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Frequency Control in Electronic Devices AN INVESTIGATION INTO THE DESIGN AND CONTROL OF TUNABLE PIEZOELECTRIC RESONATORS by Muturi G. Muriuki BS, Hope College, 1997 MS, University of Pittsburgh, 2000 Submitted to the Graduate Faculty of the School of Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2004 UNIVERSITY OF PITTSBURGH SCHOOL OF ENGINEERING This dissertation was presented by Muturi G. Muriuki It was defended on July 27, 2004 and approved by Dr James T. Cain, Professor, Electrical Engineering Dr Marlin H. Mickle, Professor, Electrical Engineering Dr Qing-Ming Wang, Assistant Professor, Mechanical Engineering Dr Jeffrey S. Vipperman, Assistant Professor, Mechanical Engineering Dissertation Director: Dr William W. Clark, Associate Professor, Mechanical Engineering ii AN INVESTIGATION INTO THE DESIGN AND CONTROL OF TUNABLE PIEZOELECTRIC RESONATORS Muturi G. Muriuki, PhD University of Pittsburgh, 2004 Piezoelectric resonators are used in electronic devices and electrical circuits as a frequency source. The most commonly used material for the piezoelectric resonators is quartz. The quartz resonator has a tunability of between 10 ppm (0.001%) and 100 ppm (0.01%) of the nominal frequency of operation. This work shows that greater tunability can be achieved using resonators made using piezoelectric materials other than quartz and a shunt-tuning technique. The tuning afforded by using lead zirconate titanate as the piezoelectric material in a cantilever type resonator is explored in detail from an analytical and experimental standpoint. It is shown that this tuning can be up to over 10,000 ppm (1%) of the nominal operational frequency in the configuration looked at, which was not optimized to maximize the tuning range. Questions of implementation of the resonator in a commonly used resonator circuit were also answered. The resonator was experimentally shown to be operable in a modified Pierce circuit with a tuning range that was analytically predicted. iii TABLE OF CONTENTS 1.0 INTRODUCTION ……………………………………………………………………………… 1 1.1 RESONATORS ………………………………………………………………………… 2 1.1.1 Resonator Techniques …………………………………………….…….……. 4 1.2 OBJECTIVES OF THE WORK …………………………………………...…………… 7 2.0 PIEZOELECTRICITY ……………………………….………………………………………… 10 2.1 THE PIEZOELECTRIC EFFECT .…...………………………………………………. 10 2.2 PIEZOELECTRIC MATERIALS ………………..…………………………………… 10 2.2.1 Crystalline Piezoelectric Materials ………………….……………..………. 10 2.2.1.1 Quartz ………………………………………….…………...………... 10 2.2.1.2 Rochelle Salt …………………………………….……………….…... 13 2.2.1.3 Tourmaline ……………………………………….………………….. 13 2.2.1.4 Gallium Orthophosphate ……………………….………………….. 14 2.2.1.5 The CGG Group …………………………………………………….. 15 2.2.1.6 Lithium Niobate and Lithium Tantalate ……………………...….. 16 2.2.1.7 Other single crystal compounds ……………………………….….. 17 2.2.2 Non-crystalline Piezoelectric Materials …………………………………… 18 2.2.2.1 Barium Titanate ………………………………………………….….. 19 2.2.2.2 Bismuth Titanate family ……………………………………….…… 19 iv 2.2.2.3 Tungsten Bronze family ……………………………………………. 19 2.2.2.4 Perovskite Layer Structure Ferroelectrics ……………………..…. 20 2.2.2.5 Lead Zirconate Titanate (PZT) ………………………………….…. 20 2.2.2.6 Aluminum Nitride ……………………………………………….…. 22 2.2.2.7 Zinc Oxide …………………………………………………………… 22 2.2.2.8 PVDF …………………………………………………………………. 22 2.3 PIEZOELECTRIC SYSTEMS, CLASSES AND MATERIAL STRAINS .………… 23 2.3.1 Piezoelectric Systems and Classes …………………………………………. 23 2.3.1.1 Orthorhombic System ……………………………………………… 25 2.3.1.2 Tetragonal System ………………………………………………….. 25 2.3.1.3 Trigonal System …………………………………………………….. 26 2.3.1.4 Hexagonal System …………………………………………….….… 27 2.3.1.5 Cubic System ………………………………………………………... 27 2.3.2 Piezoelectric Strain Matrices and Induced Strains ……………………….. 27 3.0 RESONATORS AND OSCILLATORS AND THEIR CHARACTERISTICS ..…………… 29 3.1 RESONATOR CHARACTERISTICS .………………………………………………. 29 3.1.1 Q …………………………………………………………………………….… 29 3.1.2 Stability ……………………………………………………………………..… 30 3.1.2.1 Jitter …………………………………………………………………... 30 3.1.2.2 Drift and Aging ……………………………………………………... 30 3.1.2.3 Temperature Stability ………………………………………………. 31 3.2 TYPES OF RESONATORS AND OSCILLATORS ………………………………… 32 3.2.1 Crystal Resonators and Oscillators ………………………………………... 32 3.2.2 Atomic Resonators and Oscillators …………..………………………….… 33 v 3.2.3 Microwave Resonators and Oscillators ………………………...……….… 34 3.2.4 Electrical Circuits ……………….…………………………………………… 35 3.2.5 Mechanical and Electromechanical Resonators and Oscillators …….….. 35 3.3 MODES OF RESONANCE OR OSCILLATION …………………………………... 37 3.3.1 Bulk Acoustic Waves (BAW) ………………………….……………………. 37 3.3.2 Surface Acoustic Waves (SAW) ……………….…………………………… 38 3.3.3 Direct Electrical Coupling …………………………………….…………….. 38 3.3.3.1 Microwave Cavity Modes …………………………….……………. 