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University Mtcrotilms International A Bell & Howell Information Company 300 North Zeeb Road Ann Arbor Ml 48106-1346 USA 313/761 4700 800/521-0600 O rder N um ber 8914089 A n a d v an c ed stu d y o f m icrow ave sin te rin g Watters, David Gray, Ph.D. Northwestern University, 1989 C opyright © 1988 by W atters, D avid Gray. A ll rights reserved. UMI 300 N. Zoeb Rd. Ann Arbor, Ml 48106 NORTHWESTERN UNIVERSITY AM ADVAMCKO STUDY OF MICROWAVE SIMTKRINO A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IH PARTIAL FULFILLMENT OF THE REQUIREMENTS for tha dtgrtt DOCTOR OF PHILOSOPHY Flald of Elactrlcal Englnaarlng By David Gray Wattars BVAMSTON, ILLINOIS JUNE 1989 O Copyright by David Gray Watters 1988 All Rights Reserved 11 ABSXBASX An Advanced Study of Mlcrwivt Sintering David Gray Matters An Investigation of microwave sintering of ceramics reveals that under some conditions, anomalous results, Including core melting and thermal "runaway" can occur. This research explains these phenomena using theoretical models of the electromagnetic heating process. The most lmpostant variable Is temperature dependent electrical conductivity which can Increase by several orders of magnitude during sintering. A portion of this work Is devoted to material characterization as a function of temperature. The theoretical analysis Is divided Into two cases, electrically thick and electrically thin bodies. A seml- lnflnlte space Is used as the model for thick samples. Transient analysis of the heat and wave equations, using an explicit numerical method, reveals that low thermal conductivity can cause subetantlal thermal gradients resulting In core melting. heating: There are three regions of Initially, low loss materials heat slowly; as conductivity rises with temperature, the heat generation ill rate accelerates producing a rapid temperature rise; at high temperatures, a "saturation” effect occurs, due to decreasing electrical skin depth, Increasing surface reflection and Increasing thermal radiation. Solution of the steady state equations Indicates a monotonlc relationship between applied power and steady state temperature. The fields are assumed uniform In electrically thin bodies, and a planar slab Is used as the model. Numerical analysis of transient heating reveals the possibility of thermal Instability. Non-dimensional analysis of the steady state heat equation using regular perturbation theory Indicates that there is a maximum stable steady state temperature. An equation Is derived relating applied power to steady state temperature. Above a critical applied power level heating Is unstable. The Inverse of this equation yields a formula for electrical conductivity as a function of applied power and steady state temperature. Since both power and temperature are measurable, material characterization Is possible. Material characterization at room temperature is performed using an automatic scalar network analyzer. There Is an optimum sample thickness, based on electrical conductivity, dielectric constant and frequency. Lossy samples must be electroplated to remove gaps between sample and sample holder. lv ASggfflilBgBgBXg I want to taka this opportunity to acknowledge the following people for their contribution to this effort. First, I want to express ay gratitude and sincere appreciation to ay advisers, Dr. Morris B. Brodwln, for help In defining the alcrowave problea and Dr. Gregory A. Kriegsaann, for his aatheaatical Insight. I also want to thank the aeabers of the coaalttee, Dr. Lynn Johnson, Dr. Carl Kannewurf and Dr. Allen Taflove for their tlae and Interest In this work. This work has been supported by the Materials Research Center at Northwestern under Grant DMR85-20280. Acknowledgement Is due Mr. Karla Mattar for his assistance in characterization experiaents and Dr. Y.L. Tlan for his assistance In saaple preparation and sintering experiaents. Finally, a special thanks to ay parents, W. Gray and Marilyn Hatters for their encourageaent and patient support throughout ay studies at Northwestern. David Gray Watters, 1988 Psalas 115:1 v TABLE OB CONTENTS Abstract........................................ Ill Preface.....................................................v List of Figures................ xl I. Introduc t Ion........................................... 1 II. Lltsratura Review..................................... 7 III. Theoretical Analysis................................ 14 A. General Theory................................... 14 1. Electromagnetic Aspects..................... IS 2. Thermal Aspects..............................23 3. Sintering Dynamics...........................24 4. B. Case Study................................... 