. delivering maximum power to the load, is often a better merit than minimizing the VSWR since GT and ∆GT include the effect of the tuner’s internal loss. Fig. 4.1 (a) ideal tunable low-pass pi-network and (b) equivalent circuit of practical pi-network tuner with fixed inductor and tunable capacitors. An ideal tunable low-pass pi-network topology is shown in Fig. 4.1(a) using a tunable inductor and two tunable capacitors. A practical pi-network tuner consisting of tunable capacitors may be implemented as shown in Fig. 4.1(b). In this case, since there is no practical tunable inductor available, the series equivalent inductance tuning is achieved by adjusting the tunable capacitor 48 C3 connected in parallel to a fixed inductor L [21]. As a result, the series equivalent inductance Le value is determined by Le = L 1 − ω 2 LC3 (4.4) where ω equal to 2πf is the angular frequency in (rad/s). In Fig. 4.1(b), C1 to C3 represent high-Q tunable capacitors having a finite tuning range Ck,min ≤ Ck ≤ Ck,max, (4.5) where k={1,2, or 3}. 4.3 Explanation of composite components frequently used in the algorithm The tuning algorithm for the pi-network tuner is developed based on the circuit topology, depicted in Fig. 4.2, which consists of a load impedance, ZL = RL + jXL, a source with a internal impedance Ro and the pi-network tuner. The pi-network tuner input port (source port) is defined between terminals 1-1’ while the terminals 2-2’ define the output port where the load (e.g. antenna) is to be connected. The equivalent inductance Le comprised by L and C3 and composite source/load impedances, which are frequently used in the tuning algorithm development, will be discussed in this section. Fig. 4.2 Circuit topology of pi-network tuner with driving source and load to be matched . 49 4.3.1 Equivalent Series Inductance & Its Characteristics In Fig. 4.2, Le represents the equivalent inductor of the parallel combination of C3 and L with the inductance value determined by (4.4). To obtain a desired equivalent inductance Le for a fixed L at a given frequency ω, the required C3 can be derived from (4.4) as C3 = Le − L (4.6) ω 2 Le L The series equivalent inductor Le presents a resonance at the frequency given by fr = 1 2π LC 3 (4.7) The equivalent inductance Le for L = 3 nH varies with operating frequency as shown in Fig. 4.3 when C3 takes values from 0.5 pF to 5 pF. Fig. 4.3(b) shows that Le becomes negative when the operating frequency is beyond fr. Also, for frequencies near or beyond the resonance frequency fr the effective Q of the series element will be significantly reduced. To keep Le positive, C3 should be small enough to meet the following inequality C3 < 1 (4.8) ω 2L 20 C3 Range Effective Inductance Le (nH) 18 0.500pF 0.625pF 0.750pF 0.875pF 1.000pF 1.125pF 1.250pF 1.375pF 1.500pF 16 14 12 10 8 6 4 2 800 1000 1200 1400 1600 Frequency (MHz) (a) 50 1800 2000 2200 4.3.2 Composite Load and Source Impedance As described in [6], any impedance in the Smith chart can be perfectly matched by properly choosing the appropriate L-network topology and component values. The L-network topology choice roughly depends upon RL ≤ Ro or RL > Ro (formally, they should be the ‘Yin and Yan’ areas in Smith chart [6]). Two main topologies of the L-network are used: the L-network with the shunt capacitor on the right side of the series inductor or the L-network having the shunt capacitor on the left side of the series inductor. A tunable low-pass pi-network is turned into either of the two L-networks by adjusting the shunt capacitor, C1 or C2, to zero. However, in practice this is not possible since a realizable capacitor always has parasitic capacitance making its minimum capacitance, Cmin, greater than zero. 100 Effective Inductance Le (nH) C3 Range 2pF 3pF 4pF 5pF 50 0 -50 -100 800 1000 1200 1400 1600 1800 2000 2200 Frequency (MHz) (b) Fig. 4.3 Equivalent or composite inductance Le as a function of frequency for different values of C3 ranging (a) 0.5 pF to 1.5 pF and (b) 2 pF to 5 pF. Therefore, depending on the load impedance, we either absorb C1,min into the source impedance Ro to form a composite source impedance ZS or absorb C2,min into the load impedance ZL to form a composite load impedance Ze. Thus, we can adopt the matching approaches similar to the ones used for the aforementioned L-networks. 51 From Fig. 4.2, the composite source impedance ZS is comprised of Ro and C1,min and can be expressed as ZS = R o B C 1, min Ro 1 1 = = + j 2 2 YS G o + jB C 1, min 1 + R o B C 1, min 1 + R o2 B C2 1, min (4.9) In (4.9), G o = 1 Ro , and B C 1 , min = 2 π f ⋅ C 1 , min = ω ⋅ C 1 , min (4.10) The composite load impedance Ze consists of ZL and C2,min and its expression is given in (4.11) Z e = Re + jX e = Re = Xe = 1 G L + j (BL + BC 2,min ) GL G + (B L + BC 2,min ) 2 2 L − (B L + B C 2 , min ) G + (B L + B C 2 , min 2 L (4.11) ) 2 where GL and BL are given in (4.12) YL = R X 1 = G L + jB L = 2 L 2 − j 2 L 2 ZL RL + X L RL + X L (4.12) And BC 2 ,min = 2πf ⋅ C 2 ,min = ω ⋅ C 2 , min (4.13) The portion of the Smith chart that can be effectively perfect matched is reduced after using the composite source or composite load impedance. This means that not every load impedance in the Smith chart can be perfectly matched by using a realizable pi-network tuner. If a perfectly 52 matched solution does not exist, the tuning algorithm will provide the best matching based on maximizing ∆GT or alternatively minimizing VSWR. 4.4 Formulas and algorithm for perfect match 4.4.1 Perfect Match Solutions If the perfect match solution exists, there are two sets of formulas to find pi-network component values. Either set of formulas is chosen depending on the load impedance characteristics. For a given load impedance ZL = RL + jXL, and a specific operating frequency f, the perfect match solutions for RL ≤ Ro and RL > Ro are as follows: In the case of RL ≤ Ro, setting C2 = C2,min and BC2 = ωC2,min, we have solutions for C1 and Le of (see Appendix A) C1 = 1 2π f G o (G e − G o ) ( R e ( Ro − R e ) − X e (4.14) and Le = 1 2πf ) (4.15) where Ge = 1/Re. Capacitance C3 can be obtained by substituting (4.15) into (4.16), where L is a fixed given inductance. In the case of RL > Ro (accurately, the condition should be 1/GL > Ro and in the algorithm calculation this restricted condition has been considered at step 11.), setting C1 = C1,min and BC1 = ωC1,min, the solutions for C2 and Le to achieve perfect match are (see Appendix A) C2 = 1 2πf GL 1 + Ro2 BC21,min − GL2 − BL R o ( ) and 53 (4.16) GL 1 + Ro2 BC21, min − GL2 2 Ro BC1, min 1 Ro Le = + GL 2πf 1 + Ro2 BC21, min 1 + Ro2 BC21, min Ro ( ) ( ) (4.17) Substituting (4.13) into (4.6), we can obtain the required C3 capacitance value for this match solution. 4.4.2 Algorithm for Perfect Match For a better understanding of the algorithm the reader is advised to refer to the flow chart provided in Fig. 4.4 when reading the following steps. 1. Check the real part of the load impedance, RL, first if RL ≤ Ro or RL > Ro. a. Algorithm for RL ≤ Ro (Branch1): 2. Set C2 = C2,min and let BC2 = ωC2,min. 3. Calculate GL, BL from (4.12) and use the resulting values to calculate the equivalent load impedance Ze from (4.11) 4. Utilize (4.14) and (4.15) to obtain C1 and Le values. 5. If real or real part of Le < 0 or C1 < C1,min, then go to branch ‘Algorithm for RL > Ro (Branch2)’ and follow that branch to complete the calculations. Otherwise proceed with step 6. 6. If C1 ≤ C1,max, compute C3 by using (4.6). Otherwise, set C1 = C1,max, re-compute Le by using (19) and calculate C3 with (4.6). 7. If C3 < C3,min, set C3 = C3,min and follow the algorithm presented in Section 4.5. Otherwise, we obtain solutions C1,max, C2,min, and C3 if C3 ≤ C3,max and C1 ≥ C1,max. 54 8. If C3 > C3,max, set C3 = C3,max or set C3 = C3,min only if C3 > 1/(ω2L), and then follow the algorithm given in Section 4.5. Otherwise, we obtain solutions C1, C2,min, and C3 if C3 ≤ C3,max and C1 ≤ C1,max. b. Algorithm for RL>Ro (Branch 2): 9. Set C1 = C1,min and let BC1 = ωC1,min. 10. Calculate C2 and Le values by using (4.16) and (4.17). 11. If C2 < C2,min, go to ‘Algorithm for RL ≤ Ro (Branch1)’ and follow the algorithm given in that branch. 12. If C2 < C2,max, use (4.6) to calculate C3 from the Le derived from (4.17). Otherwise, set C2 = C2,max and recompute Le by utilizing (4.19); then calculate C3 with (4.6). 13. If C3 < C3,min, set C3 = C3,min and follow the algorithm presented in Section 4.5. Otherwise, we obtain solutions C1,min, C2,max, and C3 if C3 ≤ C3,max and C2 ≥ C2,max. 14. If C3 > C3,max, set C3 = C3,max or set C3 = C3,min only if C3 > 1/(ω2L), and then follow the algorithm given in Section 4.5. Otherwise, we obtain solutions C1,min, C2 and C3 if C3 ≤ C3,max and C2 ≤ C2,max. 4.5 Formulas and algorithm for best match 4.5.1 Formulas for Best Match From the previous section, we have seen that perfect match solutions do not always exist for all the load impedances in the Smith chart. This is due to the limitations imposed by the finite tuning range of the tunable capacitors, Ck,min ≤ Ck ≤ Ck,max (k = 1,2 & 3), and the fixed value of the 55 series inductor. In this case, we can employ the best match solutions, which maximize GT or ∆GT and/or minimize input VSWR (or reflection coefficient). The expressions of the GT and ∆GT are given in (4.2) and (4.3), respectively, and the VSWR at the input port of the pi-network tuner, is defined by (4.1), where Γin has an expression (see Appendix B) Γin = Go [1− X Le (BL + BC2x )] − GL (1− X LeBC1x ) + j[GoGL X Le − BC1x − (BC2x + BL )(1− X LeBC1x )] Go [1− X Le (BL + BC2x )] + GL (1− X LeBC1x ) + j[GoGL X Le + BC1x + (BC2x + BL )(1 − X LeBC1x )] (4.18) where X Le = ωLe In Subsection 4.4.2, steps 6 and 12, we may be forced to take C1 = C1,max and C2 = C2,min, or to employ C1 = C1,min and C2 = C2,max, respectively. In either case, we need to find an equivalent series inductor Le, which minimize the input VSWR of the pi-network tuner. The Le providing the best match resulting from Le = ∂VSWR in ∂L e = 0 is presented in (4.19), [ 1 BC 2 x + BL + BC1x Ro2 (BC 2 x + BC1x + BL )(BC 2 x + BL ) + G L2 ⋅ 2πf (BC 2 x + BL )2 + GL2 ⋅ BC21x Ro2 + 1 [ ]( ) ] (4.19) where BCkx = ω ⋅ C k ,min or Ck ,max (k = 1 or 2) In the case of C3 < C3,min or C3 > C3,max as occurs in steps 7 – 8 and 13 – 14 of Subsection 4.4.2, we need to set C3 = C3,min or C3 = C3,max. If the C2 value has been previously assigned to C2,min or C2,max , the C1 value should be chosen to minimize the input VSWR of the pi-network tuner from ∂VSWR in ∂C 1 = 0 , and the C1 for the best match is C1 = 1 2πf ⋅ [ ] X Le (B L + BC 2 x ) + G L2 − (B L + BC 2 x ) 2 [1 − X Le (B L + BC 2 x )]2 + X Le2 G L2 56 (4.20) where X Le = ωLe = ωL 2 1 − ω LC3, min or ωL 1 − ω 2 LC3,max On the other hand, if the C1 value has been defined as equal to C1,min or C1,max , the C2 value that minimizes the input VSWR of the pi-network tuner derived from ∂VSWR in ∂C 2 = 0 C2 = is [ ( ] ) 1 X Le 1− X Le BL 1+ Ro2 BC21x + BC1x Ro2 (2BL + BC1x ) − Ro2 (BL + BC1x ) ⋅ 2 2 2πf X Le + Ro2 (1− X Le BC1x ) (4.21) The corresponding formulas of Le, C1, and C2 resulting from maximizing ∆GT can also be derived, but such a derivation is outside the scope of this chapter. On the other hand, we found that the final results of using the formulas derived either from maximizing ∆GT or from minimizing input VSWR differ insignificantly for low loss networks, i.e., the former set of formulas may provide only a few hundredths of a dB better average ∆GT than the latter ones. 57 C3 > 1 C3 > 1 Lω 2 Fig. 4.4 Tuning algorithm flow chart for Branch 1 and 2 58 Lω 2 4.5.2 Algorithm for Best Match 1. From step 6 in Section 4.4.2, we have C1 = C1,max and C2 = C2,min or BC1x = ωC1,max and BC2x = ωC2,min,. Use (4.19) to recalculate Le to obtain the best match. Then, follow the steps presented in Section 4.4.2. 2. From step 7, we have C1 = C1,max and C3 = C3,min if C3 ≤ C3,min. Employ (4.21) to compute C2 for the best match. 3. From step 8, we obtain C2 = C2,min and C3 = C3,max if C3 > C3,max and C3 < 1/(ω2L). Employ (4.20) to calculate C1 for the best match. We shall take C2 = C2,min and C3 = C3,min even if C3 > C3,max but C3> 1/(ω2L). Then use (4.20) to calculate C1. 4. From step 12, we have C1 = C1,min and C2 = C2,max, or BC1x = ωC1,min and BC2x = ωC2,max. Use (4.19) to recalculate Le to obtain the best match. Then, follow the steps presented in Section 4.4.2. 5. From step 13, we have C2 = C2,max and C3 = C3,min if C3 ≤ C3,min. Employ (4.20) to compute C1 for the best match. 6. From step 8, we obtain C1 = C1,min and C3 = C3,max if C3 > C3,max and C3 < 1/(ω2L). Adopt (4.21) to calculate C2 for the best match. We shall take C1 = C1,min and C3 = C3,min even if C3 > C3,max but C3 > 1/(ω2L). Then use (4.21) to calculate C2. The C1 and C2 values calculated from (4.20) and (4.21), respectively, in steps 16 – 17 and 19 – 20 may not be within their boundaries. Therefore, in the implementation of the best match algorithm, we use the following Branch 3 algorithm when the calculated C3 value for the perfect match is out of its boundaries, i.e., C 3 ≤ C 3, min or C 3 ≥ C 3, max . 59 4.5.3 Algorithm for Branch 3 The reader is again advised to refer to the flow chart given in Fig. 4.5 when reading the algorithm for this branch. The goal of the algorithm in Branch 3 is to search for the maximized GT or ∆GT based on using formulas (4.20) and (4.21). In the ∆GT calculations, S parameters of the pi-network tuner need to be used. The formulas for the S parameters of the pi-network can be found in Appendix C. a. We shall first set C3 = C3,max if C3 ≥ C3,max, or set C3 = C3,min if C3 ≤ C3,min or C3 ≥ C3,max while meeting inequality (4.8). b. Set C1 = C1,min and use (4.21) to compute C2. If C2 is within the range of C2,min ≤ C2 ≤ C2,max, we utilize (4.3) to calculate ∆ GT = ∆ GT 1 , or otherwise setting C2 = C2,min if C2 ≤ C2,min or C2 = C2,max if C2 ≥ C2,max, we then compute ∆ GT = ∆ GT 1 by using (4.3) ( ∆GTx represents a calculated results to identify from the calculated result ∆GTx). c. Set C1 = C1,max and use (4.21) to compute C2. If C2 is within the range of C2,min ≤ C2 ≤ C2,max, we utilize (4.3) to calculate ∆ GT = ∆GT 2 , or otherwise setting C2 = C2,min if C2 ≤ C2,min or C2 = C2,max if C2 ≥ C2,max, we then compute ∆GT = ∆GT 2 by using (4.3). d. Set C2 = C2,min and use (4.20) to compute C1. If C1 is within the range of C1,min ≤ C1 ≤ C1,max, we utilize (4.3) to calculate ∆GT = ∆GT 3 , or otherwise setting C1 = C1,min if C1 ≤ C1,min or C1 = C1,max if C1 ≥ C1,max, we then compute ∆GT = ∆GT 3 by using (3). e. Set C2 = C2,max and use (4.20) to compute C1. If C1 is within the range of C1,min ≤ C1 ≤ C1,max, we utilize (4.3) to calculate ∆ GT = ∆GT 4 , or otherwise setting C1 = C1,min if C1 ≤ C1,min or C1 = C1,max if C1 ≥ C1,max, we then compute ∆GT = ∆GT 4 by using (4.3). 