39 3.3.3.2 Microstrip Lines …………………………………………………….. 39 3.3.4 Mechanical Vibration …………………………………………...…………... 40 3.3.4.1 Transverse Vibration ……………………………………………….. 40 3.3.4.2 Torsional Vibration …………………………………………………. 41 3.3.4.3 Axial Vibration ……………………………………………………… 41 3.3.4.4 Membrane Vibration ……………………………………………….. 42 3.3.4.5 Plate Vibration ………………………………………………………. 42 4.0 DESIGN AND MODELING …………………………………………………………………. 43 4.1 DESIGN ...……………………………………………………………………………... 44 4.2 MODELING OF THE DESIGNS ..…………………………………………………... 47 4.2.1 One Degree of Freedom Systems ………………………………………….. 47 4.2.1.1 Simple Model from the Beam Equation ………………………….. 47 4.2.1.2 Shape Functions applied to the Hamiltonian of the Electromechanical System ......……………….…………………….. 50 5.0 BENDING FREQUENCIES OF VIBRATION .……………………………………………... 55 5.1 ANALYTICAL MODEL ……………………………………………………………... 55 vi 5.2 EXPERIMENTAL SETUP ..………………………………………………………….. 57 5.3 COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS .………. 58 5.4 SUMMARY …………………………………………………………………………… 64 6.0 TUNING RANGE …………………………………………………………………………….. 65 6.1 ANALYTICAL MODEL …………………………………………………………...… 65 6.2 EXPERIMENTAL SETUP .…………………………………………………………... 70 6.3 COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS .…...….. 71 6.4 SUMMARY …………………………………………………………………………… 85 7.0 FEEDBACK GAIN ….………………………………………………………………………… 87 7.1 ANALYTICAL MODEL ……………………………………………………………... 87 7.2 RESULTS …...…………………………………………………………………………. 90 7.3 SUMMARY ………………………………………………………………………….... 98 8.0 REACTANCE AND RESONATOR OPERATING FREQUENCIES ……………….……. 99 8.1 ANALYTICAL MODEL ……………………………………………………………. 100 8.2 EXPERIMENTAL SETUP .…………………………………………………………. 102 8.3 COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS ….…... 107 8.3.1 Resonator PS VIII …………………………………………………………... 107 8.3.2 Resonator PS XII …………………………………………………………… 113 8.3.3 Resonator PS XIII …………………………………………………………... 118 8.3.4 Resonator PS XIV .………………………………………………………….. 122 8.3.5 Resonator PS XV …………………………………………………………… 128 8.3.6 Resonator PS XVI ……………………….………………………………….. 134 8.4 SUMMARY ……………………………………….…………………………………. 144 9.0 CONCLUSIONS AND FUTURE WORK .……………….………………………………... 145 vii 9.1 CONCLUSIONS …………………………………………………………………….. 145 9.2 FUTURE WORK ..…………………………………………………………………… 148 BIBLIOGRAPHY …………….……………………………………………………………………….. 150 viii LIST OF TABLES Table 1 Piezoelectric Properties of some single crystals of the CGG group .……………………. 16 Table 2. Applied charges and induced strains ……………………………………………………... 16 Table 3. Table of material properties used in the model ………………………………….….…… 59 Table 4. Frequency of vibration from MathCAD …………………………………………………... 59 Table 5. Experimental and analytical bending mode frequencies ………………………….…….. 61 Table 6. Theoretical tuning ranges from analytical model …………………………………......…. 71 Table 7. Comparison of tuning range from SigLab and analytical data …………………….....… 72 Table 8. Resonator PS VIII inverter driven tuning and operating frequencies in kHz ……...… 108 Table 9. Resonator PS XII inverter driven tuning and operating frequencies in kHz ……........ 113 Table 10. Resonator PS XIII inverter driven tuning and operating frequencies in kHz ….…… 119 Table 11. Resonator PS XIV inverter driven tuning and operating frequencies in kHz …….… 123 Table 12. Resonator PS XV inverter driven tuning and operating frequencies in kHz …….…. 129 Table 13. Resonator PS XVI inverter driven tuning and operating frequencies in kHz, run 1 .. 135 Table 14. Resonator PS XVI inverter driven tuning and operating frequencies in kHz, run 2 .. 136 Table 15. Comparison of tuning range from Pierce circuit operation and analytical data …..... 147 Table 16. Comparison of tuning range from operation with circuit with separate gain and phase adjustment, and analytical data ……………………………………………………..…… 148 ix LIST OF FIGURES Figure 1. Simplified quartz crystal circuit diagram …….…………………………………………... 5 Figure 2. Butterworth-VanDyke equivalent circuit for crystal resonators ……………………….. 6 Figure 3. Simplified piezoelectric resonator circuit …………………………………………………. 6 Figure 4. Crystal with crystallographic axes indicated ……………………………………….…… 11 Figure 5. Q Factor ………………………………………………………………………………….…... 30 Figure 6. Examples of shear, flexure and extensional vibration modes …………………….……. 33 Figure 7. The basic electrical resonant circuit …………………………………………..………....… 35 Figure 8. Microstrip Line Oscillator - Ring Type ………………………………...……….…....…… 40 Figure 9. Plan view of the layout of the resonator ………..………………………………...……… 45 Figure 10. Side view of the layout of the resonator …………………………………………….…... 46 Figure 11. Width transformation in calculation of the centroid ……………………………..…….. 49 Figure 12. Frequency response measurement experimental setup ……………………………..…. 57 Figure 13. First five displacement mode shapes
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