28 Theoretical Heating of a Semi-Inflnlte Space 30 1. Constant Conductivity....................... 30 a. Transient Solution..................... 35 b. Steady State Solution.................. 37 c. Steady State Results................... 39 d. Transient Results...................... 42 2. Temperature DependentConductivity........... 51 a. b. C. Transient Solution..................... 52 Steady State Solution.................. 60 Theoretical Heating of a Planar Slab............. 64 vl 1. Constant Conductivity........................67 a. Transiant Solution..................... 67 b. Steady atata Solution..................71 2. Temperature Dependent Conductivity.......... 72 a. Transient Solution..................... 73 b. Steady state Solution and the Onset of Instability......................... 77 1. Shooting Method Solution..........78 11. Approximate Analytical Solution...80 111. Stability Analysis................90 3. Steady State Solution for an Insulated Planar Slab................................. 93 4. D. Steady State Solution for a Thick Slab...... 99 Theoretical Heating of an Infinite Cylinder 103 1. Transient Solution..........................105 2. Steady State Solution for Constant Conductivity............................... 105 3. Steady State Solution for Temperature Dependent Conductivity..................... 106 4. Steady State Solution for a Tube........... 108 5. Steady State Solution for a Thick Cylinder................................... 109 E. 7. Theoretical Heating of a Sphere................. Ill 1. Steady State Solution...................... Ill 2. Fields In a Sphere......................... 114 Generalized Stability Formula For Thin vii Electrical Bodies............................... 115 G. H. IV. Parameter Study................................. 118 1. Constant Conductivity......................119 2. Linear Conductivity........................119 3. Quadratic Conductivity.....................120 4. Exponential Conductivity.................. 123 5. Other Exponential Forms................... 125 6. Summary.................................... 129 "Hot wall" Applicator...........................131 1. Half Space Solution........................132 2. Planar Slab Solution...................... 134 Applicator Selection................................ 139 A. B. Experimental Apparatus..........................139 1. Slab Applicator............................141 2. Rectangular Applicator.................... 143 3. Circular Cavity Applicators............... 149 Thermal Stability in a Rectangular Cavity Applicator...................................... 152 V. Material Characterization............................. 175 A. Automatic Material Characterization Equipment... 175 1. 2. The Apparatus..............................176 a. Theory................................ 179 b. Results............................... 180 Measurement Range and Sources of Error vlll 185 a. Limitation* Based on Maxwell's Equations..... b. Gap Effects........................... 198 1. Capacitance Model................201 11. Transverse ResonanceModel........209 111. lv. c. B. 185 Techniques for Removing Gaps 214 Experimental Results.............217 Component Imperfections...............224 High Temperature Techniques.................... 232 1. High Temperature AMCE..................... 232 2. Other High Temperature Methods............ 233 3. In Situ Characterization...................234 4. A New Method for High Temperature Characterization........................... 235 VI. Conclusion and Recommendations for Future Nork Notes and References............ 246 249 APPENDIX A: Program for Computing Transient Temperature Profiles In a Semi-Infinite Space with Constant Conductivity.............257 APPENDIX B: Program for Computing Transient Temperature Profiles In Semi-Inflnlte Space with Temperature Dependent Conductivity...........259 APPENDIX C: Non-Dimensional Analysis.................... 263 lx APPENDIX D: Damped Haraonlc Oscillator...................266 APPENDIX E: Thermal Equilibrium of Semi-Infinite solid...269 APPENDIX F: Program for Computing Transient Temperature Profiles in a Planer Slab.................... 272 APPENDIX Q : Program for Computing Steady State Temperature Profiles in a Planar Slab Using a Numerical "Shooting" Method................ 274 APPENDIX H: Modes in a Dielectrically Loaded, Circular Cavity....................................... 277 APPENDIX I: Program to Study Gap Effects Using the Method of Transverse Resonance...............287 APPENDIX J: Sample Resonance in Coaxial Line............ 293 K: Electroplating Process...................... 295 Vita...................................................... 298 X LIST OF FIGURES 1. Temperature history of alumina sample during sintering. Adapted from [1].......................... 