60 f. Comparing ∆GT 1 / ∆GT 1 , ∆GT 2 / ∆GT 2 , ∆GT 3 / ∆GT 3 , and ∆GT 4 / ∆GT 4 , we choose the solutions or the setting of the tunable capacitors corresponding to the largest ∆GTx or ∆GTx among these four ∆GT values calculated from steps b to e. The solutions can be one of the following permutations sets: ( C1,min / C1,max , C2, & C 3, min / C 3, max ) or ( C1,min / C1,max , C3,min / C3,max ) C 2, min / C 2, max ,& C3,min / C3,max ), or (C1, C2,min / C2,max ,& where we should read Cx or Cy. These are the best match solutions in the terms of maximizing ∆GT. ∆GT 2 ∆GT 3 ∆GT 1−4 ∆GT 1− 4 ∆GT 1 ∆GT 4 Fig. 4.5 Algorithm flow chart for Branch 3 61 Cx / C y as 4.6 Comparison of algorithm and optimizer results In this section, different ‘experimental’ matching results calculated by using the algorithm are compared to those obtained by using a commercial and validated optimizer [19]. We first compare of the results obtained from a lossless pi-network turner and then those resulting from the same tuner but using components with finite Q factor. As an example, it is assumed that the pi-network tuner used here has the following parameters: tunable shunt capacitors C1 and C2 have a tuning range from 0.8 pF to 5 pF, C3 has a tuning range from 0.25 pF to 4 pF, and the fixed series inductor L has a value of either 6.8 nH or 2.3 nH for low frequency band (700 to 960 MHz) and high frequency band (1710 to 2170 MHz), respectively. The input VSWR contours of match tuning the load with reflection |ΓL| varying from 0.05 to 0.95 at 700 MHz by using the algorithm and utilizing the optimizer are shown in Fig. 4.6. The average VSWR over the Smith chart within the region of 0.05≤ |ΓL| ≤ 0.95 and -180o ≤ ∠ΓL ≤ 180o is 2.13 for both the algorithm and the optimizer. Fig. 4.6 Input VSWR contour plots for 0.05 ≤ |ΓL| ≤ 0.95 obtained with (a) the algorithm and (b) the optimizer simulation at 700 MHz 62 In the region of |ΓL| < 0.5, the input VSWR is low. In most cellular handset applications a VSWR < 3:1 is usually required after matching. If the load impedance is located inside the area defined by |ΓL| < 0.5, no matching task would be required.. Therefore, the most interesting area in the Smith chart to check tuner performance is within the region of 0.5 ≤ |ΓL| ≤ 0.90 and -180o ≤ ∠ΓL ≤ 180o. From this point on, we will only discuss the matching performance within this region to highlight the magnitude of improvement where a tuner is most beneficial. A comparison of the average input VSWR obtained from the optimizer and algorithm at different frequencies are given in Table 1. Table 4.1 comparison of average VSWR resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9 Frequency (MHz) 700 960 1710 2170 Optimizer Average VSWR 3.22 2.31 1.22 1.41 Algorithm Average VSWR 3.22 2.31 1.23 1.42 63 ∆VSWR 0 0 0.01 0.01 Fig. 4.7 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm and (b) optimizer at 2170 MHz for the case of lossless network. The plots of the ∆GT versus the load reflection coefficient (0.5 ≤ |ΓL| ≤ 0.9 and -180o ≤ ∠ΓL ≤ 180o) at 2170 MHz derived from the tuning algorithm and the optimizer simulation are given in Fig. 4.7 (a) and (b), respectively. The average ∆GT over the above area is found to be 3.19 dB for the algorithm and 3.20dB for the optimizer. A comparison of the average ∆GT resulting from the algorithm and the optimizer at different operating frequencies is presented in Table 2. The difference is equal to or less than 0.01 dB over the frequency from 700 MHz to 2170 MHz. In the case of the pi-network without loss, the ∆GT derived by means of minimizing input VSWR or maximizing GT or ∆GT approach is exactly the same. Table 4.2 comparison of average ∆GT resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9 Frequency (MHz) 700 960 1710 2170 Optimizer Average ∆GT (dB) 2.12 2.68 3.31 3.20 Algorithm Average ∆GT (dB) 2.12 2.67 3.30 3.19 64 ∆GT Discrepancy (dB) 0 -0.01 -0.01 -0.01 In practice, all of the components of the pi-network have a finite Q factor instead of infinite. We now compare the algorithm tuning results of a pi-network tuner having loss with those resulting from the optimizer simulations. Assuming that the tuner utilizes the components with the same tunable range and the nominal value as defined in the previous example but having a finite Q factors of QC1,2 = 100 for C1 and C2, QC3 = 150 for C3 and QL = 55 for the fixed value series inductors. In order to take the finite Q of the components into account, the final ∆GT and/or input VSWR calculations need to use the following component values and j , k= 1,2, or 3 Cˆ k = C k 1 − Q Ck ( f ) (4.22) j Lˆ = L 1 − Q L ( f ) (4.23) In the case of a tuner having loss, the plots of the ∆GT versus the load reflection coefficient (0. 5 ≤ |ΓL| ≤ 0.9 and -180o ≤ ∠ΓL ≤ 180o) at 700 MHz derived from the tuning algorithm and the optimizer simulation are presented in Fig. 4.8. The average ∆GT over the above area is found to be 1.79dB and 1.78dB for the optimizer and algorithm, respectively, a difference of only 0.01 dB. A comparison of the average ∆GT resulting from the algorithm and the optimizer at different operating frequencies is presented in Table 3. The difference is lower than 0.1 dB over the frequency from 700 MHz to 2170 MHz. In the case of finite but high Q components, the ∆GT obtained from minimizing input VSWR or maximizing GT or ∆GT approach is slightly different, and the later approach provides higher average ∆GT than the former one. Therefore, for a network tuner with loss, achieving the perfect match does not maximize the GT or ∆GT to the load. From Table 3, we notice that the average ∆GT resulting from the algorithm for a lossy tuner is always slightly less than the average ∆GT obtained from the optimizer. The reason is that the 65 algorithm’s main goal is to achieve a perfect match (VSWR = 1) and only in the case of having no perfect match situation, to then maximize the ∆GT. In addition, the algorithm first determines the tunable component values for matching based on an assumption of a lossless network tuner and then computes the final ∆GT and/or input VSWR with finite Q components. Contrarily, the optimizer takes the losses into account from the beginning of the process. However, it requires approximately one hour and fifty minutes generating the results shown in Fig. 4.8(b) when using commercial RF CAD optimizers on a typical PC, while it takes less than five seconds to obtain the same plot by utilizing the algorithm on the same PC. Thus tuning using this analytical algorithm is approximately one thousand times faster than the optimizer simulation with negligible errors for high Q components. This implies that the time saved by using this approach can significantly reduce the network design cycle duration. The algorithm can be coded for fast closed loop control of any tunable pi-, L- or certain double pi-network, independent of the tuning technology. (a) 66 (b) Fig. 4.8 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm and (b) optimizer at 700 MHz. Table 4.3 Comparison of ∆GT resulting from algorithm and optimizer at different frequencies for low loss network case. Frequency (MHz) 700 960 1710 2170 Optimizer Average ∆GT (dB) 1.79 2.10 2.29 2.29 Algorithm Average ∆GT (dB) 1.78 2.05 2.22 2.22 ∆GT Discrepancy (dB) -0.01 -0.05 -0.07 -0.07 4.7 Algorithm extension for different Pi-network topologies. The present algorithm can be easily extended for network circuit topologies different from that presented in Fig. 4.1(b) by reformulations. Two different topology examples are introduced in this section. 67 4.7.1 Shunt Tunable Capacitor Having a Fixed Inductor Connected in Parallel. A shunt inductor Ls may be connected in parallel with C2 and/or C1 in the network shown in Fig. 4.1(b) to make the minimum capacitance of the composite equivalent tunable capacitor formed by Ls||C2 or C1 lower than the original C1,min or C2,min.. Assuming that the shunt tunable capacitor C2 on the load side of the pi-network has a fixed inductor Ls connected with C2 in parallel as shown in Fig. 4.9, the equivalent capacitance formed by the LS and C2 at frequency ω can be expressed as 1 C 2e = C 2 1 − 2 ω LS C 2 , k= 1,2, or 3 (4.24) The algorithm presented in Sections 4.2 to 4.5 can then be used for the pi-network configuration shown in Fig. 4.9 as long as the C2 value in the algorithm is replaced by the C2e value defined by (4.24) while ensuring C2e is positive, i.e., C2 > 1 2 ω LS (4.25) Fig. 4.9 Pi-network shunt C2 having a fixed inductor Ls connected in parallel. 4.7.2 Series portion of a pi-network consisting of series LC combination The series portion of the Pi-network in Fig. 4.1(b) can be changed to a series LC combination as shown in Fig. 4.10. This is an alternative method achieving an equivalent effective inductance. The main reason for such a change is to increase the network frequency tuning range. However 68 this series configuration needs to use larger inductors and much larger tunable capacitors than the original parallel structure for the same operation frequencies. Fig. 4.10 Pi-network with a series LC circuit in its series path. In this case, the equivalent series inductance can be expressed as 1 Le = L1 − 2 ω LC 3 (4.26) Then, the C3 calculation must be done using (4.27) instead of (4.6) C3 = 1 ω (L − Le ) 2 (4.27) To keep Le positive, C3 should be large enough to meet the inequality C3 > 1 ω 2L (4.28) All other formulas given in the previous sections can then be reused in the tuning algorithm for the network topology shown in Fig. 4.10. 4.8 Application examples. 4.8.1 Replacing Optimizer to Analyze Matching Performance As was shown in 4.6, the analytical tuning algorithm presented in this chapter can be used to analyze the matching performance of the pi-network tuner in terms of transducer gain GT or relative transducer gain ∆GT. The use of this algorithm requires considerably less time than an iterative optimizer. More examples will not be introduced here for space limitations. 69 4.8.2 Predetermining the Match Tuning Setting for Known Antenna Load Impedance In the case of an antenna impedance match based on open loop control, the antenna impedance is premeasured under different conditions over different frequencies. The analytic tuning algorithm can be adopted to determine the component setting of the matching network for the measured antenna impedance immersed in different scenarios at each frequency. Fig. 4.11 presents the VSWR versus frequency curves for a mobile phone antenna before and after using a pi-network tuner. In this example, a pi-network as shown in Fig. 4.1(b) containing a series inductor L=6.8 nH, and capacitor values C1,min = C2,min = 0.8 pF, C1,max = C2, max = 5 pF, C3,min = 0.25 pF and C3,max = 4 pF is employed as the impedance matching network and it is tuned by means of our algorithm. From Fig. 4.11, we can see the input maximum VSWR after using the tuner is reduced from a VSWR = 5.8:1 reduces to a VSWR of 1.2:1 over the frequency range from 820 MHz to 970 MHz. The relative transducer gain ∆GT and resulting tunable capacitor C1 ~ C3 settings at frequencies 850, 880, 925, and 970 MHz, are given in Table 4. If instead of a continuous capacitance setting, the tunable capacitors are adjusted in discrete steps of 0.125pF, the matching performance degradation is shown in Table 5. The ∆GT degrades only 0.1 dB or less, and the VSWR increases 0.1. Usually, in this frequency range the matching performance is not highly sensitive to component value quantization. The capacitance step size for commercially available surface mount chip capacitors may be larger than this particular example. 70 7 At Tuner Input 6 At Ant. Input VSWR 5 4 3 2 1 0 820 850 880 910 940 970 Frequency (MHz) Fig. 4.11 VSWR versus frequency of a given antenna impedance with and without using the Pi-network tuner. The tuner has components L= 6.8 nH, C1,min = C2,min = 0.8 pF, C1,max = C2, max = 5 pF, C3,min = 0.25 pF and C3,max = 4 pF and the values of the capacitor settings have been found using the presented tuning algorithm. Table 4.4 Matching performance and tunable capacitor settings considering matching network with continuous capacitance. Freq (MHz) 850 880 925 970 VSWR 1.04 1.10 1.05 1.06 ∆GT (dB) 1.63 1.17 0.55 2.16 C1 (pF) 5 0.8 0.8 5 C2 (pF) 0.80 2.86 2.84 0.80 C3 (pF) 2.45 2.66 1.73 1.45 Table 4.5 Matching performance and tunable capacitor settings considering matching network with discrete capacitance steps. Freq (MHz) 850 880 925 970 VSWR 1.11 1.13 1.08 1.16 ∆GT (dB) 1.55 1.11 0.54 2.06 C1 (pF) 5 0.875 0.875 5 C2 (pF) 0.875 2.875 2.875 0.875 C3 (pF) 2.5 2.75 1.75 1.5 4.8.3 Control Algorithm for Closed loop Impedance Matching Since the algorithm deterministic nature provides much faster tuning settings than its optimizer counterpart, it is a good candidate to be used as a control algorithm for closed loop dynamic antenna impedance match closed loop control. A conceptual block diagram of the closed loop impedance matching system utilizing this algorithm is depicted in Fig. 4.12. 71 The objective of this loop system implementation is the determination of the antenna complex input impedance through the measurement of input and output voltages, Vin and Vout. The antenna impedance ZAnt can be determined by the initial values Y2 and Y3 in the pi-network and the voltages Vin and Vout by using the following expression Z Ant = 1 (Vin / Vout − 1)Y3 − Y2 (4.29) Vin Vo ut ZAnt= 1 (Vin/Vout−1)Y3 −Y2 ∠(Vin Vout ) Fig. 4.12 Conceptual block diagram of closed loop impedance matching control system. The proposed algorithm is used here to adjust the tunable capacitor settings once the antenna complex impedance is detected. This type of complex impedance determination using both, voltage and current measurements has been proposed in [7] for an L-type matching network. Because we are dealing with a different topology (no series component is connected directly to antenna to measure its current), the impedance determination is based only on two node voltage measurements. The analytical tuning algorithm can then be applied to calculate the required tunable capacitor values, C1 ~ C3 for the best match as long as the instant antenna impedance is known. This block diagram is given as an illustrative example of a particular application of the algorithm however specific implementation details are outside the scope of this chapter. 72 4.9 Conclusions. This chapter presents a novel tuning algorithm that has been shown to be very effective in determining the optimum tuning settings for a pi-network tuner with finite tuning range capacitors. Perfect match solutions are obtained if the values C1~C3 derived from the tuning algorithm are within the available tuning range. Otherwise, the algorithm always achieves the best possible match solutions. Formulas for the component values that achieve the perfect or the best match solutions are presented along with flow chats depicting the operation of the algorithm. Based on the description of the algorithm presented herein, a software version of this algorithm can be implemented in any suitable programming language. Using the algorithm to find the network component values resulting in the best possible match for any given load impedance is more than a thousand times faster than the time required to find an equivalent solution by using a commercially available optimizer. Furthermore, it was demonstrated that the match tuning accuracy derived from the tuning algorithm method is as good as that resulting from the optimizer simulation approach, thus the algorithm represents a practical and a useful method for closed feedback control of pi-network tuners. The proposed algorithm can be possibly extended to a network tuner topology comprised of four or less tunable components having a limited tuning range if these circuit structures can be transformed into an equivalent pi-network topology. Appendix A. Derivation of formulas (4.14)-(4.17) The derivation of equations of (4.14) – (4.17) needs to be split into two cases, i.e., R L < Ro and R L > Ro , then the derivation can be done by means of two different configurations of the equivalent circuit topologies. For (4.14) and (4.15), our derivation is based on the equivalent 73 circuit configuration shown in Fig. 4.13. C2 in the pi-network tuner has been absorbed by Xe and Re as given in (4.11). Z left = 1 Go + jBC1 Fig. 4.13 Equivalent circuit for (4.14) and (4.15) derivations. The impedance on the left side of the dashed line is given by Z left = G B 1 = 2 0 2 − j 2 C1 2 Go + jBC1 Go + BC1 Go + BC1 (4.30) In the conjugation match condition, we should have (4.31) and (4.32) satisfied. G0 = Re G o2 + B C2 1 (4.31) BC1 = X e + 2πf ⋅ Le G + BC21 (4.32) 2 o where BC1 = 2πf ⋅ C1 .From (4.31), we obtain (4.14) and from (4.32) we derive (4.10) as, C1 = Le = 1 2πf 1 1 BC1 = 2πf 2πf Go 1 11 1 − − Go2 = Go (Ge − Go ) = Re 2πf Ro Re Ro BC1 1 − Xe = G2 + B2 2πf o C1 Re Go 1 − Go2 − Xe = G R 2πf o e (4.33) ( R (R − R ) − X ) e o e e (4.34) In the derivation of (4.16) and (4.17), the equivalent circuit as depicted in Fig. 4.14 is used. For R L > Ro , C1 is set to minimum Cmin. Zleft and Zright in Fig. 4.14 can be expressed respectively as, 74 Z left Z right Fig. 4.14 Equivalent circuit for (4.16) and (4.17) derivations. Z left = Ro BC1 min Ro2 1 = − j Go + jBC1 min 1 + Ro2 BC21 min 1 + Ro2 BC21 min (4.35) and Z right = R L2 (B C 2 + B L ) RL 1 = − j G L + j (B C 2 + B L ) 1 + R L2 (B C 2 + B L )2 1 + R L2 (B C 2 + B L )2 (4.36) where BC1min = 2πf ⋅ C1,min and BC 2 = 2πf ⋅ C 2 In conjugation match condition, we should have the real part of (4.35) equal to Ro RL = 2 2 2 1 + Ro BC1 min 1 + RL (BC 2 + BL )2 (4.37) and the imaginary part of (4.35) without the negative sign equal to the following expression B C 1 min R o2 R L2 (B C 2 + B L ) = 2 π f ⋅ L − e 2 1 + R o2 B C21 min 1 + R L2 (B C 2 + B L ) (4.38) From (4.37), after manipulating we derive (4.16) C2 = 1 2πf GL 2 2 2 R 1 + Ro BC1,min − G L − BL o ( ) (4.39) and from (4.38) and (4.16) we obtain (4.17) GL 1 + Ro2 BC21,min − G L2 Ro2 BC1, min 1 Ro Le = + GL 2πf 1 + Ro2 BC21, min 1 + Ro2 BC21, min Ro ( ) ( ) 75 (4.40) Appendix B. Derivation of input reflection coefficient. Z in Zx Fig. 4.15 Equivalent circuit for Γin derivation The input reflection coefficient Γin is defined as Γin = Z in − Ro Z in + Ro (4.41) where Zin is the input impedance as shown in Fig. 4.15, and it can be expressed as Z in = where B C1 = 2πf ⋅ C1 and BC 2 = 2πf ⋅ C 2 (4.42) Yx is expressed as Yx = where 1 1 = jBC1 + 1 Z x jBC1 + Y x and X Le = 2πf ⋅ L e . Z in = G L + j ( BC 2 + B L ) 1 = Z x 1 − X Le (BC 2 + B L ) + jG L X Le (4.43) Substituting (4.43) to (4.42), we obtain Zin to be 1 − X Le (BC 2 + BL ) + jGL X Le GL (1 − X LeBC1 ) + j[BC1 + (BC 2 + BL )(1 − X Le BC1 )] (4.44) Finally, plugging (4.44) into (4.41), we derive the input reflection coefficient Γin (4.18) to be expressed as, Γin = Go [1− XLe(BL + BC2x )] −GL (1− XLeBC1x ) + j[GoGL XLe − BC1x −(BC2x + BL )(1− XLeBC1x )] Go[1− XLe(BL + BC2x )] +GL (1− XLeBC1x ) + j[GoGL XLe + BC1x +(BC2x + BL )(1− XLeBC1x )] 76 (4.45) Appendix C. Parameters of Pi-Network tuner. Fig. 4.16 Equivalent circuit of pi-network tuner The equivalent circuit of the tunable pi-network tuner is simply depicted in Fig. 4.16, which consists of an equivalent inductor Le and two tunable capacitors C1 and C2. These components are better expressed as impedance ZLe and admittance YC1 & YC2 if the loss of the components needs to be considered. The loss of the components can be characterized by a resistance or conductance as (4.46) – (4.48) Z Le = RLe + jX Le (4.46) YC 1 = G C1 + jBC1 (4.47) And YC 2 = GC 2 + jBC2 (4.48) or otherwise by the Q factor of the components as 1 Z Le = 2π f ⋅ L e + j Q Le ( f ) 1 YC 1 = 2π f ⋅ C 1 + j QC1 ( f ) (4.49) (4.50) And 1 YC 2 = 2πf ⋅ C 2 + QC 2 ( f ) j The S parameters of the pi-network in Fig. 4.16 are derived as 77 (4.51) S11 = ( ) [ ( + Y + (1 + Y ) ] )⋅ Z − YC1 + YC 2 + 1 + YC 2 − YC1 − YC1 YC 2 ⋅ Z Le 2 + YC1 S 21 = S12 = C2 C1 + YC 2 + YC1 YC 2 2 ( Le ) 2 + YC1 + YC 2 + 1 + YC1 + YC 2 + YC1 YC 2 ⋅ Z Le (4.52) (4.53) And S 22 = ( ) [ ( + Y + (1 + Y ) ] )⋅ Z − YC1 + YC 2 + 1 − YC 2 − YC1 − YC1 YC 2 ⋅ Z Le 2 + YC1 C2 C1 + YC 2 + YC1 YC 2 (4.54) Le where Z Le = Z Le , Ro YC 1 = YC1 ⋅ Ro , and YC 2 = YC 2 ⋅ Ro (4.55) References [1] H. Song, et al., "Automatic antenna tuning unit for software-defined and cognitive radio," in Antennas and Propagation Society International Symposium, 2007 IEEE, 2007, pp. 85-88. [2] P. Sjoblom and H. Sjoland, "An adaptive impedance tuning CMOS circuit for ISM 2.4- GHz band," Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 52, pp. 11151124, 2005. [3] J.