ABSTRACT WILKERSON, JONATHAN RYAN. Passive Intermodulation Distortion in Radio Fre- quency Communication Systems. (Under the direction of Professor Michael B. Steer and Kevin G. Gard). Passive intermodulation distortion can interfere with intended communications signals limiting the capacity and range of a communications system. Many physical mecha- nisms have been suggested as causes of passive intermodulation distortion. The description of these mechanisms are generally limited to empirical or behavioral models rather than physical descriptions due to the di±culty in isolating passive intermodulation mechanisms. Measurement of passive intermodulation distortion is complicated by the weakly nonlinear behavior of passive components, inhibiting the physical isolation of passive intermodulation producing mechanisms. The dynamic range required to measure the weak nonlinearities of these components can often exceed 100 decibels. A broadband measurement system based on feed-forward cancellation possessing dynamic range in excess of 113 decibels is constructed to overcome passive intermodulation measurement di±culties. Electro-thermal distortion is found to be a dominant passive intermodulation source with a de¯ned non- integer order Laplacian behavior. This behavior results in long-tail transients and a well de¯ned thermal dispersion characteristic in the generated passive intermodulation distortion that cannot easily be explained by integer order di®erential equations. A fractional calculus description of the phenomena is introduced, accurately modeling both long-tail transients and thermal frequency dispersion. The physics behind electro-thermal distortion is derived analytically for general lumped, lossy microwave components, transmission lines, and anten- nas. Microwave attenuators, terminations, integrated circuit resistors, transmission lines, and antennas are manufactured to isolate the electro-thermal phenomena. The developed high dynamic range measurement system is used to characterize the thermal dispersion characteristic in the generated passive intermodulation distortion for each manufactured component. Electro-thermal conductivity modulation, dependent only on material param- eters, is shown to be a dominant passive intermodulation source in all passive microwave circuits. °c Copyright 2010 by Jonathan Ryan Wilkerson all rights reserved Passive Intermodulation Distortion in Radio Frequency Communication Systems by Jonathan Ryan Wilkerson A dissertation submitted to the Graduate Faculty of North Carolina State University in partial full¯llment of the requirements for the Degree of Doctor of Philosophy Electrical Engineering Raleigh, NC 2010 APPROVED BY: Dr. Douglas Barlage Dr. Mohammed Zikry Dr. Michael B. Steer Dr. Kevin G. Gard Chair of Advisory Committee Co-Chair of Advisory Committee ii BIOGRAPHY Jonathan Ryan Wilkerson was born in Greenville, NC. He received the B.S. degree in both Electrical Engineering and Computer Engineering in 2005, and the M.S. degree in Electrical Engineering in 2006, from North Carolina State University, Raleigh, NC. Since 2005 he has been a graduate Research Assistant with the ERL Laboratory, Electrical and Computer Engineering Department, North Carolina State University. Mr. Wilkerson's research interests include electro-thermal physics, nonlinear microwave and RF circuits, wave propagation and antennas, thermo-acoustics, electro-acoustics, high dynamic range measurement, and physically based nonlinear modeling. iii ACKNOWLEDGMENTS I would like to express my appreciation for the support and guidance provided by my ad- visors, Dr. Michael Steer and Dr. Kevin Gard, during my PhD research. I would like to thank Dr. Michael Steer for lending me his considerable knowledge in physics, mathemat- ics, and microwave and RF circuits. He de¯ned my research and guided me through my publications and presentations. I am grateful for access to Dr. Steer's professional contacts that funded my research, namely ARO. I would like to thank Dr. Kevin Gard, who was an invaluable resource in this work for research direction, measurement methods and ideas, his considerable knowledge of analog and RF circuits, and his assistance in all my publications. I would also like to thank the members of my committee, Dr. Mohammed Zikry and Dr. Doug Barlage, for their comments and discussions on my research. I would like to thank my graduate student colleagues Gregory Mazzaro, Theodore Robert Harris, Glenwood Garner III, Peter Lam, Chris Saunders, Dr. Frank Hart, Dr. Sonali Luniya, and Zhiping Feng for stimulating conversation and an exchange of knowledge that extended my own research and experience. Through their humor and creativity they provided an enjoyable working environment and the encouragement needed to ¯nish. I extend a special thanks to Robert Harris, Peter Lam, and Dr. Steve Lipa who assisted me greatly in the development of several projects. I also would like to thank Dr. Aaron Walker and Dr. Mark Bu® for teaching me to use the laboratory equipment in the Electronics Research Laboratory when I ¯rst joined the Ph.D program. Lastly, I would like to thank my family for providing me encouragement and support, instilling in me a work ethic, and making me believe I could achieve any goal. Their support made this work possible. iv TABLE OF CONTENTS LIST OF FIGURES . ix LIST OF TABLES. xii 1 Introduction . 