16 2. a) loss tangent and b) dielectric constant of alumina as a function of temperature at 3.5 GHz. Adapted from [32]..................................... 20 3. Percent absorbed power vs. conductivity for a planar slab of thickness d ■ 0.95 cn, f ■ 2.45 GHz, €r ■ 10................................................ 22 4. Thermal conductivity vs. temperature for various ceramics. Adapted from [38]........................... 25 5. Heat capacity vs. temperature for various ceramics. Adapted from [38].......................... 26 6. Ceramic particles In contact, showing a grain boundary. Adapted from [40].......................... 27 7. Geometry of planar half space......................... 31 8. Percent power absorbed vs. conductivity for a planar half space, f * 2.45 GHz, €r ■ 10...............34 9. Transient temperature profiles In a planar half space; o ■ .2 S/m, P - 50 W/cmz, a)500 sec, b)1500 sec, c)steady state........................................ 44 10. Transient temperature profiles In a planar half space; o “ .002 S/m, P ■ 5000 W/cmz, a) 100 sec b)400 sec........................................ 45 11. Transient temperature profiles in a planar half space; o ■ .02 S/m, P * 500 W/cmz, a)100 sec b )400 sec............................................ 46 12. Transient temperature profiles in a planar half space; o ■ .2 S/m, P ■ 100 W/cm2, a)100 sec, b)200 sec, c )300 sec, d)400 sec...................... 47 13. Transient temperature profiles in a planar half space; o ■ 2 S/m, P ■ 50 W/cmz, a)100 sec b)200 sec, c)292sec, d)1256 sec...................... 48 14. Transient temperature profiles in a planar half space; a)solid, o ■ .2 S/m, k ■ 20 W/m-K, P ■ ✓ xi 100 W/cm2, t * 400 sec, b)dash, o * .2 S/m, k ■ 20 W/m-K, P - 200 W/cm2, t ■ 200 sec, c)dot, o ■ .2 S/», k ■ 10 W/a-K, P » 100 W/cm2, t ■ 400 sec.............................................. 50 15. Transient temperature profiles In a planar half space with temperature dependent conductivity, a • .002 exp(.0026T), P ■ 500 W/cm2, a)1000 sec, b)1500 sec, c)1700 sec, d)1800 sec, e)1900 sec.................................................. 55 16. Surface temperature vs. time for Transient heating of a planar half space with temperature dependent conductivity, o ■ .002exp(.0026T), P - 1000 W/cm2......................................... 57 17. Transient conductivity profile for planar half space after 1900 sec, o - .002exp(.0026T), P - 1000 W/cm2......................................... 58 18. Steady state surface temperature vs. Incident power (W/cm2) for planar half space with temperature dependent conductivity, o * .002exp( .0026T)......... 63 19. Geometry of planar slab...............................65 20. Steady state temperature profiles for a 6mm planar slab, with a).002 S/m, b).02 S/m and c).2 S/m. Power Is 500 W/cm2....... :.......................... 69 21. Steady state temperature profiles for a 6mm planar slab, with a)2 S/a, b)20 S/m and c)200 S/m. Power is 500 W/cm2................................... 70 22. Surface temperature vs. time for transient heating of planar slab with temperature dependent conductivity, o ■ .002exp(.0026T), a) 100 W/cm2, b) 150 W/cm2 c)200 W/cm2................74 23. Heating of sintered alumina sample, surface temperature vs. time................................. 75 24. Normalized steady state surface temperature vs. normalized applied power for planar slab, with f(v) • exp(.78v), c, ■ .002, cz ■ .00018, using a numerical "shooting method.".................81 25. Normalized steady state temperature profile In a planar slab, \ - .005, f(v) « exp(.78v), ct xll .0 0 2 , a n d Cj ■ . 0 0 0 1 8 ..........................................................................................8 5 26. Noraallzad staady atate surface taaparatura vs. noraallzad appllad power for planar slab, with f(v) ■ axp(.78v), c, - .002, c2 ■ .00018, using an approxlaata analytical solution............ 86 27. Staady state surface taaparatura vs. appllad power; un-noraallzed from Fig. 25.......................... 88 28. Noraallzad staady state surface taaparatura vs. noraallzad appllad power for planar slab, with f(v) ■ exp(.78v), c, ■ .002, c2 ■ .00018, coaparlng a)solld, approxlaata analytical solution and b)dash, nuaerlcal "shooting" aathod............................................... 