-S. Fu, et al., "A ferroelectric-based impedance tuner for adaptive matching applications," in Microwave Symposium Digest, 2008 IEEE MTT-S International, 2008, pp. 955-958. [4] K. Brito de Brito and R. Nunes de Lima, "Impedance Network for an Automatic Impedance Matching System," in Microwave Conference, 2007. APMC 2007. Asia-Pacific, 2007, pp. 1-4. 78 [5] J. de Mingo, et al., "An RF electronically controlled impedance tuning network design and its application to an antenna input impedance automatic matching system," Microwave Theory and Techniques, IEEE Transactions on, vol. 52, pp. 489-497, 2004. [6] R. Rea. (2006, The Yin-Yang of Matching: Part 1: Basic Matching Concepts. pp. 16 – 25. Available: www.highfrequencyelectronics.com [7] A. van Bezooijen, et al., "Adaptive Impedance-Matching Techniques for Controlling L Networks," Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 57, pp. 495-505, 2010. [8] P. Smith, Electronic Applications of the Smith Chart, 2nd ed. Raleigh, NC: SciTech/Noble, 1995. [9] J. R. Moritz and Y. Sun, "Frequency agile antenna tuning and matching," in HF Radio Systems and Techniques, 2000. Eighth International Conference on (IEE Conf. Publ. No. 474), 2000, pp. 169-174. [10] Y. Sun and J. K. Fidler, "Component value ranges of tunable impedance matching networks in RF communications systems," in HF Radio Systems and Techniques, Seventh International Conference on (Conf. Publ. No. 441), 1997, pp. 185-189. [11] M. Thompson and J. K. Fidler, "Determination of the impedance matching domain of impedance matching networks," Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 51, pp. 2098-2106, 2004. [12] Y. Sun and J. K. Fidler, "Design method for impedance matching networks," Circuits, Devices and Systems, IEE Proceedings -, vol. 143, pp. 186-194, 1996. [13] Y. Sun and F. J. K., "Determination of the impedance matching domain of passive LC ladder networks," Journal of the Franklin Institute, vol. Vol. 333(B), p. 15, 1996. 79 [14] M. Thompson, "Application of Multi Objective Evolutionary Algorithms to Analogue Filter Tuning," in Evolutionary Multi-Criterion Optimization. vol. 1993/2001, ed Berlin/Heidelberg: Springer, 2001, pp. 546-559. [15] Y. Sun and W. K. Lau, "Evolutionary tuning method for automatic impedance matching in communication systems," in Electronics, Circuits and Systems, 1998 IEEE International Conference on, 1998, pp. 73-77 vol.3. [16] Y. Sun and W. K. Lau, "Antenna impedance matching using genetic algorithms," in Antennas and Propagation, 1999. IEE National Conference on., 1999, pp. 31-36. [17] M. Thompson and J. K. Fidler, "Fast antenna tuning using transputer based simulated annealing," Electronics Letters, vol. 36, pp. 603-604, 2000. [18] M. Thompson and J. K. Fidler, "Application of the genetic algorithm and simulated annealing to LC filter tuning," Circuits, Devices and Systems, IEE Proceedings -, vol. 148, pp. 177-182, 2001. [19] (Feb. 5 2009). AWR: Microwave Office. Available: http://web.awrcorp.com [20] G. Gonzalez, "Microwave Transistor Amplifiers Analysis and Design," 2nd ed Upper Saddle River, NJ: Prentice-Hall, 1997, pp. 183-184. [21] A. S. Morris III, "Tunable Matching Network Circuit Topology Selection," US Patent 2009 0267705 A1, 2009. 80 5. TUNABLE MEMS NOTCH FILTER AND ITS FREQUENCY CONTROL LOOP FOR WIRELESS TRANSCEIVERS 5.1 Introduction In a frequency division duplex (FDD) wireless transceiver, such as CDMA and WCDMA, the transmitter and receiver sections of a mobile station must operate simultaneously. A typical RF FDD front-end single-band block diagram of the wireless transceiver is shown in Fig. 5.1 [1]. The duplexer in the RF front-end is used to separate the transmission and reception signals. The duplexer specification requirements on suppressing unwanted signal and/or interference are very high. Typically 55dB or greater of isolation is required to suppress the transmission signal leaking into the receiver and a minimum 45 dB is required to suppress the transmitter noise in the receiver frequency band. Excessive transmission leakage through the duplexer to the receiver will cause inter-modulation and/or cross-modulation interference desensitizing the receiver [1]. An external SAW filter with modest rejection level (typically 20dB), is often placed after the LNA as shown in Fig. 5.1 to relax the mixer linearity and duplexer rejection requirements. However SAW filters historically have shown resistance to integration and frequency tunability, thus increasing the size, component count and cost of the overall transceiver. In order to address these problems, tunable solutions such as YIG filters have been proposed [2], which exhibit low loss and broad tuning bandwidth characteristics, but require an externally applied magneto-static field, suffer from slow tuning times due to hysteresis effects and exhibit high power consumption. 81 Fig. 5.1 Block Diagram of an Iso-frequency repeater. Distributed filter designs using coupled sections of resonant printed structures such as loaded combline filters [3], loaded loop resonators [4]-[5] or interdigitated filters [6], have been reported in literature. However, the large footprint required becomes the main disadvantage for any distributed implementation when designed for operation at typical cell phone frequency bands (700MHz-2.7GHz). In general, either acoustic or tunable lumped element filters must be used to meet cell phone real estate constraints. In [7], a non tunable notch filter using bond wire inductors operating at the IMT band (TX: 1.92-1.98GHz, RX: 2.11-2.17GHz) was reported. However, low suppression level (approx. 12dB) and high insertion losses (3dB) were measured. In addition, due to the non tunable nature of the design, the suppression level shows high variation within in the operating frequency band. In [8], a tunable notch filter based on RF MEMS showed 1.2dB insertion loss in the pass band and 25.5dB notch rejection. In this chapter, an improved version of the notch filter of [8], having less than 0.8dB pass band insertion losses is presented. Section 5.2 presents the theoretical analysis of the filter while Section 5.4.1 focuses on its implementation and measurement results. In the application of wireless mobile station transceivers, the narrow band notch filter frequency needs to have frequency accuracy within 100 kHz for 2.5G and 3G mobile stations where the 82 channel spacing is 200 kHz or 50 kHz for the long term evolution (LTE) mobile stations where channel spacing is 100KHz. Therefore, accurate frequency selection and tracking capabilities are required. A novel filter frequency control loop based on sensing the reflection phase of the leaking transmitter signal has been designed for this objective. The filter frequency control loop uses the transmitter carrier itself as a reference signal to keep the filter tracking the transmitter operation frequency. A mathematical model is developed to theoretically analyze the notch filter frequency control loop. A closed form solution to this frequency control loop equation is derived in section 5.3. Circuit model simulations in section 5.3.3 are employed to validate the theoretical solution, to investigate the loop control transient behavior, and to consider design parameters tradeoff. Finally, section 5.4.2 describes the implementation and performance assessment of the control loop and its interaction with the tunable filter to conform the overall autonomous tunable filter system. 5.2 Tunable Notch filter design In practical notch filter designs, a single bandpass is usually designed to coexist near the notch frequency. When this is the case, the design is also referred as single-pole single-zero (SPSZ) filter denoting a unique pole and zero in the filter transfer function. The proposed notch filter consist of a series-LC resonator (dashed box in Fig. 5.2) providing the signal notch or rejection at a transmitter frequency, and two shunt capacitors C3 and C4 that combine with the excess reactance of the series-LC resonator to form a low loss passband at the corresponding receiving frequency. The proposed tunable notch filter circuit using tunable components is shown as Fig. 5.2. This topology allows for the notch filter suppression and the bandpass filter insertion loss to be tuned separately. All the tunable capacitors used in this filter are RF MEMS capacitors [9]. 83 This series-LC block comprising !1 , 1 and 2 resonates at the notch frequency. 3 1 4!1 3 (5.1) where Cr represents the resonator composite capacitance resulting from the series connection of 1 and 2 can be expressed as 3 1 2 1 5 2 (5.2) Fig. 5.2 Circuit schematic of the SPSZ tunable filter design. At resonance (#3 ), the series-LC block presents a short circuit to ground reflecting most of the signal traveling along the transmission line. The values of external components, !1 and 1 must be chosen in order to obtain a resonance frequency higher than 1.98 GHz when all MEMS capacitor cells used in 2 are in the Off state (minimum capacitance). By tuning the value of 2 the resonance frequency can be dynamically adjusted to generate a notch at one of the transmitter operating frequencies. The center frequency of this notch filter should be capable of being tuned to cover the entire IMT transmission frequency band (from 1.92 to 1.98 GHz). On the other hand, the bandpass filter comprised of the series-LC in combination with the tunable capacitors 6 and 7 which are located in a symmetrical fashion on both sides of the 84 resonator. This arrangement minimizes the insertion loss of bandpass filtering and makes impedance matching identical seen from ports 1 and 2. The total capacitance || of 6 plus 7 for resonating at the receiver frequency ωrx can be obtained from (5.3). 9 1 2 3: !1 ; 1 3 (5.3) The S-parameters for the notch and bandpass combination filter can be derived and expressed as ; 3 (26 32 !1 1 2 ; 26 2 ; 26 1 ; 1 2 '=> <11 ? Where <12 <21 ;2@( 32 !1 1 2 ; 2 ; 1 ' ? ? (2@ 32 !1 1 2 5 2@2 5 2@1 5 2 36 => 6 !1 1 2 ; 2 3 => 6 62 ; 2 3 => 6 1 ; 3 => 1 2 ' (5.4) (5.5) (5.6) where => is a reference impedance. It is expected that <11 <22 (symmetric matching condition) only if the capacitors 6 and 7 are identical. 5.3 Notch filter frequency control loop based on reflected reference signal 5.3.1 System Description. The notch filter in the previous section provides an inherent narrow rejection bandwidth. The filter center notch frequency needs to be tuned with an accuracy of 100 kHz or less for 2.5G and 3G mobile systems or 50 kHz for the LTE system. To achieve the accurate frequency tuning of the narrow band notch filter, a frequency automatic control loop is required. This loop utilizes the transmitter carrier as a reference signal and uses the reflection phase change of the reference 85 signal from the notch filter to tune the filter frequency and to track carrier frequency that the mobile station transmitter operates on. To the authors knowledge, the frequency control loop for automatically tuning a narrow bandwidth notch filter has not been reported to date, although the frequency automatic control loop for the MMIC bandpass filters has been discussed in [10]-[13]. The bandpass filter frequency control loops are usually based upon sensing the transmission coefficient phase (or <21 phase). One key difference from the bandpass filters is that the signal passing through the notch filter should be suppressed to a very weak level. Additionally, the notch filter S21 presents 180o phase jump at its notch (or center) frequency (Fig. 5.3). As a consequence, the transmission phase information is difficult to utilize for the frequency control loop to tune the filter frequency. In the case of notch or other narrowband rejection filters, we can use the phase information of the reflected reference signal from the filter (i.e., the phase information of the S11 around the notch frequency). The advantage of sensing S11 is that the magnitude of the reflection coefficient of the notch filter near its notch frequency is very high. In addition, the S11 reflection phase behavior versus frequency is continuous across its notch frequency and quasi linear fashion as shown in Fig. 5.3. Fig. 5.3 S-parameter response of notch filter without the nearby co-exisisting pass band. 86 The block diagram of a proposed novel tunable notch filter automatic frequency control loop utilizing the reflected reference signal is shown in Fig. 5.4. This control loop will tune the filter notch frequency to suppress the transmitter leakage by utilizing the carrier of the transmitter leakage with a frequency ωr as the reference signal. A cos(ωot + m(t ) + ϕ o + ∆ϕ ) A cos( ω o t + m ( t ) + ϕ o ) Fig. 5.4 Block diagram of tunable notch filter automatic frequency control loop utilizing reflected reference signal. The basic operation principle can be described as follows: The reference injected (leakage) signal having a frequency ωr passes through a directional coupler. Most of the leakage signal is forwarded to the input port of the tunable filter while a small fraction of the signal is coupled to the left logarithmic amplifier of Fig. 5.4. The forward signal is reflected by the notch filter and has a phase change ∆φ different from 180o if the leakage signal carrier frequency ωr is different from the notch filter originally tuned frequency ωo. The reflected signal comes back to the directional coupler and it is coupled to the right logarithmic amplifier of Fig. 5.4. 87 The signal amplified by the left logarithmic amplifier passes through a phase shifter, which is used to equalize the group delay of the two paths, and arrives at the first input of the phase detector (PD) (see point “B” of Fig. 5.4). The reflected signal with an extra phase shift ∆φ after amplification by the right logarithmic amplifier reaches the second input port of the PD (see point “A” of Fig. 5.4). The phase shifter is used to adjust the initial phase difference between the two PD inputs to 90o. The output of the PD contains a low frequency voltage Vc(t) proportional to sin (∆D' and some high frequency products. The high frequency products are filtered by a low-pass RC Fig. 5.5 Mathematical model of notch filter frequency automatic control loop utilizing the reflection signal. loop filter. Only the low frequency output passes through the lowpass filter and it is digitized by a analog-to-digital converter (ADC). The ADC and next step is not necessary if the tunable filter comprises of analog tuning devices. The digital signal is then coded by an encoder that creates a set of proper tuning code words that tune the MEMS capacitors in the notch filter and make the notch filter frequency aligning with the leakage carrier frequency ωr. 88 5.3.2 Mathematical Model and Formulation Under the assumption that the notch filter can be modeled as voltage controlled analog tunable filter, the notch filter automatic frequency control loop can be fully characterized by the mathematical model as presented in Fig. 5.5. The transfer function EFF (@ ' of the notch filter reflection versus input reference signal has an expression (see Appendix A for its derivation) 2 (-' ;1 V ;G3 32 Q S EFF (@ ' *K @$-O G M U 3 2 2 (-' L HI 2 ; 1J 5 G32 P RT 3 N Where G3 => 2! 