1 1.1 Motivations . 2 1.2 Overview . 2 1.2.1 Passive Distortion Modeling . 3 1.2.2 Passive Intermodulation Measurement . 3 1.2.3 Objective: Explain and Model the physical source of PIM . 4 1.3 Original Contributions . 5 1.3.1 Broadband High Dynamic Range Measurement System . 5 1.3.2 Nonlinear Electro-Thermal Theory . 5 1.3.3 Fractional Electro-Thermal Circuit Model . 5 1.3.4 Foster Expansion Electro-Thermal Circuit Model . 6 1.3.5 Distributed PIM Model . 6 1.3.6 Resonant PIM Model . 6 1.3.7 Electromagnetic and Acoustic Anechoic Chamber . 6 1.4 Dissertation Outline . 6 1.5 Published Works . 8 1.5.1 Journals . 8 1.5.2 Conferences . 8 1.6 Unpublished Works . 9 2 Review of Nonlinear Analysis and Measurement Techniques . 10 2.1 Introduction . 11 2.2 Passive Distortion . 12 2.2.1 Metal-Insulator-Metal and Metal-Metal Contact Nonlinearities . 12 2.2.2 Ferromagnetic Material Nonlinearity . 20 2.2.3 Piezoelectric Material Nonlinearity . 21 2.2.4 Acoustic Nonlinearities . 23 2.2.5 Electrical Conductivity Nonlinearity . 24 2.3 Distortion Analysis Methods . 25 2.3.1 Power Series . 26 2.3.2 Volterra Series . 26 2.3.3 Behavioral Models . 27 2.3.4 Physical Models . 28 2.4 Distortion Measurement . 29 v 2.4.1 AM-AM and AM-PM Characterization . 29 2.4.2 THD Characterization . 30 2.4.3 Two-Tone Characterization . 32 2.4.4 Multi-Tone and Band-Limited Continuous Characterization . 34 2.4.5 Passive Distortion Measurement . 34 2.5 Fractional Calculus . 38 2.5.1 Fractional Calculus Natural Functions . 39 2.5.2 Fractional Integral De¯nition . 41 2.5.3 Fractional Derivative De¯nition . 42 2.5.4 Fractional Di®erential Equations . 45 2.5.5 Numerical Methods . 45 2.6 Conclusion . 49 3 Broadband High Dynamic Range Measurement . 51 3.1 Introduction . 52 3.2 Feed-Forward Cancellation Theory . 53 3.3 Linear Feed-Forward System Design . 58 3.4 Component Linearity . 61 3.4.1 Ampli¯ers . 61 3.4.2 Circulators . 65 3.4.3 Terminators and Attenuators . 69 3.4.4 Cables and Connectors . 72 3.4.5 Summary of Component Linearity . 74 3.5 High Dynamic Range Design . 75 3.5.1 Mixing E®ects in Ampli¯ers . 75 3.5.2 Radiative Coupling E®ects and Sources . 78 3.5.3 Spurious Frequency Content . 80 3.5.4 Reflection E®ects on Nonlinearities . 84 3.5.5 Summary of Nonlinear System Design . 86 3.6 System Applications . 86 3.6.1 Dynamic Range Enhancement for PIM Measurement . 87 3.6.2 Digitally-Modulated Signals . 88 3.6.3 Summary of System Applications . 91 3.7 Conclusion . 91 4 Electro-Thermal Passive Intermodulation Distortion . 94 4.1 Introduction . 95 4.2 Heat Conduction and Electro-Thermal Distortion . 96 4.2.1 Electrical and Thermal Coupling . 97 4.2.2 Fractional Time Evolution . 100 4.2.3 Fractional Heat Conduction System for Lossy Lumped Components 103 4.2.4 Summary of Electro-Thermal Nonlinearity . 108 4.3 Electro-Thermal Circuit Models . 108 4.3.1 Fractional Compact Circuit Model for Electro-Thermal PIM . 110 vi 4.3.2 Foster Approximation . 113 4.3.3 Summary of Electro-Thermal Modeling . 116 4.4 Case Study: Microwave Terminations . 118 4.4.1 Thermal Coe±cient of Resistance Characterization . 118 4.4.2 Thermal Transient Characterization . 120 4.4.3 Two-Tone PIM Characterization . 122 4.4.4 Summary of Electro-Thermal Distortion in Terminations . 125 4.5 Case Study: Platinum Attenuator . 125 4.5.1 Thermal Parameter Characterization and Foster Model . 126 4.5.2 Platinum Electro-Thermal Dispersion . 126 4.5.3 Summary of Electro-Thermal Distortion in Attenuators . 129 4.6 Case Study: Integrated Circuit Distortion . 131 4.6.1 Electro-Thermal Dispersion Measurement . 132 4.6.2 Wide Thermal Bandwidth Devices . 133 4.6.3 Narrow Thermal Bandwidth Devices . 140 4.6.4 Summary of Electro-Thermal Distortion in IC's . 148 4.7 Conclusion . 150 5 Distributed Passive Intermodulation Distortion . 152 5.1 Introduction . 153 5.2 Distributed Electro-Thermal Theory . 154 5.2.1 Heat Conduction on Transmission Lines . 155 5.2.2 Electro-Thermal PIM of a Finite Element . 160 5.2.3 Distributed PIM Interference . 163 5.2.4 Summary of Distributed Electro-Thermal PIM Theory . 167 5.3 Nonlinear Conductivity Isolation.