89 29. Geoaetry of Insulated planar slab.................... 94 30. Staady state taaparatura profile for an Insulated slab....................................... 98 31. Noraallzad staady state surface taaparatura vs. noraallzad appllad power for Insulated planar slab, with f(v) ■ exp(.78v), c, - .002, c2 ■ .00018 and a ■ 0.1, a) solid, uninsulated slab and b)dash. Insulated slab.............................. 100 32. Geoaetry of Infinite cylinder........................104 33. Noraallzad steady state surface taaparatura vs. noraallzad applied power for planar slab, with f(v) ■ 1 + .3V2, a)a ■ .002, b)<* ■ .003, c )ot - .004...........................................122 34. Noraallzad staady state surface taaparatura vs. noraallzad appllad power for planar slab, with f(v) ■ exp(.78v), a)c, > 0 , c2 ■ .00018, b)c, ■ .002, c2 ■ .00018, c)c, ■ .002, c2 ■ 0 ...............................................124 35. Noraallzad steady atate surface taaparatura vs. noraallzad applied power for planar slab, c, ■ .002, cz - .00018, a)k -0.68, b)k - 0.78, c)k - 0.88...............................................126 36. Noraallzad staady state surface taaparatura vs. noraallzad appllad power for planar slab, with f(v) ■ 1 ♦ 600 exp(-13.3/v), c, ■ .002, c2 ■ .00018...............................................128 xlll 37. Normalized staady stats surface tsapsrature vs. normalized applied power for planar slab, with f(v ) - 1 378.5/V exp(-4.67/v), c, - .002, C, - .00018......................................... 130 38. Transient hot wall heating of planar half space with no microwave radiation, Taat ■ 1300K, a)100 sec, b)400 sec................................ 133 39. Surface temperature vs. time for hot wall heating of planar half space, P ■ 500 W/cm*, o - .002 exp(.0026T), a)T, t - 300K, b)T„t - 1300K........... ."........................... 135 40. Surface temperature vs. time for hot wall heating of planar slab, Ta>1 * 1300K, o ■ .002 exp(.0026T) a)P - 0, b)P - 100 W/cm*, c)P - 500 W/cm*......................................... 136 41. Temperature profiles In hot wall heating ofan 8mm planar slab, o ■ .002 exp(.0026T), k ■ 5 W/mK, P - 500 W/cm*, T „ t - 1300K, a)40 sec, b)140 sec, c)200‘sec, d)240 sec................ 138 42. Block diagram of microwave sintering apparatus...... 140 43. Slab applicator geometry.............................142 44. Rectangular applicator geometry..................... 144 45. T equivalent network for cylindrical rod............ 146 46. Equivalent circuit for rectangular applicator....... 148 47. Propagation constant for sample-filled TEllt circular waveguide, where b/a Is the ratio of the dielectric radius to the cavity radius. Adapted from [69]................................... 151 48. TE1(1 applicator geometry.............................154 49. Equivalent circuit of cylindrical rod using Marcuvltz model..................................... 157 50. Variation of rod Impedance, Ra, with conductivity........................................ 158 51. Variation of rod Impedance, Xa, with conductivity........................................ 159 xlv 52. Variation of rod Impedance, R*, with conductivity........................................ 160 53. Variation of rod Impedance, , with conductivity........................................ 161 54. Simplified circuit for low lose, thin rods..........162 55. Cavity equivalent circuit........................... 164 56. Normalized steady state surface temperature vs. normalized applied power for cylindrical rod In rectangular applicator with f(v) - exp(.78v), c, ■ .002, cz ■ .00018, a) critical coupling at 300K, b) free space cylinder..................... 168 57. Normalized steady state surface temperature vs. normalized applied power for cylindrical rod In rectangular applicator with f(v) - exp(.78v), c, ■ .002, cz - .00018, a) critical coupling at 2300K, b) free space cylinder.................... 170 58. Normalized steady state surface temperature vs. normalized applied power for cylindrical rod In rectangular applicator with f(v) - exp(.78v), c, ■ .002, cz ■ .00018, a) critical coupling at 1300K, b) free space cylinder.................... 171 59. Percent power absorbed vs. normalized temperature for different coupling arragements, a)solld, 300K, b)dash, 1300K, c)dot, 2300K..........173 60. Block diagram of Automatic Material Characterization Equipment (AMCE)................... 177 61. AMCE Sample holder, showing a) end view and b) cross-sectional view of the modified APC-7 connector........................................... 178 62. Reflection and transmission vs. frequency for Stycast H1K, 0 - Is reflection data, □ - Is transmission data................................... 182 63. Reflection and transmission vs. frequency for silicon, o - 0.017 S/cm, 0 - Is reflection data, □ - Is transmission data............................ 184 64. Sum of magnitudes and sum of squares of magnitudes of reflection and transmission vs. conductivity........................................ 189 XV 65. Reflection coefficient vs. dielectric conetent for fixed frequency and conductivity................192 66. Transalsslon ve. conductivity for a) d ■ 4.56aa, b) d ■ 0.3 a a ....................................... 