3 (5.7) (5.8) Note that for the mathematical description in this section a simple notch filter is considered to simplify the expressions. However the same procedure can be made analogously in the case of having a nearby passband working from (5.5) at expenses of a considerable increase in the solution complexity. From the frequency control loop mathematical model of Fig. 5.6 we can derive the automatic frequency control loop equation of the notch filter as (see Appendix B for derivation) 1 2 (-' (-' 5 W - ; GX Y W Z>2 2 2 (- ; ' ; 1` ] G3 32 \ _ sin G \ _ 3 2 2 ([ ^ HI 2; ' ; 1J 5 G32 3 89 (5.9) where W the low pass filter cut off frequency W 1⁄a , and τ the MEMS capacitor actuation delay time. Equation (5.9) is a second order nonlinear differential equation. From this equation, we can see that the output signal level from the phase detector (P.D.) is not only dependent on the phase difference between the reference signal and the reflected signal by the notch filter, but also upon the magnitude of the reflected signal. However, the magnitude variation near the frequency ωr within notch filter bandwidth is very small. In addition, the logarithmic amplifier further reduces the PD output signal amplitude difference. Considering !1 and 3 in the filter both having a finite Q factor, they can expressed as !c1 !1 d1 ; @/fF g (5.10) h3 3 d1 ; @/W g i And > Substituting (5.12) and (-' 3 5 ∆ (-' (5.11) fF Wi fF 5 Wi (5.12) into (5.9) and considering the magnitude of the PD inputs being constant, (-' j 3 and > k 1, we obtain a differential equation of ∆ (-' as 1 2 (-' (-' 5 l W - GX Y W Z>2 ; sin m 2 2∆ (- ; ' G ; 23 3 > n 1 G3 ; > (5.13) With the above assumptions, the right side of (5.13) can be further linearized as (5.14) 1 2 (-' (-' 2> ∆ (- ; ' G3 5 l ;∆ op: GX q ; r W G3 > ; 1 3 2(G3 > ; 1' 90 (5.14) where ∆ op: is the maximum frequency control range achieved when |sin( ∆D'| 1, which is determined by the overall loop gain and the PD output low frequency signal level ∆ op: Y W Z>2 2 (5.15) The process of solving equation (5.14) by using Laplace transformation is not described here in detail as it involves lengthy manipulations. During this solution process, in order to find a closed form solution for the inverse Laplace transform we need to assume that the MEMS capacitor actuation delay is small (to linearize exp(;' j 1 5 '. Equation (5.14) can be approximately solved by using Laplace transformation and the final time domain solution is ∆ (-' v Where ∆ op: G3 GX W 1 ,1 ; exp(;2 -'/ ; 2 ,1 ; exp(;1 -'/ · 2(G3 > ; 1' 1 (2 ; 1 '2 exp(;1 -' (1 ; W ' ; exp(;2 -' (2 ; W ' 5 x > q r 2 ; 1 1,2 ; y (1 ; 3 GX ' z 4 y2 (1 ; 3 GX ' ; 4 y 3 GX 2 3 2∆ op: > (G3 > ; 1' 3 (5.16) (5.17) (5.18) and ∆ > is the initial filter frequency error when t = 0. The frequency control error of the filter control loop depends upon the Qo factor of the notch filter and the value of the fixed inductor !1 . The final frequency error can be obtained by using (5.16) and letting - j ∞. ∆ (- j ∞' => 8!1 > (5.19) Note that (5.19) can be also interpreted as the final frequency error related to the notch/rejection bandwidth if we consider } ~ 1/> . The gain loop parameters GX , Y , and Z> can be set during 91 the design or selection of the building blocks in the control loop (integrator, phase detector and amplifier). The parameter y can be derived as follows. The capacitance versus control voltage Zy (-' equation can be expressed as (-' Zy (-' 5 > (5.20) Using (5.20) the filter instantaneous notch frequency can be written as (-' 1 4!1 (-' 1 4!1 (Zy (-' 5 > ' 1 q 1 4!1 > 41 5 Zy(' /> r (5.21) where > is the capacitance when the control voltage Zy (-' 0 (i.e 3 ) and is a constant that depends on the MEMS tunable capacitor design that relates capacitance to applied bias voltage. We can linearize (5.21) using Taylor expansion series as: (-' l 1 4!> ; 1 Z (-' 2 4!>6 y (5.22) The terms in (5.20) can be associated with those from the frequency voltage tuning equation (-' 3 5 y Zy (-' to conclude that y l ; 1 2 4!1 >6 (5.23) 5.3.3 Calculations and Simulations of Notch Filter Frequency Control Loop Performance In this subsection the effect of several parameters in the performance of tunable notch filter and control loop system (locking time, final error and exponential behavior) are studied in detail. The notch filter under the study is implemented by using a fix value inductor !1 10.6 and a 3 0.6& which produces a notch frequency at 2GHz. 92 When y 20, > 1000 and 0 ; 10 (close to real MEMS actuation delay), the transient responses of this frequency control to an initial frequency offset (∆ > 15) derived from (5.16) and ADS simulations are shown in Fig. 5.6. Good agreement of the results obtained from both approaches can be observed for both cases of actuation delay, which validates the theoretical approach solution. 4 2 0 ∆ω(t) in MHz -2 -4 -6 -8 -10 τ=0µs (ADS Sim.) τ=0µs (Calculation) τ=10µs(ADS Sim.) τ=10µs (Calculation) -12 -14 -16 0 100 200 300 400 500 600 Time (µs) Fig. 5.6 Comparison of Closed form expressions and ADS simulations for the second order filter response with In this experiment, 20, ∆ > 15 > 1000 and 0 10 Using the same loop parameters as above, the locking time responses for the filter > =1000, 100, 80 and 40 are shown in Fig. 5.7. All responses have similar exponential behavior but the final frequency error decreases with a > increase. Table I shows the numerical final frequency error for all considered > values. The low pass filter cut off frequency y or the loop bandwidth impacts the filter frequency locking time. The locking time responses for different y and Qo = 80, A = 0.12, ∆ > 15, and τ = 0 are shown in Fig. 5.8. Small y values create damped oscillations in the transient responses and therefore, longer frequency locking times. To achieve a shorter frequency convergence time for this loop, we choose the cut-off frequency y 10~20 kHz in the final design. 93 2 0 -2 Qo=1000 Qo=100 Qo=80 Qo=40 ∆ω(t) in MHz -4 -6 -8 -10 -12 -14 -16 0 100 200 300 400 500 600 700 800 900 1000 Fig. 5.7 Simulated filter frequency control loop transient response for different > values with y =20KHz, A=0.12 and ∆ > =15MHz, τ=0us. Time (µs) > 1000 100 80 40 Table 5.1 Final Frequency Error versus Value Final Frequency Error (KHz) 11 87 148 627 Relative Error (%) respect to ∆ > 0.073 0.58 0.98 4.18 The overall loop gain (controlled by parameter ‘A’) impacts the locking time and oscillatory behavior. A high loop gain may cause frequency divergence situations. Fig. 5.9 shows the filter frequency control error transient response for different values of ‘A’. The number of oscillations and locking time increases with an increasing gain. It was found that a gain value higher than A>4.1 results in a divergent situation. Due to the loop gain importance, good care must be given not to exceed the convergence threshold during the control loop design. The capacitor actuation delay time can also affect the frequency locking time of the filter and may originate a divergent solution. The filter frequency control error transient response for different MEMS capacitor actuation delays is shown in Fig. 5.10. The time scale has been expanded to 3ms in order to observe convergence. For the delay times larger than 150 s, the filter frequency control loop becomes unstable and the frequency is divergent. It is important to 94 choose the overall loop gain in consonance with the actuation delay time to make the loop operate in convergent region with the fastest locking time. A low loop gain should be chosen for large actuation delays. In this case the price of achieving the convergence is a longer locking time. 15 ωc=20 KHz 10 ωc=10 KHz ωc=5 KHz ∆ω(t) in MHz 5 ωc=1 KHz 0 -5 -10 -15 -20 0 100 200 300 400 500 600 700 800 900 1000 200 225 250 Fig. 5.8 Simulated filter frequency control loop transient response for different y values with > =80, A=0.12 and ∆ > =15MHz, =0us. Time (µs) 30 A=0.3 ∆ω(t) in MHz 20 A=1 A=3 10 0 -10 -20 0 25 50 75 100 125 150 175 Fig. 5.9 Simulated filter frequency control loop transient response for different A values with y =20KHz, > =80 and ∆ > =15MHz, =0us Time (µs) 95 Fig. 5.10 Simulated filter frequency control loop transient response for different values with > =80, y =20KHz and ∆ > =15MHz , =0.03 5.4 Implementations and measurements In this section, the standalone tunable filter implementation is first discussed in detail. Measurement results from tuning range, bandwidth, rejection and insertion loss values are presented. The filter is then integrated with a discrete implementation of the control loop to conform the overall tunable filter system. The total system performance is evaluated through the filter transient response and reference signal tracking capability. 5.4.1 Standalone Tunable Filter An existing high Q tunable digital capacitor array (TDCA) (see Fig. 5.11) flip chip solution from Wispry Inc [9] was utilized in the design. The TDCa consists of twenty tunable capacitor cells of nominal value 1pF or 0.875pF depending on the specific cell. The minimum capacitance step resolution is 0.125pF. The cells in the TDCa can be interconnected on the PCB level in order to achieve any desired topology. The Q of the die level capacitors was measured to be greater than 150 at 2GHz, allowing low insertion loss designs. In addition, the value of capacitance is highly repeatable, which is an important feature for narrowband tunable filtering circuits. The IP3 level for this device is 96 65dBm. The group delay distortion is below 1ns in the received signal pass band. The CMOS biasing circuitry is integrated in the same chip and transforms a 3.3V supply voltage to the required 35V voltage actuation level. The power consumption is 6 µA and 90 µA in the sleep and the active mode (charge pump on), respectively. A Serial Peripheral Interface (SPI) is used to control the capacitor banks states. A USB port is used to transmit the tuning commands from PC control software. Fig. 5.11 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell; b) 3D image of capacitance bits Two identical filters were designed using a single TDCA die. The objective is to involve one of the filters in the control loop system and replicate any action taken in the remaining filter that is used for monitoring and reproducibility testing purposes. The final PCB layout distribution can be seen in Fig. 5.12. The red and green color rectangles represent cells that were used for the notch and bandpass sections, respectively. Fig. 5.12 also provides information on how the cells are interconnected underneath the chip showing that half of the TDCa cells where used for each filter. The SPSZ filter of section 5.2 was fabricated using a 0.254mm thick Rogers 4003C substrate (εr=3.55, tanδ=0.0021 at 2.5GHz) backed with 14µm thick copper (Fig. 5.13). The 50 ohms microstrip line width was calculated to be 0.58mm for this substrate. A bottom layer of thick FR4 was used for mechanical stability purposes and to allocate the control circuitry and SPI 97 buffers. Side launch SMA connectors were used to connect the structure to a VNA through 3.5mm coaxial cables. The components used for the filter are SMD 0402 Murata high Q multilayer ceramic capacitors and 0603 chip CoilCraft inductors. Fig. 5.12 Detailed TDC layout. Red cells and green cells are used for the resonator and insertion loss blocks, respectively Fig. 5.13 Fabricated SPSZ tunable filter prototype board and filter structure detail. 98 The measured transmission and return loss of the filter for 39 different tuning states of the resonator block are shown in Fig. 5.14. The measured suppression in the transmitter band is equal or higher than 22 dB within a 5MHz bandwidth, and the insertion loss in the receiver band is less than 0.8dB within the reception signal bandwidth. The return loss is better than 20dB in the pass band on the receiver side. The suppression level is almost constant over the entire transmitter operating band as shown in Fig. 5.14 The filter tuning range is 90MHz which covers the transmitter band (1920MHz-1980MHz) and the receive band (2110MHz-2170MHz). Fig. 5.15 shows the insertion and return loss tuning using capacitor 6 and 7 banks when 2 is in the minimum capacitance state. As expected, this tuning does not affect the position of the resonator block resonant frequency. Therefore, this tuning capability can be used to compensate for changes in the insertion and return loss values as shown in while tuning the notch. Finally, Table II summarizes the suppression and insertion loss levels achieved by tuning the TDCA chip for the two limiting frequency pairs 1.92GHz-2.11GHz and 1.98GHz-2.17GHz. S21/S12 C2 ↑↑ S11/S22 C2 ↑↑ Fig. 5.14 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when tuning resonator block capacitor 2 . 99 Fig. 5.15 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when tuning insertion loss block capacitors 6 and 7 . Table 5.2 Summary of Suppression and insertion losses within 5MHz Bandwidth Tx Leakage Suppression RX Insertion Loss IMT Band Lower Pair Tx: 1.92GHz, Rx: 2.11GHz 22.1 dB IMT Band Upper Pair Tx: 1.98GHz, Rx: 2.17GHz 21.3 dB 0.8 dB 0.7 dB 5.4.2 Notch Filter Control Loop The notch filter control loop was implemented using discrete components as shown in Fig. 5.16 to prove the concept (The integrated circuit implementation is reserved as future work). A 20dB directional coupler (Meca 722S-20-1.950) distributes the input and reflected leakage signals. An analog adjustable phase shifter (Narda 3752) was used to provide the 90 degrees phase difference at the phase detector input when the transmit leakage and notch frequency are aligned. The phase detector board (AD8302) includes the logarithmic amplifiers and provides a single ended output with dynamic range between 0 and 2 volts. The integrator is implemented using and operational amplifier (THS3091DDA). The low pass filter is a single stage RC circuit with a cutoff frequency of ωc=20KHz. The analog to digital convertor (ADC)-(NI USB-6009) is used to 100 convert the output voltage from the integrator into a digital signal. A PC was used here as an encoder in order to generate the tuning words that actuate the tunable RF MEMS capacitors. The communication link between the PC and the TDCA die is based on SPI commands sent via USBSPI interface (Total Phase Cheetah). The left tunable filter ports are connected to a vector network analyzer for monitoring purposes. Due to the unknown delay associated with the ADC and PC processing speed, it is very difficult to accurately determine the fast filter locking response time. Therefore, in order to check the filter convergence the time axis is considered here as number of iterations (or tuning words) required for the filter to achieve the frequency tracking. The locking time can then be estimated by multiplying the number of iterations by the MEMS actuation delay (typically 10µs with this technology). The final estimated value would be reasonably close to the expected locking time in a future IC implementation. Fig. 5.16 Discrete component implementation of the proposed notch filter control loop. 101 The loop performance and locking time will be evaluated in high or low loop gain conditions. Due to the digital nature of the tunable notch filter, only certain discrete frequency states are possible, which explains the expected step behavior of locking time response curves Fig. 5.17 shows the filter transient response when the initial frequency offset ∆ > 49 and the loop gain 0.4. In this case, approximately 35 iterations where needed to achieve convergence yielding an estimated locking time of 350µs. This matches reasonably well with the predicted behavior of Fig. 5.9. Fig. 5.18 shows the transient response when the gain is increased to 3. The system then presents oscillations that eventually are damped to reach convergence in approximately 125 iterations (1250µs). Gains higher than 3 resulted in non convergent situations. From Fig. 5.17 and Fig. 5.18 we can conclude that, as we expected from simulation, it is important to properly set the overall loop gain in order to obtain the fastest locking time, smoothest exponential behavior and frequency convergence. In addition, high loop gains may potentially create undesired frequency fluctuations after convergence in case of system noise pick up. 0 ∆ω(t) in MHz -10 -20 -30 -40 -50 -60 5 10 15 20 25 30 35 40 45 50 Number of Iterations 55 60 65 70 75 80 Fig. 5.17 Measured filter frequency control loop transient response for different A=0.3 and ∆ > =49MHz. 102 40 ∆ω(t) in MHz 20 0 -20 -40 -60 0 20 40 60 80 100 120 140 Number of Iterations Fig. 5.18 Measured filter frequency control loop transient response for different A=3 and ∆ > =50MHz. The estimation of the filter frequency error in this practical discrete implementation is a very challenging task for several reasons: (a) the tuning resolution of the present TDCA being 0.125 pF, the tunable digital filter can only achieve certain frequencies, (b) the noise pick-up of the discrete system is not negligible and (c) the analog-to-digital convertor number of bits is limited. The final frequency error of this experimental filter frequency control loop will not exclusively depend on the quality factor of the components but will be considerably affected by the before mentioned factors. Assuming these limitations we tried to estimate the final frequency for our discrete implementation by changing the filter external components to !1 27, 1 0.1&, which reduces the step resolution to a maximum of 500KHz at expenses of reduced notch tuning range of 5.6MHz. The loop gain was chosen as in Fig. 5.17 to avoid oscillations in the response. An experiment was conducted choosing different ∆ > 5 MHz. The maximum recorded frequency error is 277KHz for this particular implementation. The final frequency control error could be further reduced if the filter frequency tuning resolution and the overall Q factor increase. 