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" STUDY OF THE VISCOELASTIC PROPERTIES OF LIQUIDS " " AT MICROWAVE FREQUENCIES " A thesis submitted for the degree of Doctor of Philosophy, by Herb Joseph John Seguin Electrical Engineering Department, Imperial College of Science and Technology University of London November,1963. 2 ABSTRACT The development of an " S " band system for use in viscoelastic relaxation investigations in liquids at high frequencies is discussed. The experimental microwave ultrasonic system ,operating at a frequency of 3000 meagacycles per second , employed piezo- electric single quartz crystals,operating in a nonresonant, surface excitation mode, as the electromechanical transducers. The real and imaginary components of the shear mechanical impedance,of various test liquids , were obtained from measure- ments of the relative amplitude and phase change of an acoustic decay pattern,initiated within the quartz crystal by pulse excitation, upon application of the test liquid to the crystal surfaces. Theoretical and practical measurement accuracies of the apparatus are investigated, along with the limitations of the system as a tool in the studies of the fundamentals of the liquid state. Possible avenues of improvement are suggested as are possible extensions of the basic system design to the higher microwave frequency spectrum. 3 ACKNOWLEDGMENTS The author wishes to express deepest thanks and appreciation to Professor J. Lamb and Dr. J. Barlow for the original presentation of the project and for continued guidsnce,assistance and encouragement, throughout this work. The author is also indebted to Mr. J. Collins for continued interest and assistance in the design and development of the experimental system . Sincere appreciation is expressed to Mr.J. Ritcher for help in the theoretical acoustic aspects of the project. A vote of thanks is given to Mr.R. Hutchins for help and encouragement in the construction of the experimental system. The generosity and kind assistance of the engineering staff of Ferranti Ltd. of Edinburgh, Mircowave Division, is also acknowledged. Special thanks is also given to Mr. R. Watters for assistance in the preparation of this thesis. Appreciation is also expressed to the Athlone Fellowship Managing Committee and the National Research Council of Canada for providing financial support during the execution of this research project, 4 CONTENTS ITEM PAGE ABSTRACT 2 ACKNOWLEDGEMENTS 3 CONTENTS 4 UNITS AND SYMBOLS 12 PART I 17 FIELD OF STUDY CHAPTER 1 " INTRODUCTION " 18 1;1 Historical 18 1;2 Present work 21 1;3 Presentation 22 CHAPTER 2 " VISCOELASTIC THEORY " 23 2;1 General 23 2;2 Viscoelastic models 24 2;3 Maxwell liquid 24 2;4 Shear wave propagation in Maxwell viscoelastic 30 media. 2;5 Complex rigidity modulus G 36 2;6 Reduction principle 39 2;7 Distribution of relaxation times 43 2;8 Derivation of distribution function 47 PART II 50 EXPERIMENTAL SYSTEM CHAPTER 3 " DESCRIPTION OF APPARATUS 51 3;1 Introduction -51 3;2 General description of apparatus 51 3;3 General description of system components and 56 operation. 5 ITEM PAGE CHAPTER 4 " DETAILED ANALYSIS OF INDIVIDUAL COMPONENTS IN 65 TRANSMITTER UNIT. 4;1 General 65 4;2 Oscilloscope 65 4;3 Low level pulse generator 66 4;4 High level pulse generator 70 4;5 High power modulator 81 4;6 Magnetron and associated equipment - 86 86 4;6;1 Magnetron 4;6;2 Constant current electromagnet power supply 87 90 4;6;3 Magnetron temperature controller 4;7 Launching section 90 4;8 Waveguide twist section 93 4;9 Precision power level setting attenuator 93 4;10 Magnetron mean power monitor 93 4;11 H plane bend 96 4;12 Ferrite resonance isolator 96 4;13 Magnetron frequency and pulse shape monitor loo 4;13;1 General description loo 4;13;2 Crossed directional coupler lo2 4;13;3 Broad band crystal mount lo3 4;13;4 Temperature stabilised wavemeter 1o4 CHAPTER 5 " DETAILED ANALYSIS OF INDIVIDUAL COMPONENTS IN 107 BRIDGE UNIT. 5;1 General 107 5;2 Bridge magic T l07 5;3 Bridge reference channel. llo 5;3;1 General llo 6 ITEM PAGE 5;3;2 Precision level setting attenuator and phase llo shifter. 5;3;3 Squeeze section phase shifter. 115 5;3;4 Waveguide to coaxial line transformer. 12o 5;4 Bridge test channel 121 5;4;1 General 121 5;4;2 Precision four screw tuner._ 122 5;4;3 Precision rotory attenuator. 122 5;4;4 Precision level setting phase shifter. 127 5;4;5 Precision measurement phase shifter. 127 5;4;6 Waveguide to coaxial line transformer. 127 CHAPTER 6 " DETAILED ANALYSIS OF INDIVIDUAL COMPONENTS IN 129 RECEIVER SYSTEM. 6;1 General 129 6;2 H plane bend 129 6;3 Double directional coupler. 129 6;4 T.R. cell. 131 6;5 Waveguide to coaxial line transformer. 132 6;6 Parametric amplifier system. 