194 67. Reflection and tranaalsalon va. frequency for a)o ■ .01 S/ca, b)o ■ .1 S/ca....................... 196 68. Obaerved dielectric conetant va. true dielectric constant for an air gap of a).l ua, b)l urn, c)10tia, d )100 ua.................................... 200 69. Geoaetry of the gap................................. 202 70. Equivalent circuit for the gap using a distributed capacitance aodel....................... 204 71. Observed conductivity vs. true conductivity for an air gap of a).01 ua, b).l ua, c)l ua. d)10 ua, e) 100 ua........................................ 206 72. Observed dielectric constant vs. frequency for an air gap of a).01 ua, b).l ua, c)l ua, d)10 Ua...................................................207 73. Observed conductivity vs. frequency for an air gap of R/a).l ua, b)l ua, c)10 ua, d)100 ua..........208 74. Equivalent network for Transverse Resonance Model (TRM)......................................... 210 75. Observed conductivity vs. true conductivity for a) TRM aodel, b)capacitance aodel................... 215 76. Observed conductivity vs. frequency for a)TRM aodel, b) capacitance aodel......................... 216 77. The effect of air gaps on the aeasureaent of Stycast H1K; a) dash, aeasureaent with gap, b) solid, aeasureaent with filled gap.................. 220 78. The effect of air gaps on the aeasureaent of .017 S/ca silicon; a) dash, aeasureaent with gap, b) solid, aeasureaent with filled gap...............221 79. Transalsslon vs. frequency for high loss (ooc ■ .28 S/ca) silicon using an electroplated saaple. Figure shows theoretical curves for .2, .25 and .3 S/ca............................................. 223 xvi 80. Signal flow graph raprasantation of AMCE............ 225 81. Block dlagraa of In altu charactarizatIon aathod using a tranaait/racaiva tuba....................... 236 82. Axial taaparatura proflias in a alnterad alualna saapla for various powar levels..................... 238 83. Axial taaparatura profiles in a sintered alualna-TlC (30%) saapla for various power levels...239 84. Percent error in conductivity aeasureaent vs. noraallzad taaparatura for a 5% error in c,......... 242 85. Percent error in conductivity measurement vs. noraallzad taaparatura for a 5% error in c,......... 243 86. Percent error in conductivity measurement vs. noraallzad teaperature for a 5% error in v .......... 244 87. Geoaetry of sample in TEMl xv 11 cavity.................. 278 CHAPT1R I INTRODUCTION Microwave sintering Is a valuable aethod for processing ceramic materials. It produces samples with Improved microstructure [1] at high processing rates, with greater energy efficiency than conventional sintering techniques. Prior work at Northwestern has demonstrated the feasibility of sintering a variety of materials; [2] yet In some cases, anomalous results are observed, Including core melting, localized "hot spots" and thermal "runaway." [3] This study has been undertaken to further our understanding of the sintering process. The objectives of this work are: to develop theoretical models to predict temperature profiles and stability In the microwave sintering of materials of various geometries; and to describe material characterization techniques to determine how sample properties change during sintering. The results of this work apply to electromagnetic heating in general, but we concentrate on microwave sintering. In Chapter II, the microwave sintering literature Is reviewed. [4-20] He survey the current progress In the heating of ceramics with a particular emphasis on the problem of thermal runaway. In Chapter III, we describe the general theoretical 1 2 problea of altctroMgnttic hosting. Wo discuss tho toaporature dopondonco of aaterlal paraastors such as oloctrlcal conductivity; and tho relationship between absorbed power and conductivity. We then consider the solution of the heat and wave equations In soae slaple geoaetrlea. An explicit nuaerlcal solution Is used to solve the transient heat equation. The steady state solution Is used to deteralne tne stability of the heating process. The slaplest geoaetry is the seal-lnflnlte space.[21] We show the laportance of theraal conductivity In deteralnlng thereal gradients. For aaterlals with teaperature dependent electrical conductivity, transient analysis Indicates three regions of heating; a) at low teaperatures the saaples heat slowly; b) at lnteraedlate teaperatures the saaples heat rapidly, due to increased heat generation; c) at high teaperatures there Is a "teaperature saturation" caused by decreased field penetration. In the seal-infinite slab there is a aonotonlc relationship between absorbed power and steady state teaperature, [22] and furtheraore, the solution is stable even when the electrical conductivity Increases with teaperature.