103 5.5 Conclusions A complete tunable filter system comprising of an RF MEMS tunable notch filter and its associated frequency control loop has been presented. The theoretical analysis and the derived closed form solutions and formulas have been proved very useful to understand the design and provide an implementation of the tunable filter system. This tunable filter can be practically used in the transceivers of wireless mobile stations. It can be possibly integrated into mobile transceiver RF ICs. The notch filter frequency control loop formulation developed here is not only applicable to this specific filter topology but can be applied to any narrowband band-stop tunable filters. Concepts of this tunable filtering system may be also used in the design of a more complex future tunable duplexer system. Appendix A. Derivation of the transfer function of the notch filter. A simplified notch filter circuit topology (excluding the insertion loss block) is shown Fig. 5.19. The ABCD matrix of this filter can be expressed as: where >y 1 } I >y >y @ 3 1 ; 3 2 ! 0 J 1 (5.24) @ 3 (-' 3 2 > 1; (-'2 (5.25) And (-' 1 (-' ; > H ! 4!(-' 104 (5.26) Fig. 5.19 Simple notch filter circuit topology We can obtain the S-parameters matrix from (5.24) and (5.25) as: I <11 <21 1 <12 ; = J I >y > <22 2 2 5 >y => 2 ;>y =3 J (5.27) Substituting (5.25) into (5.27) we can derive the transfer function of <11as (5.7) Appendix B. Derivation of the frequency control loop differential equation. We derive the differential equation of the notch filter frequency control loop starting with the low pass loop filter and integrator as shown in Fig. 5.20. From this figure, we can obtain the following equations: Z (-' ; ZX (-' (-' a ZX (-' 1 (-' (5.28) (5.29) ZW (-' GX ZX (-'- (5.30) 2 (-' Z (-' GX - 2 W (5.31) 105 Ve (t ) = ki ∫ Vi (t )dt Vi (t ) Vo2 AK d ⋅ H S 11 ( jω ) ⋅ sin (∠H S 11 ( jω )) 2 VC (t ) Fig. 5.20 Low pass loop filter and integrator Substituting (5.29) and (5.31) into (5.28) and rearranging its terms, we have the second order non linear differential equation 1 2 ZW (-' 5 ZW (-' GX Z (-' 2 W - where W 1 a (5.32) (5.33) The frequency voltage tuning equation of the notch filter is (-' 3 5 W ZW (- ; ' And ZW (- ; ' (-' ; 3 W (5.34) (5.35) Substituting (5.35) into (5.32), we derive the differential equation (5.9): 1 2 (-' (-' GX Y W Z>2 5 EFF sin( ∠ (<11 '' W - 2 2 (5.36) Finally, using the transfer function (5.7) of the notch filter reflection coefficient, we finally obtain the differential equation (5.9). 106 References [1] Q. Gu, “RF System Design of Transceivers for Wireless Communications,” Springer, 2005 [2] W. J. Keane, “YIG Filters aid wide Open Receivers”, Microwave Journal. Vol. 17, no 8, Sept 1980. [3] I. C. Hunter, J.D. Rhodes, “Electronically Tunable Microwave Bandpass Filters”. IEEE Trans. on Microwave Theory and Techniques. Vol. 30, no 9, Sept. 1982. [4] S. J. Park, K. Y. Lee and G. M. Rebeiz, “Low-Loss 5.15-5.70GHz RF MEMS Switchable Filter for Wireless LAN Applications”, IEEE Trans. on Microwave Theory and Techniques. Vol 54, no 11, Nov. 2006 [5] B. Jitha, P. C. Bybi, C. K. Aanandan, P. Mohanan, “Microstrip Band Rejection Filter using Open Loop Resonator”, Microwave and Optical Technology Letters. Vol. 50, no 6, June 2008. [6] A. R. Brown, G. M. Rebeiz, “A Varactor Tuned RF Filter”, IEEE Trans. on Microwave Theory and Techniques. Vol. 48, no 7, Jul. 2000. [7] H. Khatri, L. E. Larson, D. Y. C. Lie, “On-chip Monolithic Filters for Receiver Interference Suppression using Bond-Wire Inductors”, Student paper, Silicon Monolithic Integrated Circuits in RF Systems (SiRF), Jan 2006. [8] J De Luis, J. R.; Morris, A. S.; Gu, Q.; De Flaviis, F.; , "A tunable asymmetric notch filter using RFMEMS," Microwave Symposium Digest (MTT), 2010 IEEE MTT-S International , vol., no., pp.1-1, 23-28 May 2010 [9] Wispry Inc. Tunable RF solutions. www.wispry.com. Email: rf_mems@wispry.com. 20, Fairbanks. Suite 198. 92618 Irvine, CA. USA. 107 [10] P. Katzin; B. Bedard; Y. Ayasli; , "Narrow-band MMIC filters with automatic tuning and Qfactor control," Microwave Symposium Digest, 1993., IEEE MTT-S International , vol., no., pp.403-406 vol.1, 1993 [11] Aparin, V.; Katzin, P.; , "Active GaAs MMIC band-pass filters with automatic frequency tuning and insertion loss control," Solid-State Circuits, IEEE Journal of , vol.30, no.10, pp.1068-1073, Oct 1995 [12] Quintanel, S.; Serhan, H.; Jarry, B.; Billonnet, L.; Guillon, P.; , "Theoretical and experimental implementation of microwave active filters using automatic frequency control techniques," High Power Microwave Electronics: Measurements, Identification, Applications, 1999. MIA-ME '99. Proceedings of the IEEE-Russia Conference , vol., no., pp.IV7-I12, 1999 [13] L. Billonnet, et al., “Recent Advances in Microwave Active Filter Design, Part 2: Tunable Structure and Frequency Control Techniques,” Int. Journal RF and Microwave CAE 12, pp. 177 – 189, Wiley Periodicals, Inc., 2002. 108 6. APPROXIMATE CLOSE FORM DESIGN EXPRESSIONS FOR CAPACITIVELY LOADED PLANAR INVERTED-F ANTENNA 6.1 Introduction External monopoles used as antennas in earliest handset devices experienced a fast migration towards the dominant use of internal antennas. These antennas feature a lower cost, lower profile, simpler fabrication and, to some extent, easier integration with the rest of the RF front end modules. Most internal antenna designs consist of multi-resonant printed structures usually referred as planar inverted F antennas (PIFAs). A PIFA used for cell phone applications as shown in Fig. 6.1 can be viewed as a quarter wavelength narrow patch where the shorting pin/wall serves the purpose of ‘built-in’ matching network [1],[2]. Efforts have been focused over the past years in increasing the number of operating frequency bands [3], minimizing the overall antenna size and ground plane dimensions [4] and human body interaction effects [5]. One common technique in handset antenna design to achieve wider impedance bandwidth and higher radiation efficiencies is to use air as substrate below the antenna or to print the patch on top of a very thin layer of low loss and low permittivity dielectric material providing mechanical support. Multiple antenna arms of different lengths can be designed to obtain multiple coupled or uncoupled resonances to cover multiple cell phone frequency bands [6]. 109 Various simulation experiments and prototypes have demonstrated that a capacitively loaded PIFA as the one in Fig. 6.1 can achieve additional size reduction [2] and/or frequency agility [7],[8]. However, the capacitive loaded PIFA antenna concept has been only studied for very specific topologies and its fundamental tradeoffs with other important antenna parameters have not been clearly understood. The material in the following sections will show how this approach provides the mentioned advantages at expenses of degradation in the antenna impedance bandwidth and radiation efficiency. Previous works have focused on the determination of the resonant frequency of conventional (non loaded) PIFAs, which was approximately predicted using modal theory [9] or transmission model analysis [10],[11]. However, to our knowledge, an extended set of closed form expressions to quantitatively predict other important antenna parameters is not available to date. The objective of this chapter is to present a set of approximate closed form expressions to predict the most important parameters for capacitively loaded handset PIFAs such as input impedance, resonant frequency, radiation efficiency, quality factor and impedance bandwidth. Without losing generality, the same expressions can be applied for conventional PIFA design assuming absence of loading capacitance. This set of approximate closed form expressions will yield insight into the effects of the several design parameters over the antenna performance and allow the assessment of fundamental tradeoffs decisions in the early stages of antenna design. In order to illustrate the usefulness of these expressions, section 6.2 is concluded with a set of guidelines for a hypothetical example design of a tunable PIFA under certain specified conditions of frequency range, radiation efficiency and impedance bandwidth. Finally, section 6.3 presents a comparison of measurements and full wave simulation results in order to validate the proposed approximate closed form expressions. 110 6.2 Closed form expressions for capacitively loaded PIFA design A canonical representation of a capacitively loaded tunable PIFA antenna of patch length ! , width and height is shown in Fig. 6.1. A tunable (or switchable) shunt capacitor with capacitance is connected between the antenna radiating open edge and its ground plane. The shorting wall is located in close proximity to the feeding probe and typically has the same width as the antenna patch. Fig. 6.1 Capacitively loaded tunable PIFA 6.2.1 Transmission line model for a capacitively loaded PIFA The antenna in Fig. 6.1 can be modeled using a transmission line equivalent circuit shown in Fig. 6.2. Because of its simplicity, this model can be easily implemented in any commercial microwave circuit simulator, and provides practical intuition into the basic antenna behavior. Fig. 6.2 Transmission line equivalent model for the capacitively loaded PIFA In Fig. 6.2, Gr is the radiation conductance associated with the power radiated by the antenna open edge and Bs is the radiation susceptance related to the energy stored in the antenna near field. 111 ay is the short circuit wall impedance. Expressions for the model components will be given later in this section. Note that in the transmission line model, the tunable capacitor is connected in parallel with the radiation admittance. The total patch length (! ) is the sum of the transmission line length from the feed point to the antenna open edge (! ) and the distance between the feed and short wall (! ). => is the characteristic impedance of the antenna patch, which is related to the patch width, height over the ground plane and permittivity (εr) of the supporting substrate [12]. The edge capacitive load effect can be graphically explained in the simplified transmission line model of Fig. 6.3. The capacitor in Fig. 6.3(a) is replaced by an equivalent open circuited transmission line of length !W in Fig. 6.3(b). The value of the equivalent transmission line length can be can be found using microwave theory as: !W 1 1 - G => 2#3 (6.1) Where G=2/ and is the effective wavelength. The new circuit for the antenna in Fig. 6.3(b) will now resonate when the equivalent length ! !W 5 ! , is approximately quarter wavelength at the operating frequency. In other words, the capacitive loading is acting as an effective electrical length extension of the antenna. This model reveals two important features that are valid for any edge capacitively loaded PIFA: 1. For a constant operating frequency (#3 ), the required physical patch length (! ) in order to achieve resonance will be less than quarter wavelength due to the capacitive loading length extension effect. This allows for antenna size miniaturization using a fixed capacitor loading and can be used in non tunable antenna designs [13]. 112 2. For a fixed ! , the antenna operating frequency can be decreased by increasing the amount of capacitive loading. Therefore, tunable antenna designs can make use of a variable (or switchable) capacitor to achieve frequency agility. Fig. 6.3 (a) Equivalent transmission line model of a capacitively loaded PIFA, (b) the capacitor has been replaced by an open circuited section of transmission line By looking at points 1 and 2, related exclusively with size and operation frequency, a capacitive loaded antenna is an attractive candidate for reducing the size and/or tuning the frequency of handset antennas. However, these two advantages will produce a decrease in impedance bandwidth and radiation efficiency as it will be shown in the following sections. In the following subsections, different approximate closed forms expressions will be presented to predict the behavior of input impedance, resonant frequency, radiation efficiency, quality factors and impedance bandwidth versus different capacitive loading conditions. All expressions can be applied to standard non loaded PIFAs by considering the particular case of 0. 6.2.2 Input impedance. An approximate expression for %3 and } can be found in [14] and [12] respectively. To find %3 , a slot of length parallel to the ground plane was chosen to model the radiation from the open patch edge as 113 %3 Where 12¡¢ £¤ 4¥¦§§ 2 1 2 & v 120 2 2 2 90 (6.2) is the equivalent width that models the spreading of the slot field from the strip edge. &2 is the integral defined in [14]. The approximation that simplifies (6.2) performed over &2 integral in is more accurate for << . On the other hand, the equivalent electrical length extension (Δ©' due to the open edge fringing fields is used to model the radiation susceptance as } > tan (GΔ©' where Δ© ¬ (6.3) 0.95 0.075(+3 ; 2,45' ; 1 5 0.85G 1 5 10G (6.4) and where > 1/=> . The short circuit radiation resistance for narrow patches can be modeled as that of a small element of constant current [15] while the loss resistance is determined by the material and operation frequency. where ² and ayi®¯ # 2 #3 H 80 ; ay °> > ± ² 2 are the material conductivity and permeability, respectively and (6.5) > is the speed of light. The short circuit wall impedance of Fig. 6.2 can be considered to be the sum of radiation and loss resistances ay ay 3pY 5 ay °> . The short circuit loss resistance will be in most cases negligible compared to the short circuit radiation resistance. The latter will increase with the patch height and operation frequency and will be valid as long as the current can in the short wall can be considered constant ( ³ ). 114 The input admittance (X ) of the equivalent model can be found as the sum of admittances 1 and 2 in Fig. 6.2 X > ,f 5 @> tan(´! '/ > ,y 5 @> tan(´! '/ 5 > 5 @f tan(´! ' > 5 @y tan(´! ' (6.6) where f %3 5 @} 5 @2#3 is the capacitively loaded edge admittance, y 1/ay and ´ is the complex propagation constant of the line. The reflection coefficient calculated with this model is compared with simulation results obtained from method of moments (Zeland IE3D) in Fig. 6.4 and Fig. 6.5. Two different heights 2, 4 are considered with lengths !± 34.5 and !