132 6;6;1 General 132 6;6;2 Parametric amplifier signal input and output 133 circuit. 6;6;3 Basic parametric amplifier unit. 135 6;6;4 Parametric amplifier pump circuit. 138 6;7;1 R.F. varactor switch system. 139 6;7;2 Varactor swtich. 14o 6.7;3 Varactor switch pulses. unit. 141 6;8 Coaxial line to waveguide transformer. 144 6;9 Parametric amplifier performance and alignment 144 monitor. 7 ITEM PAGE 6;10 Ferrite resonance isolator. 145 6;11 Balanced mixer system. 145 6;11;1 General 145 6;11;2 Balanced mixer magic T. 147 6;11;3 Local oscillator channel. 148 6;11;4 Mixer channels. 148 6;12 Intermediate frequency amplifiers. 149 6;12;1 Head amplifier. 149 6;12;2 Crystal current monitor. 15o 6;12;3 Second intermediate frequency amplifier. 152 6;12;4 Third intermediate frequency amplifier. 152 6;13 Receiver system noise figure. 155 6;14 Oscilloscope. 156 CHAPTER 7 " DETAILED DISCUSSION OF INDIVIDUAL COMPONENTS IN 157 MICROWAVE ULTRASONIC TRANSDUCER UNIT. 7;1 General 157 7;2 Resonant microwave cavities. 157 7;2;1 General. 157 7;2;2 Cavity number one. 158 7;2;3 Cavity number two. 16o 7;2;4 Cavity number three. 162 7;2;5 Cavity number four. 165 7;2;6 Cavity number five. 167 7;3 Microwave cavity calculations. 169 7;3;1 General. 169 7;3;2 Coaxial cavity number one. 169 7;3;3 Coaxial cavity number three. 171 7;3;4 Radial cavity number three. 172 7;3;5 Comparison of cavity parameters. 178 8 ITEM PAGE 7;3;6 Modified radial cavity 179 7;4 Quartz crystal transducer. 183 7;4;1 General 183 7;4;2 Choice of crystal cut. 185 7;4;3 Crystal calculations. 188 7;4;4 Acoustic attenuation in quartz. 191 7;5 General microwave ultrasonic transducer 193 calculations. CHAPTER 8 " AUXILIARY ITEMS OF EQUIPMENT." 197 8;1 Thermostating bath. 197 8;2 Noise source. 197 CHAPTER 9 " THEORY AND MATHEMATICS OF MEASUREMENT " 2o2 9;1 General considerations. 2o2 9;2 Typical measurement. 2o7 CHAPTER 10 " SYSTEM ACCURACY CONSIDERATIONS. " 210 10;1 General. 210 10;2 Transmitter considerations. 210 10;3 Microwave bridge considerations. 211 10;4 Receiver system considerations. 212 10;5 Microwave ultrasonic transducer. considerations. 213 10;5;1 Resonant cavity considerations. 213 10;5;2 Quartz crystal considerations. 214 10;5;3 Test liquid considerations. 218 10;6 Discussion. 221 CHAPTER 11 " RESULTS " 223 11;1 Numerical results. 228 9 ITEM PAGE 11;2 Discussion. 23o PART III 233 EXPERIMENTAL SYSTEM THEORY APPENDIX A " SUMMARY OF VECTOR IDENTITIES " 234 APPENDIX B " ELECTROMAGNETIC WAVES " 236 B;1 Field vector symbols and units. 236 B;2 Electromagnetic equations of state. 237 B;3 The wave equation. 237 B;4 Solutions to the wave equation. 238 B;4;1 T.E.M. modes in coaxial cylinders. 238 B;4;2 Propagation in rectangular waveguides. 24o B;4;3 Propagation in circular waveguides. 242 B;4;4 Propagation in nonuniform radial waveguides. 245 APPENDIX C " GENERAL TRANSMISSION LINE EQUATIONS." 25o C;1 Coaxial line relations. 25o APPENDIX D tll ELASTIC WAVES " 251 D;1 Stress-strain components. 251 D;2 General equations of motion in anisotropic media. 258 D;3 Energy flow in anisotropic elastic materials. 263 D;4 Elastic wave propagation in piezoelectric media. 265 D;5 Elastic constants for rotated axes. 271 D;6 Elastic constants for quartz with rotated Y cut. 272 D;7 Reflection and refraction of elastic waves. 273 D;8 Excitation of Elastic waves. 282 D;9 Electromechanical conversion efficiency. 286 10 ITEM PAGE APPENDIX E " RESOPANCE ISOLATOR THEORY " 290 E;1 Magnetodynamic equations. 290 E;2 Precessional motion. 291 E;3 Ferrite equations of motion. 293 E;4 Permeability tensor. 294 E;5 Demagnetisation factors and resonance fields. 296 E;6 Waveguide circular polarisation. 298 E;7 Practical considerations. 300 APPENDIX F " MICROWAVE RESONANT CAVITY THEORY " 3o1 F;1 Coaxial cavity. 310 F;1;1 Cavity dimensions and frequency. 3o2 F;1;2 Cavity Q 303 F;1;3 Cavity power level. 307 F;1;4 Cavity fields. 308 F;1;5 Cavity efficiency. 317 F;2 Radial Cavity. 32o F;2;1 Cavity dimensions and frequency. 32o F;2;2 Cavity fields. 33o F;2;3 Cavity Q 334 F;2;4 Cavity efficiency. 338 APPENDIX G. PARAMETRIC AMPLIFIER THEORY 342 G;1 General considerations. 342 G;2 Power relations. 346 G;3 Gain considerations. 348 G;4 Amplifier bandwidth. 357 G;5 Noise figure of parametric amplifier. 359 G;6 Parametric amplifier alignment procedure. 365 11 ITEM PAGE APPENDIX H " NOISE FIGURE CONSIDERATIONS " 368 H;1 General noise formulae. 368 H;2 Measurement of receiver noise figure. 371 REFERENCES 374 12 UNITS AND SYMBOLS The centimeter-gram-second system of units has been adopted in chapter 2 , which deals with the viscoelastic aspect of this work, in order to be consistant with the system of units normally found in the literature.
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