[23,24] The electrically thin, planar slab geoaetry Is considered next. Nuaerlcal analysis of the tlae dependent heat equation, when electrical conductivity Increases rapidly with teaperature, Indicates the possibility of Instability. Analysis of the steady state equation using a 3 nuaerlcal "shooting" asthod rsvsals the prsssncs of Instability when ths Incident power exceeds a critical value. A perturbation analysis of the steady state solution yields a relationship between Incident power and steady state teaperature.[29] Solution of the Inverse problea gives electrical conductivity as a function of Incident power and steady state teaperature. Since surface teaperature and power are aeasurable values, In situ characterization of electrical conductivity Is possible. In the slab geoaetry, we consider two special cases: the perturbation solution Is applied to an Insulated saaple to reveal stability lnforaatlon, and the solution for the thin slab Is extended to Include slabs whose thickness Is on the order of a wavelength. The Infinitely long, thin, cylinder Is described, Indicating slallar stability behavior to the planar slab. The saae Inverse solution applies.[26] The special case of a tubular saaple Is considered, as well as that of the thick cylinder. When this analysis Is applied to spherical geoaetry the saae solution Is found. The relationship between absorbed power and teaperature derived for the slab holds for cylindrical and spherical eaaples ae well. A correction factor Is Introduced based on the surface area to voluae ratio of the saaple. A general expression Is derived for the theraal stability of electrically thin bodies. 4 A parameter atudy Is performed to determine how electrical and thermal properties affect thermal stability. If electrical conductivity Increases at a linear rate or less, heating is stable. For conductivities that vary quadratlcally with temperature, there Is low temperature and high temperature stability, but an Intermediate range of temperatures that are unstable. Exponential conductivity variation, which models many ceramic materials, experiences stable heating up to a maximum steady state teaperature. This maximum value can be controlled by changing the heat transfer conditions, or by doping the saaple to change the teaperature dependence of electrical conductivity. A nuaerlcal study of the "hot wall" applicator reveals enhancement of low teaperature heating rates up to the wall teaperature. In Chapter IV, the experimental apparatus for microwave sintering is discussed. Various applicators are reviewed, Including slab, rectangular and circular, both TM0I0 and TEllt . He describe the fields In the circular applicator and show how the Impedance analysis [18] for the rectangular applicator can be simplified to a cavity equivalent circuit for thin low loss samples. The thermal stability analysis Is applied to cylindrical rods In the rectangular cavity. During sintering, proper control of cavity parameters (coupling and tuning) Is essential to maintain stable heating. Chapter V daals with tha problem of Material characterization. He first deecrlbe an automatic technique using a scalar network analyzer to determine the permittivity and conductivity as a function of frequency and temperature.[27,28] Uncertainties In the measurement are discussed, Including high and low loss limits, gap effects [29] and device Imperfections. From Maxwell's equations we show that when conduction current dominates, changss In displacement current (dielectric constant) are difficult to measure; and when displacement current dominates, changes In conduction current (conductivity) are obscured. Small gaps between the sample and the walls of the sample holder produce noticeable errors. We use a distributed capacitance model and a transverse resonance analysis to study the effect of the gap, and conclude that In order to obtain accurate measurements, all gaps must be eliminated. We describe a method for electroplating samples to eliminate gaps. Multiple reflections between components produces measurement errors. A signal flow graph analysis Is used to determine the most significant errors. Some of these errors are systematic and can be removed through analysis of the data. A technique is needed to measure sample properties at high temperatures. We review several existing methods and 6 propose an in situ ssthod based on prior theoretical Insights. He conclude that a sea11 spherical saaple say be aost useful because the teaperature profile is likely to be unifora. The conclusion addresses areas of future work. In the theoretical area, this Includes expansion of