± 33.5 chosen to make the antenna resonate at 2GHz with 0& for each case. From these results, the frequency decrease provided by the increased capacitance is clearly visible. It is apparent that the accuracy of the transmission line model is slightly better for lower antenna heights. The frequency error increases for higher capacitive loading. Despite these differences and the limitations imposed by the approximations, this model is a good tool to obtain a quick representation of the loaded PIFA behavior in typical cell phone profile sizes. 6.2.3 Determining the patch length for a given operation frequency and loading capacitance. For design purposes, it is desired to determine the physical patch length (! ) required to make the antenna operate at a certain frequency (#3 ) using a fixed value of capacitive loading (e.g. if antenna size reduction is intended). This could be done by solving for ! in (6.6) after imposing the condition µ¶X · 0. However, although a solution can be obtained, it results in a very long expression that provides very little physical insight or practical application. Therefore, a different, more practical 115 approach will be used here. In [16], an approximate expression for the resonant frequency of a PIFA operating in the fundamental mode and valid when the short wall length is equal to the antenna width is given as Fig. 6.4 Magnitude of the reflection coefficient using the transmission line model and method of moments simulations (IE3D) for a PIFA over air with 2, ± 4, !± 34.5, ! 1. Error refers to the absolute value of the difference in frequency between simulations and closed form expressions. Fig. 6.5 Magnitude of the reflection coefficient using the transmission line model and method of moments simulations (IE3D) for a PIFA over air with 4, ± 4, !± 33.5, ! 1. Error refers to the absolute value of the difference in frequency between simulations and closed form expressions. 116 #3 > 4(! 5 '√+3 (6.7) Where ! in [16] was the total metallic patch length of a non loaded PIFA. In our case, substituting ! in (6.7) by the equivalent length ! !W 5 ! and using (6.1), the expression for the new required physical patch length as function of the desired operating frequency and loading capacitance can be obtained as 1 > - ; 2#3 √+3 2=> #3 ! 2#3 √+3 Fig. 6.6 (6.8) shows the required patch length in terms of wavelength versus loading capacitance in order to operate at a fix resonant frequency of #3 2% for the cases of 2, 4, 6 obtained by method of moments (Zeland IE3D) simulations and closed form expression (6.8). It is observed that the patch length decreases in an exponential manner with increasing capacitance. As expected, the required patch length for zero capacitive loading is close to quarter wavelength (particular case of non loaded PIFA antenna) and is shorter for higher values of due to the increased presence of fringing field effects. In addition, Fig. 6.6 curves obtained from expression (6.8) are in concordance with the intuitive explanation offered by the transmission line model: the reduction in patch effective length with an increase capacitive loading at a fix operating frequency is clearly visible. Full wave simulations and calculations from (6.8) are shown in good agreement. 117 0.25 h=2mm h=2mm h=4mm h=4mm h=6mm h=6mm Patch Lenght LT (λ) 0.20 0.15 Closed Form Sim. IE3D Closed Form Sim. IE3D Closed form Sim. IE3D 0.10 0.05 0.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Loading capacitance C (pF) Fig. 6.6 Simulations and closed form (CF) expressions results for the total patch length (LT) required to make the PIFA resonating at 2GHz versus capacitve loading for different height values and ± 4 6.2.4 Required loading capacitance for a given length and operating frequency. An expression for the required capacitance to operate at a certain frequency for a fix patch length can be of interest for tunable PIFA designs. This expression could be used to find a proper finite tuning range of the tunable/switchable capacitor/s. The value of required capacitance for a fix patch length and some desired resonant frequency can be found by substituting ! in (6.7) by the equivalent length ! !W 5 ! , using (6.1) and solving for . The resulting capacitance value is then 1 2#3 √+3 (! 5 ' q- ¬ => #3 r > 2 1 Fig. 6.7 (6.9) compares the required capacitance value using versus resonating frequency using (6.9) and 118 method of moments simulations (Zeland IE3D) for a fix patch length (! was chosen to make the antenna resonate at 2GHz when 0&) and different 1, 2, 4, 6. This graph shows once again the exponential behavior of the frequency with capacitive loading. An important conclusion can be extracted from Fig. 6.7; In case the capacitance is to be increased in discrete steps (i.e. tunable digital capacitor/switched capacitor banks), a finer capacitance step resolution would be specially desirable for higher frequencies (low capacitive loading) in order to avoid coarse frequency jumps. 2.0 h=1mm Closed Form h=1 Sim. IE3D h=2mm Closed Form h=2mm Sim. IE3D h=4mm Closed Form h=4mm Sim. IE3D h=6mm Closed Form h=6mm Sim. IE3D Resonance frequency, fr (GHz) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Capacitive loading, C(pF) Fig. 6.7 Simulations and closed form (CF) expressions results for the resonance frequency versus capacitive loading for different height values. The patch total length was chosen to resonate at #3 2% in absence of capacitive loading ( 0&'. 6.2.5 Radiation efficiency and quality factor. Radiation efficiency is a key antenna parameter and is paid special attention in cell phone antenna design. The radiation efficiency quantifies how much of the energy accepted by the antenna is radiated to the space. The radiation efficiency can be expressed in terms of the quality factors of the antenna as [12] ¹3 > > 5 3 119 (6.10) where > is the loss quality factor associated to the conductor and dielectric loss dissipation mechanisms and can be found as [12] With 1 1 1 5 > y Y (6.11) y 4#3 > ² (6.12) Y 1 1;» 1 5 -º »+3 (6.13) Where ² is the metal conductivity, -º is the dielectric loss tangent and » (0.5 ¼ » ¼ 1) is the dielectric filling factor representing the fraction of total fields in the dielectric. Due to the fact that most practical PIFA antennas for cell phone applications are usually built over a relatively thick air gap using a very thin layer of supporting substrate [6], we will not consider the effect of surface waves or dielectric losses in this study. 3 in (6.10) is the antenna radiation quality factor and is a uniquely defined parameter [17] given by 3 2#3 ½ 03 (6.14) where ½ represents the total energy stored in the antenna and 03 is the total power radiated to the space. From (6.9) it is clear that in absence of losses, or when > k 3 , the radiation efficiency approaches one. In [18], an approximate expression for ½ and 03 in a half wavelength microstrip antenna was found using cavity mode analysis and the expressions from the radiation of a Hertzian dipole 120 on a grounded layer. Using the same approach and assuming a fundamental mode surface current distribution flowing in the y-axis direction (see Fig. 6.8) as ¾¿ (À' 1¡ 2 2À @ > > > (6.15) where 1¡ is the amplitude of the dominant mode-field that depends on the patch length to width aspect ratio and feed position [19],[20]. The power radiated by the half wavelength antenna was found [18] to be approximately Áf/2 1 2 2 2 (G 03 v 2 > ' 80 ¬ ¾¿ (À'À > f/2 2 (6.16) where ! /2 was the total length of the patch that serves as limits for the current integration as shown in Fig. 6.8(a). On the other hand, the total energy stored in the half wavelength antenna can be found at resonance, when the electric (½ ) and magnetic (½o ) energies are equal, as Áf/2 1 2 ½ 2½o > ¾¿ À 2 f/2 Fig. 6.8(b) (6.17) shows the case of the capacitively loaded PIFA which is resonating with a physical length less than quarter wavelength. In this case, the current integration in (6.16) and (6.17) needs to be performed over a smaller interval [0,! ], and we will use the ! formula that was found in (6.8). The final expressions after performing the integration within the new intervals are: 03 320 2 2 1¡ 2 2 2> ¹>2 (4=>2 2 #32 2 5 1' 121 (6.18) ½ 2 ! 1¡ 1 4! q 25 r 2 > 4#3 > > 4> Where ¹> is the free-space wave impedance and 4! /> is an empirical constant. By substituting (6.18) and (6.19) into (6.14) the general expression for the capacitively loaded PIFA radiation quality factor is obtained as 3 1 1 Â2 - 2560 2=> #3 1 6> ¹>2 (4= 2 2 #32 2 5 1' 5 sin I2 - JÃ 2=> #3 > > 4! 2 (6.19) Fig. 6.8 (a) Surface current distribution in the fundamental mode opeation for a regular half wavelength microstrip antenna, (b) current distribution in a capacitively loaded PIFA antenna. The shadowed region indicates the current integration area. When 0, the obtained radiation quality factor corresponds to the one of a regular quarter wavelength PIFA antenna. Radiation efficiency can now be calculated combining (6.20), (6.11) and (6.12) into (6.10). A comparison of the PIFA radiation efficiency using different heights (h = 1mm, 2mm, 4mm, 6mm) versus capacitive loading from 0pF to 2pF obtained using method of moments simulations 122 (IE3D) and the final closed form expression is shown in Fig. 6.9(a). On the other hand, Fig. 6.9(b) shows the efficiency versus antenna resonant frequency for the mentioned loading capacitance range. 100 90 Radiation Efficiency ηr (%) 80 70 60 50 40 h=1mm h=1mm h=2mm h=2mm h=4mm h=4mm h=6mm h=6 CF 30 20 10 0 0.00 0.25 IE3D CF IE3D CF IE3D CF IE3D 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Capacitive Loading (pF) (a) 100 Radiation Efficiency (%) 80 60 40 h=1mm h=1mm h=2mm h=2mm h=4mm h=4mm h=6mm h=6mm 20 IE3D CF IE3D CF IE3D CF IE3D CF 0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Frequency (GHz) (b) Fig. 6.9 (a) Radiation efficiency versus capacitive loading obtained by MoM simulations (Zeland IE3D) and obtained closed form (CF) expressions. The antenna has parameters ± 4, #3 2% ()* 0&', +3 1, ,1, - 6/ (b) Radiation efficiency versus resonant frequency with ,0&, - 2&/. 123 The simulated and obtained radiation efficiencies differ less than 6% for all cases. An important observation is that the efficiency decreases considerably with capacitive loading. In [13] it is stated that the efficiency of a capacitively loaded PIFA would be slightly less than that of a non loaded PIFA. Here we point out that this will only be true in cases of small capacitive loading. In order to have reasonably high radiation efficiencies the antenna must be only ‘lightly’ loaded. However, this approach will in turn reduce the overall antenna frequency tuning range (according to Fig. 6.7) or restrict the patch length decrease (according to Fig. 6.8). Therefore, this is a fundamental tradeoff the designer must be aware of. For design purposes is interesting to observe the effect of the antenna dimensions ( and ' on the radiation efficiency versus capacitive loading so we define the efficiency reduction factor as Δ¹3 1 ; ¹3 ( : ' ¹3 ( 0' (6.20) where ¹3 ( : ' represents the radiation efficiency when the antenna is loaded with a capacitance : . Fig. 6.10 shows the effect of changing the patch width ( ) in the radiation efficiency for different loading conditions when 2 and #3 changes as indicated in Fig. 6.7. It is observed than a wider antenna patch provides better radiation efficiencies. Also, the degree of linear behavior in the efficiency reduction curve is increased for greater . For 31.1 the radiation efficiency reduction factor behavior becomes quasi linear with capacitive loading. Fig. 6.11, shows the effect of varying in the radiation efficiency reduction factor for different loading conditions when 4, #3 2%. It can be observed that the efficiency increases considerably with higher patch heights. 124 In summary, from Fig. 6.10 and Fig. 6.11 we can conclude that the patch width and height should be chosen to be as high as possible always that the space allocated for the antenna is sufficient, in order to minimize the efficiency reduction due to capacitive loading. Radiation Efficiency reduction, ∆ηr (%) 100 Wp=31.1 mm Wp=13.2 mm Wp=7.5 mm Wp=4.7 mm Wp=3.2 mm Wp=2.1 mm Wp=1.5 mm Wp=1.0 mm Wp=0.7 mm Wp=0.5 mm 90 80 70 60 50 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fig. 6.10 Efficiency reduction versus capacitive loading for different values of ± with 2. Capacitive Loading,C(pF) 100 h=1mm h=2mm h=4mm h=6mm Radiation Efficiency reduction, ∆ηr 90 80 70 60 50 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fig. 6.11 Efficiency reduction versus capacitive loading for different values of with ± 4, Capacitive Loading,C(pF) 125 6.2.6 Impedance bandwidth. The impedance bandwidth of the antenna can be estimated from the total quality factor defined as 1 1 1 5 3 > (6.21) A good approximation for the fractional (or relative) impedance bandwidth for narrowband matched antennas is given in [21] as }3 v Z<a ; 1 √Z<a 1.6 √2.6 )* Z<a 2.6 (6.22) where a maximum VSWR of 2.6 (return loss of 7dB) has been selected as typical for cell phone applications. If we compute the total quality factor using expressions (6.22), (6.20) and (6.11), we can obtain the relative bandwidth for the capacitive loaded microstrip antenna using (6.23). Fig. 6.12(a) shows the comparison of calculated and simulated antenna input impedance bandwidth (when VSWR=2.6) for different values of capacitive loading. Fig. 6.12(b) plots bandwidth versus resonant frequency for the loading capacitance range 0& - 2& It is observed that the bandwidth reduces quickly with increased capacitance and more abruptly for higher antenna heights. This feature ads and additional design tradeoff between capacitive loading and impedance bandwidth that needs to be paid special attention if minimum operational bandwidth is required. 126 50 h=2mm IE3D h=2mm CF h=4mm IE3D h=4mm CF h=6mm IE3D h=6mm CF Bandwidth (MHz) 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 Capacitive Loading (pF) (a) 50 h=2mm IE3D h=2mm CF h=4mm IE3D h=4mm CF h=6mm IE3D h=6mm CF Bandwidth (MHz) 40 30 20 10 0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (GHz) (b) Fig. 6.12 a) Impedance bandwidth reduction versus capacitive loading for different values of with 4. (b) Impedance bandwidth reduction versus frequency when loading capacitance is changed from C=0pF to 2pF. The patch total length was chosen to resonate at #3 2% in absence of capacitive loading ( 0&' for each case. It is interesting to study the relative bandwidth reduction in a similar manner than was done with radiation efficiency. The relative bandwidth reduction and can be expressed as Δ}3 1 ; }3 ( : ' }3 ( 0' 127 (6.23) Fig. 6.13 shows the relative bandwidth reduction dependence on with different loading conditions when 2 changes as indicated in Fig. 6.7. Similarly, Fig. 6.14 shows the effect of varying in the relative bandwidth reduction for different loading conditions when 4 and #3 2%. From Fig. 6.13 it can be observed that the wider the antenna patch, the less severe the bandwidth reduction is. On the other hand, from Fig. 6.14 we can see how the bandwidth reduction factor is lower when the patch height is also lower. However, is important to keep in mind that the absolute bandwidth when 0 will be also be lower for lower values of as seen in Fig. 6.12. Impedance bandwidth reduction, ∆ΒW r (%) 100 90 80 70 60 Wp= 31.1 mm Wp= 13.2 mm Wp= 7.5 mm Wp= 4.7 mm Wp= 3.2 mm Wp= 2.1 mm Wp= 1.5 mm Wp= 1.0 mm Wp= 0.7 mm Wp= 0.5 mm 50 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fig. 6.13 Relative impedance bandwidth reduction versus capacitive loading for different values of ± with 2 Capacitive Loading,C(pF) 6.2.7 Design Guidelines. Subsections 6.2.1 to 6.2.6 summarized the most important parameters usually found in specifications for cell phone antenna design. The conclusion from this study is that size reduction and/or frequency agility can be achieved using a capacitively loaded PIFA at expenses of lower radiation efficiency and impedance bandwidth. As direct consequence, a practical antenna to be 128 used in cell phone applications will have to be only ‘lightly’ loaded. The scope of this chapter is not focused on the design of a particular antenna (as this is regarded as future work) but to introduce the different design expressions. In order to illustrate how these tools can be used in practice, a set of guidelines for a design example of a capacitively Impedance bandwidth reduction, ∆ΒW r (%) 100 90 80 70 60 50 40 30 h=1mm h=2mm h=4mm h=6mm 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fig. 6.14 Relative impedance bandwidth reduction versus capacitive loading for different values of with ± 4. Capacitive Loading,C(pF) a. Decide antenna parameters , as high as possible accounting for the antenna allocated space constraint. b. Find the required patch length (! ) for an specific minimum capacitance and #K using (6.8). In a realistic implementation, the capacitance will be always greater than zero due to the inherent device shunt parasitics. c. Substitute the ! value found in (6.8) and calculate the maximum capacitance K required to operate at the minimum frequency (#) using (6.9). Steps (a) and (b) will determine the required capacitor tuning range (~K). The required capacitor tuning ratio can be also calculated by manipulating (6.9) in terms of a desired frequency range as 129 K -((k op: (! 5 '' ∆ ∆# - (GoX (! 5 '' (6.24) With ∆# #K/#,k op: 2#K√+3 /> and k oX 2#√+3 /> . d. Calculate the radiation efficiency and impedance bandwidth values for the pairs [, #K] and [K, #] using (6.10) and (6.23) respectively. Check if these values meet the efficiency and bandwidth design goals. If the specifications are not met iterate to (a), by increasing , or accept a decrease in frequency tuning range. 6.3 Measurement results To validate the above expressions and full wave simulations, a PIFA antenna prototype was built as shown in Fig. 6.15. The antenna square ground plane is made of copper having a side dimension of 230mm (1.5> at 2GHz). The PIFA metallic strip has a width of ± 4. The patch is 2 away from the ground plane and is supported by foam material (+3 v 1'. The prototype’s height value was chosen to be the most convenient to match the length of the available surface mount ceramic capacitors, Murata 0805 GQM series, of chip size 1.25mm (width) by 2mm (length). By choosing the same capacitor length as the patch height we avoid additional parasitic effects that could be introduce from any interconnecting path distorting the main purpose of the experimental results. The patch length is chosen to make the antenna operate at 2GHz for absence of loading capacitance ( 0&). The measured resulting patch length is ! 35 (0.233> '. This result is in good agreement with the calculated patch length of ! 35.5 (0.236> ' using (6.8). Then, the capacitor is soldered between the antenna radiating edge and the ground plane as indicated in Fig. 6.1 and Fig. 6.15. For this experiment different available capacitor values will be 130 used to compare the closed form expressions presented in the previous section with the measurements results. Fig. 6.16 shows the antenna reflection coefficients for different values of capacitor loading. The bandwidth reduction and frequency decrease with capacitive loading is apparent from this plot. Tables I and II summarize the values for the resonant frequency and impedance bandwidth respectively. Good agreement is observed between closed form expressions, IE3D simulations and measurements. The average error between calculations and measurements is 30MHz and 2.75MHz for the resonant frequency and impedance bandwidth respectively. Fig. 6.15 Fabricated PIFA prototype over large ground plane. The ceramic capacitor is loaded at the antenna radiating edge. Table 6.1 antenna resonant frequency versus capacitive loading C (pF) 0.2 0.5 1.2 1.6 1.8 Closed Form Expression (6.8) (GHz) 1.754 1.498 1.161 1.046 1.001 IE3D Simulation (GHz) 1.757 1.458 1.130 1.016 0.972 131 Measurements (GHz) 1.727 1.442 1.132 1.027 0.980 Fig. 6.16 Measured magnitude of the PIFA reflection coefficient for different capacitive loadings. Table 6.2 antenna impedance bandwidth versus capacitive loading C (pF) 0 0.2 0.5 1.2 1.6 1.8 Closed Form Expression (6.22) 16.5MHz (0.83%) 10MHz (0.57%) 5.5MHz (0.37%) 2.5MHz (0.21%) 2MHz (0.19%) 1.7MHz (0.17%) IE3D Simulation 16MHz (0.80%) 11MHz (0.62%) 6MHz (0.41%) 2.5MHz (0.22%) 2MHz (0.19%) 1.9MHz (0.19%) Measurements 22MHz (1.1%) 12MHz (0.69%) 8MHz (0.55%) 5MHz (0.44%) 4MHz (0.40%) 3.5MHz (0.36%) The efficiency was calculated using the generalized Wheeler cap method proposed by Johnston and McRory [22] and further applied on [23]. For this measurement a cylindrical Wheeler cap made of aluminum material with internal diameter of 90 mm and height of 150 mm was built. Fig. 6.17 Fabricated Wheeler Cap covering the PIFA antenna for radiation efficiency measurement. 132 The resulting radiation efficiencies are compared in Table III with those obtained using the closed form expressions and IE3D simulations. Good agreement is observed between the results with an average error between measured and calculated radiation efficiencies of 6.7%. Table 6.3 antenna radiation efficiency versus capacitive loading C (pF) 0 0.2 0.5 1.2 1.6 1.8 Closed Form Expression ((6.13,(6.17,(6.18) 90.0% 84.7% 74.4% 46.0% 32.7% 29.1% IE3D Simulation 89.1% 81.9% 70.1% 44.1% 31.9% 28.5% Measurements 86% 77% 66% 43% 27% 24% 6.4 Conclusions In order to predict the resonant frequency, quality factor, radiation efficiency and impedance bandwidth for an edge capacitively loaded PIFA, a set of useful approximate closed form expressions have been presented. The effects of the antenna loading over the resonant frequency or antenna size and related tradeoffs with impedance bandwidth and radiation efficiency have been elaborated. In order to meet typical cell phone antenna specifications, it was shown that the capacitive loading must be relatively light for an end-loaded PIFA, restricting the frequency tuning range or achievable size reduction. Some design guidelines have been presented as example of the usefulness of these expressions in a practical antenna design example. The results from closed form expressions have been validated using full wave simulations as well as measurement results. 133 References [1] S. Best, "Electrically Small Multiband Antennas " in Multiband Integrated Antennas for 4G Terminals, D. A. Sanchez-Hernandez, Ed., ed Norwood, MA: Artech House, 2008. [2] K. R. Boyle and P. G. Steeneken, "A Five-Band Reconfigurable PIFA for Mobile Phones," Antennas and Propagation, IEEE Transactions on, vol. 55, pp. 3300-3309, 2007. [3] K. R. Boyle and P. J. Massey, "Nine-band antenna system for mobile phones," Electronics Letters, vol. 42, pp. 265-266, 2006. [4] J. A. C. Picher, A. Cabedo, C. Puente, and S. Kahng, "Multiband handsrt antenna usign slots on the ground plane: considerations to facilitate the integration of feeding transmission line," Progress in Electromagnetic Research, vol. 7, p. 14, 2009. [5] C.-M. Su, et al., "User's hand effects on EMC internal GSM/DCS mobile phone antenna," in Antennas and Propagation Society International Symposium 2006, IEEE, 2006, pp. 20972100. [6] K.-L. Wong, Planar Antennas for Wireless Communications. Hoboken, NJ: John Wiley & Sons, 2003. [7] J. T. Aberle, et al., "Reconfigurable antennas for wireless devices," Antennas and Propagation Magazine, IEEE, vol. 45, pp. 148-154, 2003. [8] S.-H. Oh, et al., "Electronically Tunable Antenna Pair And Novel RF Front-End Architecture For Software-Defined Radios," EURASIP JASP, vol. 16, p. 6, 2005. [9] M. Martinez-Vazquez, "Miniaturized integrated multiband antennas," in Multiband integrated antennas for 4G terminals, D. A. Sanchez-Hernandez, Ed., ed Norwood, MA: Artech House, 2008. 134 [10] W. Yu-Shin, et al., "Two PIFA-Related Miniaturized Dual-Band Antennas," Antennas and Propagation, IEEE Transactions on, vol. 55, pp. 805-811, 2007. [11] V. Antonchik and R. G. Vaughan, "Spatial Aspects of the Transmission Line Model for Rectangular PIFA," in Electrical and Computer Engineering, 2007. CCECE 2007. Canadian Conference on, 2007, pp. 808-811. [12] P. B. R.Garg, I. Bahl, A. Ittipiboon, Microstrip Antenna Design Handbook: Artech House antennas and propagation, 2001. [13] C. R. Rowell and R. D. Murch, "A capacitively loaded PIFA for compact mobile telephone handsets," Antennas and Propagation, IEEE Transactions on, vol. 45, pp. 837-842, 1997. [14] H. Sobol, "Radiation Conductance of Open-Circuit Microstrip (Correspondence)," Microwave Theory and Techniques, IEEE Transactions on, vol. 19, pp. 885-887, 1971. [15] C. A. Balanis, Advanced Engineering Electromagnetics, 1989. [16] M. H. K. Hirasawa, Analysis, Design and Measurement of Small and Low Profile Antennas: Artech House, 1992. [17] J. S. McLean, "A re-examination of the fundamental limits on the radiation Q of electrically small antennas," Antennas and Propagation, IEEE Transactions on, vol. 44, p. 672, 1996. [18] D. R. Jackson and N. G. Alexopoulos, "Simple approximate formulas for input resistance, bandwidth, and efficiency of a resonant rectangular patch," Antennas and Propagation, IEEE Transactions on, vol. 39, pp. 407-410, 1991. [19] Y. Lo, et al., "Theory and experiment on microstrip antennas," Antennas and Propagation, IEEE Transactions on, vol. 27, pp. 137-145, 1979. 135 [20] W. Richards, et al., "An improved theory for microstrip antennas and applications," Antennas and Propagation, IEEE Transactions on, vol. 29, pp. 38-46, 1981. [21] A. D. Yaghjian and S. R. Best, "Impedance, bandwidth, and Q of antennas," Antennas and Propagation, IEEE Transactions on, vol. 53, pp. 1298-1324, 2005. [22] R. H. Johnston and J. G. McRory, "An improved small antenna radiation-efficiency measurement method," Antennas and Propagation Magazine, IEEE, vol. 40, pp. 40-48, 1998. [23] C. Mendes and C. Peixeiro, "Radiation Efficiency of Several Handset Antennas Obtained with a Modified Wheeler Cap Method," in Antennas and Propagation, 2007. EuCAP 2007. The Second European Conference on, 2007, pp. 1-4. 136 7. TUNABLE DUPLEXING ANTENNA SYSTEM FOR WIRELESS TRANSCEIVERS 7.1 Introduction With the increase in number of wireless bands and standards, tunable antennas can play an important role in reducing the overall system size and enhancing the power transfer to/from the transceiver. Traditionally, non-tunable cell phone antennas have been designed to cover multiple frequency bands by establishing tradeoffs with other important parameters such as antenna efficiency or overall size [1]. Designing and developing non-tunable small multiband antennas with reasonable efficiency becomes a more difficult task with the increase of allocated frequency bands, especially those at lower frequencies. As an example, Table 7.1 summarizes some of the current wireless standards frequency specifications ranging from 700MHz up to 2700MHz. In addition, it is well known that the user interaction with the handset affects the antenna performance considerably [2, 3]. In particular, the antenna is detuned (mismatch increases) and its efficiency decreases when the user hand/fingers/head move close to the antenna location leading to a reduced sensitivity (possible dropped calls) and increasing the power consumption (shortening the battery life). Tunability within the radiating element can be achieved by using tunable capacitors [4],[5],[6] or RF switches [7],[8]. For example, if frequency agility is exploited, the tunable antenna impedance bandwidth can be relaxed leading to a potential size decrease and a reduction of design cycle time. In addition, the antenna can be re-tuned in order to mitigate user or other loading effects, thus optimizing the power transfer. 137 As an enabling technology, RF MEMS tunable capacitors are able to meet the stringent requirements of handset front ends such as low loss, high linearity, and low power consumption [9]. While designing tunable handset antennas, two different design approaches can be considered: • Design a single antenna with broad enough bandwidth to cover the transmit and receive channels of each operating band simultaneously. • Design an antenna pair with dedicated transmitter and receiver antennas respectively. The minimum bandwidth of these antennas is the operating channel bandwidth. Both of these approaches are compared in Table 7.2 in terms of impedance bandwidth requirements for different wireless standards versus conventional non tunable antennas. Band Designation Band 17 (LTE) US-CELL E-CELL DCS PCS IMT Band 7 (LTE) Band Designation Band 17 (LTE) US-CELL E-CELL DCS PCS IMT Band 7 (LTE) Table 7.1 Wireless standards frequency specifications Transmit Uplink Receive Downlink Channel Frequency Frequency BW (MHz) (MHz) (MHz) 704~716 734~746 1.4~20 824~849 869~894 0.2~5 880~915 925~960 0.2 1710~1785 1805~1880 0.2 1850~1910 1930~1990 0.2~5 1920~1980 2110~2170 5 2500-2570 2620-2690 1.4 - 20 TX~RX Offset (MHz) 30 45 45 95 80 190 120 Table 7.2 Requirement for cell phone antenna bandwidths Conventional Single Tunable Antenna Tunable Antenna Pair Antenna BW Minimum BW Minimum BW (MHz) (MHz) (MHz) 42 (5.8%) 32.8~50(4.3~6.9%) 1.4~20 (0.19~2.7%) 70 (8.1%) 45.4~55 (5.3~6.4%) 0.2~5 (0.025~0.58%) 80 (8.7%) 45.4 (4.9%) 0.2 (0.022%) 170 (9.5%) 95.4 (5.3%) 0.2 (0.012%) 140 (7.3%) 80.4~90 (4.2~4.6%) 0.2~5 (0.010~0.26%) 250 (12.2%) 200 (9.8%) 5 (0.24%) 190 (7.3%) 90 (3.4%) 20 (0.7%) From Table 7.2, the lower instantaneous bandwidth requirements of tunable antennas are apparent compared to conventional antennas. Further, the tunable antenna pair is shown to have far lower 138 required impedance bandwidth. Note that the tuning resolution using tuned pairs must be high enough to allow for optimal tuning for each different communications channel (spacing as low as 100KHz). On the other hand, a single tunable antenna with broader bandwidth can be implemented with fewer RF MEMS switches or lower resolution RF MEMS varactor devices. Additionally, according to classic fundamental limits for small antennas [10, 11], a narrower impedance bandwidth antenna can be designed to occupy a smaller size than its conventional broadband antenna counterpart. In this context, the reduction in bandwidth requirement shown in Table 7.2 may allow for smaller tunable antenna footprints. The objective of this chapter is to examine the feasibility of the antenna pair concept by means of simulations and measurements of a fabricated prototype. Section 7.2.1 and 7.2.2 presents the proposed antenna pair solution and highlights some its additional advantages (such as the built-in filtering characteristics) from a system point of view. Section 7.2.3 analyzes the isolation level between both antennas for different arrangements within the handset. Through section 7.2, simulations will be supported by measurements on the fabricated prototype. Finally, section 7.2.4 presents the results obtained from the radiated test performed at UCI´s anechoic chamber over the single antenna element. 7.2 Duplexing Antenna System 7.2.1 Advantages from a system perspective An additional advantage of using an antenna pair with narrow impedance bandwidth is the ability to enhance the isolation levels between transmit and receive antennas [5, 12-15]. If the antenna pair configuration is properly designed, the isolation level between both elements may reach up to 25dB. Due to this inherent ‘filtering’ ability, the antenna pair is given the name 139 of duplexing antenna. If these antennas were combined with the tunable notch filter presented in a companion paper [16] as shown in Fig. 5.12(a), the overall isolation level between transmit and received branches would approach nearly 50dB levels. This value of isolation is comparable to a conventional full duplex front end solution shown in Fig. 5.12(b) comprising of a broadband multiband tunable antenna, diplexer and external duplexer module based on non integratable FBAR technology. Contrarily to external duplexers, RF MEMS tunable filters using RFCMOS process are compatible with on-chip integration. In addition, because of the desired narrowband characteristics of the duplexing antenna elements, the antenna pair may be made smaller than its broadband antenna counterpart. These features added to the advantage of multiband operability, can reduce the number of components, overall size and cost of cell phone front ends. The following sub-sections will evaluate the concept of tunable duplexing antenna in a realistic handset platform. Fig. 7.1 Duplexing antenna tunable front end concept depicting tunable narrowband antennas in combination with tunable notch filter, (b) Conventional non tunable front end comprising of a multiband broadband antenna with diplexer and external multiple duplexer modules. 140 7.2.2 Single Tunable Element Design Planar inverted-F antennas (PIFAs) have been widely used as internal mobile phone antennas due to their low profile, easiness of fabrication and relatively small footprint. PIFAs are a particular type of microstrip antenna where the resonating patch length is approximately a quarter wavelength at the operating frequency. This is accomplished by placing short circuit plate/pins in one of the antenna edges and placing the feeding probe near the shorting pins in a location selected for good impedance matching. One possible option for designing a tunable PIFA is to load a shunt tunable capacitor on its open end as shown in Fig. 7.2. Fig. 7.2 Tunable PIFA antenna concept with shunt capacitor loading on antenna open edge. In this case, when the capacitance is minimum (high impedance) the antenna resonates at the highest operation frequency. On the other hand, when the capacitance is maximized, the capacitor acts as a longer equivalent length extension of PIFA arm, making the antenna to appear electrically longer and resonating at the lowest frequency. This allows for the antenna to be considerably smaller than quarter wavelength at low frequency. The antenna size reduction allowed by the loading capacitor is accompanied by a reduction in the impedance bandwidth and radiation efficiency. Narrow impedance bandwidth (as long as is 141 greater than channel bandwidth) is a desired feature in the duplexing antenna concept to enhance the transmit-receive isolation level, although not the only factor. On the other hand, radiation efficiency reduction comes from the fact that the loaded small antenna radiation resistance decreases and starts to be comparable to the loss resistance. Efficiency values greater than 50% are usually required in cell phone applications, therefore, care must be given to avoid excessive efficiency drop by limiting the amount of capacitive loading, which in turns imposes limit on the overall frequency tuning range. In this chapter, an existing high Q tunable digital capacitor array (TDCA) (see Fig. 7.3) flip die solution from Wispry Inc. [9] was utilized in the designs. The TDCA consists of twenty tunable capacitor cells, each with a nominal tuning range of 1pF or 0.875pF. Only two 1pF cells will be utilized in this specific antenna application (avoiding excessive capacitive loading). Nonetheless, for product applications, much smaller custom TDCAs would be utilized. The minimum capacitance step resolution of each cell is 0.125pF in this chip. The cells in the TDCa can be interconnected on the PCB level in order to achieve any desired amount of capacitance and circuit topology. Fig. 7.3 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell; b) 3D image of capacitance bits. 142 The Q of the die level capacitors was measured to be greater than 100 at 2GHz with a value of capacitance highly repeatable. The IP3 level for this device is greater than +65dBm. The CMOS biasing circuitry is integrated within the same chip and transforms a 3.3V regulated supply voltage coming from the USB port to the required 35V actuation voltage. The current consumption is 6 µA and 90 µA in the sleep and the active mode (charge pump on), respectively. A Serial Peripheral Interface (SPI) is used to control the capacitor bank states. A USB port is used to issue the tuning commands from PC control software. Fig. 7.4 shows the designed shunt loaded PIFA. The antenna length was adjusted to its final value of Lp=15mm for operation at highest frequency when C=Cmin. The antenna width was chosen to be Wp=2mm to occupy the smallest area while providing sufficient radiation efficiency. Placing the tunable capacitor flip chip at the open edge of the antenna patch requires additional routing for the SPI control lines and via/pins connecting to the ground plane below as shown in Fig. 7.4. A total of eight control lines are then routed to the opposite layer of the PCB (made of FR4 material) where the control circuitry is located. In order to shield the low frequency lines from the RF signal and serve as ground connection for the tunable capacitors, a row of four grounding pins are located between the antenna structure and the eight control lines. A tunable capacitance ratio of C=0.26-1.86pF (Cmin-Cmax) corresponding to two parallel capacitor cells is used to cover the desired frequency range. The antenna patch is printed on top of a supporting dielectric material (40mm by 20mm, Rogers 5880 εr=2.2, tanδ=0.009 at 10GHz, 0.381mm thick) as shown in Fig. 7.5. The ground plane has dimensions 40mm by 90mm and is located at 4 mm distance from the antenna patch. Two pins are used as shorting point separated 0.5 mm from coaxial line fed in order to match the antenna at the operating frequency. 143 Fig. 7.4 Tunable PIFA with two shunt tunable capacitors loading the antenna radiating edge and a series external chip capacitor in series with the tunable capacitors. Fig. 7.5 Designed tunable single element and PCB dimensions. To increase the resolution of the antenna operating frequency steps, an external surface mount capacitor (Murata SMD 0402) has been added in series with the tunable capacitor die as shown in the detail of Fig. 7.4. This will produce finer capacitance step size at expenses of reducing the capacitance tuning range. The design has been simulated using Ansys HFSS finite element method solver. The lumped components such as series capacitor or tunable cells have been 144 modeled in this tool using horizontal lumped ports. The multiport S-parameter matrix resulting from full wave simulations was loaded with the proper values of fixed and tunable capacitance using AWR Microwave Office circuit simulator. A prototype using the design dimensions was built and assembled as shown in Fig. 7.6 and Fig. 7.7. In this design, the integrity of the simulated geometry of Fig. 7.5 was preserved with minor additions mainly from the DC control circuitry in the opposite side of the PCB. The control circuitry consist of connection pins for the SPI lines, voltage regulator and a few surface mount components such as capacitors/resistors. Fig. 7.6 Top and bottom views of the fabricated prototype. The top view shows the duplexing antenna structure, tunable capacitor dies and RF ground plane. The bottom side contains the control circuitry and interfacing multi-pin connector.. 145 Fig. 7.7 Picture of the duplexing antenna built prototype The simulation and measured results for different cases of capacitive loading are in good agreement as shown in Fig. 7.8. These representative cases are interesting to observe the effect of different capacitive loading on antenna resonance frequency and are listed from higher lo lower resonance frequency as: (a) absence of surface mount fix capacitor or tunable capacitors with antenna resonating at 3.32GHz, (b) a 0.5pF surface mount capacitor with absence of tunable capacitors and resonance at 2890MHz, (c) a 0.5pF surface mount capacitor and tunable capacitors set to C=Cmin=2x0.13pF with resonance at 2170MHz and (d) a 0.5pF surface mount capacitor and tunable capacitors set to C=Cmax=2x0.93pF operating at 1850MHz. From Fig. 7.8, it is also observed that if the tunable capacitors minimum capacitance was reduced to zero, the antenna tuning range would increase considerably. However, a residual parasitic capacitance is always present in any realistic device. With the current configuration the antenna tuning range is 2170MHz-1850MHz (16%).This is sufficient for operation in the IMT-I and PCS bands (refer to Table 7.1). If the capacitance step resolution was finer than the current 0.125pF (i.e 146 tunable capacitors were customized for this application) and therefore the series SMD capacitor was not needed, the same capacitance tuning range (2x0.13pF to 2x0.93pF) would allow for a resonance tuning range of 1990MHz to 990MHz (67%). Fig. 7.8 Simulated (HFSS) and Measured reflection coefficient of the single element antenna for different values of capacitive loading. Fig. 7.9 Measured reflection coefficient of the single element antenna for different values of capacitive loading. 147 Fig. 7.9 shows the measured single antenna reflection coefficient for all the possible tunable capacitor states. An overall tuning range of 1850MHz to 2170MHz with a frequency step ranging from 55MHz to 15MHz and a measured impedance bandwidth from 54MHz to 25MHz (at VSWR=2.6). 7.2.3 Antenna Pair Design Once the single element was designed, the duplexing antenna system is formed using two identical tunable antennas. One of the most important features of this system is the isolation between transmit and received antennas. This parameter is also of high importance in conventional non tunable antenna designs and depends not only on the far field antenna characteristics but also in the coupling of currents induced in the PCB ground plane. Several studies have studied the isolation between different handset antennas by analyzing different antenna orientations within the PCB [17, 18], using ground plane notches [19, 20], connecting strips and resonating elements [17, 21]. A generalization is difficult in these cases because the isolation level ultimately depends on the specific antenna topology, size of the PCB ground plane, proximity of other cell phone components and user interaction. In general, a specific study must be performed for each particular device platform. In practice, it is usually challenging to achieve more than 15dB isolation between broadband elements in a realistic handsets. 148 Fig. 7.10 Different configurations considered in the duplexing antenna isolation study. The distance ‘d’ is varied in discrete intervals and the S21 parameter is recorded for each variation at high and low frequency pair situations. Full wave simulations provide some quick understanding of the best antenna geometry for a specific platform. In this work, several geometries were considered as shown in Fig. 7.10(a-d). The main goal was to allocate the two antennas within the same region of the handset (top portion) and decide the separation distance and relative orientation. Besides the isolation level, the other important factor is the ability to tune both antennas independently. These two concepts are ultimately related as shown in Fig. 7.11 (a) where the antenna with port 1 is acting as transmitting antenna at 1850MHz while port 2 corresponds to the receiving antenna at 1930MHz. The distance between elements is d=11.8mm (referring to Fig. 7.10(a)). In this case, a second resonance appears in the transmit antenna at the receive frequency and vice versa, providing a dual frequency behavior. Besides the low isolation level under these circumstances, a change in the capacitive loading value on any of the two antennas will strongly affect the opposite antenna resonance frequency. It is obvious that this tuning dependency is not 149 a desired effect due to the fact that the control of both antennas becomes unfeasible. On the other hand, Fig. 7.11 (b) shows the same frequency pair situation when both antennas are separated a greater distance of d=36.8mm. In this case, not only the isolation level between elements will be much higher but the coupled dual resonance behavior does not appear and both antennas can be tuned independently. In order to find configurations with higher isolation, a simulation parametric study has been performed using HFSS with the topologies considered in Fig. 7.10. The separation distance ‘d’ for each case was varied in incremental steps. The resulting S-parameters matrix was then capacitively loaded in AWR Microwave office to make the antenna operate at two different frequency pairs: high frequency pair (1980MHz, 2170MHz) and low frequency pair (1850MHz, 1930MHz). The value of the S21 parameter versus separation distance for cases of Fig. 7.10(a-d) at these four frequencies are shown in Fig. 7.12(a-d), respectively. For the cases Fig. 7.12(a-b), the isolation between antennas increases with the separation distance. In particular the higher isolation level is obtained for the case of Fig. 7.12(a) when d=36.8mm and both antennas are located close to opposite PCB edges. In that situation 26.6dB and 21.8dB are the minimum isolation levels achieved for the high and low frequency pairs, respectively. On the other hand, in Fig. 7.12(c) the antenna the isolation is higher for the cases of d=0mm and d=36.8mm when the respective feeding point and radiating edge of both antennas are further away. In the intermediate cases, from approximately d=4mm to d=24mm it was not possible to tune the antennas at the specific frequencies due to the previously mentioned tuning dependency between elements. This seems to indicate that when one antenna’s radiating edge is closer to the other antenna’s feeding point the existing coupling between elements is maximum. 150 (a) (b) Fig. 7.11 Simulated S-parameters resulting from configuration of Fig. 7.10(a) when the two antennas are separated a distance (a)11.8mm and (b)36.8mm. Antenna with port 1 and port 2 are operating at 1850MHz (transmit antenna) 1930MHz (receive antenna), respectively. 151 Finally, as intuitively expected, the case of Fig. 7.12(d) where both antennas are oriented in orthogonal polarizations exhibits higher isolation level and slower variations with distance. However, it was not desirable to choose this option for final implementation as the transmit and receive antennas in the system are desired to operate at the same polarization in the communication link. For this reason, the configuration with maximum isolation (Fig. 7.10(a)) with d=36.8mm was chosen for the final implementation. The duplexing antenna prototype was built as shown in Fig. 7.7. Both antennas were then connected to the two-port vector network analyzer and tuned to operate at the high and low frequency pairs while the S21 isolation parameter was measured as shown in Fig. 7.13 For the high (Tx-Rx offset=190MHz, Fig. 7.13(a)) and low (Tx-Rx offset=80MHz, Fig. 7.13(b)) frequency pairs, the worst obtained isolation level was 18.5dB and 22.5dB, respectively. This value is slightly lower than the simulated isolation level and may be due to additional coupling created by the DC control circuitry that was not included in simulations and VNA measurement setup. However, the obtained values of isolation are still particularly high and remain as a valid proof of concept for the duplexing antenna system. 7.2.4 Single Antenna Radiated Test The single antenna was measured in UCI’s far field anechoic chamber facility in order to obtain the radiation pattern and radiation efficiency. The normalized measured radiation pattern corresponding to the two extreme capacitive cases C=Cmax and C=Cmin for the main plane cuts are shown in Fig. 7.14(a) and (b), respectively. The results are in reasonably good agreement with simulations. The obtained peak gain was 2.95dBi and 2.25dBi for the C=Cmax and C=Cmin cases, respectively. 152 The radiation efficiency was measured in UCI’s far field anechoic chamber at the optimum matched frequency for each one of the cases shown in Fig. 7.8 and compared to those obtained by HFSS simulations in Table 7.3. In general the measured efficiency is lower than the predicted by simulations and this may be due to additional losses introduced due to solder joints and other fabrication imperfections. In general, the efficiency values are still above the target 50% for all frequencies within the tuning range which was within the acceptable efficiency threshold predominantly admitted for cell phone applications. 7.3 Conclusions A duplexing antenna concept based on a narrowband antenna pair with high isolation between elements has been presented. The feasibility of this system for the use in handset applications has been validated with simulations and measurements. This work demonstrated the potential advantages of using tunable handset antennas for next generation reconfigurable front ends. A modified topology that allows for an increase in the number of covered bands and decrease in the frequency resolution steps by using a chip with smaller capacitance increments is regarded as future work. Table 7.3 antenna radiation efficiency versus capacitive loading SMD Chip Capacitor (pF) 0 0.5 0.5 0.5 Tunable Capacitor (pF) 0 0 0.26 1.86 Antenna operating frequency (MHz) 3320 2890 2170 1850 153 Simulated Radiation Efficiency Measured Radiation Efficiency 92% 91% 74% 68% 89% 83% 68% 55% (a) (b) (c) (d) Fig. 7.12 Simulated parametric study of isolation (S21) between the two antennas operating at high (1980MHz, 2170MHz) and low (1850MHz, 1930MHz) frequency pairs for the configurations depicted in Fig. 7.10. (a) (b) Fig. 7.13 Measured reflection coefficients and S21(dB) between the two antennas operating at (a) high (Tx-Rx offset=190MHz) and (b) low (Tx-Rx offset=80MHz) frequency pairs. 154 Fig. 7.14 Simulated (blue solid) and measured (red dashed) normalized radiation patterns for three orthogonal planes at two different loading conditions: (a) C=Cmax and (b) C=Cmin.. 155 References [1] K.-L. Wong, Planar Antennas for Wireless Communications. Hoboken, NJ: John Wiley & Sons, 2003. 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