the analysis to Include skin effects In the slab, and sintering dynaalcs. Bxperlaentally, an apparatus to perfora high teaperature characterization should be lapleaented. CHAPTER II LITERATURE REVIEW In this chapter the microwave sintering literature is reviewed. He first survey some of the fundamental works, where the process la developed, and then focus on the problem of thermal runaway. Microwave heating In general and sintering In particular can be performed In a variety of applicators, both traveling wave and cavity type. Thermal runaway effects are observed when the electrical losses Increase rapidly with temperature. Surface temperature feedback Is used to control the Input power and prevent runaway. Numerical analysis reveals that In some cases the system becomes unstable. The first known reference to microwave sintering of ceramics Is [4]. Bertaud and Badot have examined the sintering of refractories In a single mode cavity. They use an Iris coupled rectangular applicator In the TE10, mode at 2450 MHz. The saaple, a long rod, Is translated vertically, parallel to the electric field vector. is observed In the central region of the rod. Heating Monitoring equipment Includes an IR pyrometer to measure surface teaperature and a detector that is magnetically coupled to the sidewall of the applicator to measure the resonant behavior of the cavity. They report a cavity Q of 3000 for 7 * cold alumina rod. Upon firing, tha Q raducas to approxlaataly 200 at 1000*C. Thla lndlcataa an lncraaaa In alactrlcal conductivity aa tha taaparatura rlaaa. Thay raqulra high power for Initial haatlng (400 W ) , but only 100 W to maintain alntarlng taaparatura. application of 400 W cauaad tha aaapla to malt. adjustments wars accomplishad manually. Continual All powar Results show that sintering took place In 10 to 19 minutes at an efficiency of 90*. Mataxas and Meredith [5] discuss tha microwave haatlng problem In detail. Tha text provides an extensive bibliography and dlscusaea the practical electromagnetic aspects of the heating problem, Including applicator selection, heating rates and numerous experimental examples. The focus Is on the drying of materials, nevertheless the Information Is of value for sintering efforts as well. 0 Krage [6] consider the sintering of ferrites in a microwave oven. The study was performed using 1.17 cm x 0.3 cm disk-shaped samplea of barium ferrite. frequency was 2450 MHz. The operating The sample was placed on a turntable within a thermally insulated container. Sintering wae accomplished at a heating rate of 9°C per minute up to a stable temperature of 1230*C. observed. No cracking was Total sintering time was 135 minutes, Including burnout of the binder. Phyelcal strength, percent shrinkage, density and magnetic properties were comparable to conventionally sintered ferrites. Schubrlng [7] reports on the microwave sintering of alumina spark plugs. A conventional (2450 MHz) microwave oven surrounds an Infrared cavity which Is used to contain thermally radiated energy from the sample. The Infrared cavity Is fabricated from millboard and Is transparent to lcrowaves. Temperatures of 1600°C were easily obtained, and feedback control was needed to prevent runaway. The author obtains low quality product and attributes this to a short sintering time. Work at Los Alamos [8] Involves the use of both conventional microwave oven techniques at 2.45 GHz, and a 60 GHz, 200 kW gyrotron source. Initial results presented for alumina using the gyrotron and a TE0Z circular cavity show final densities between 80 and 94* from green densities between 50 and 59*. They have also attempted to sinter composite materials (aluminum with silicon nitride whlskeres, etc.) In a microwave oven. Samples are placed In a thermally insulated enclosure within the oven. Coupling agents such as glycerol are needed to start the heating process. Temperatures near 1700°C have been obtained. Microwave sintering research at Oak Ridge National Laboratory [9,10] involves a 28 GHz, 200 kW gyrotron source and a large, untuned cavity. The higher frequency Is 10 preferred because the loss tangent is higher for many ceramics and consequently, Initial heating Is easier. The untuned cavity Is large compared to the wavelength of the source, and the measured field strength varies by only 4%. The sample is enclosed In a thermally insulating blanket, and a thermocouple Is used to measure sample temperature. Heating rates of 50°C/rain have been obtained for alumina. A comparison of microwave sintering with conventional sintering shows faster processing times and lower sintering temperatures. Tlnga [11] reviews microwave heating and sintering up to 1986. He discusses the runaway problem, noting that feedback can be used to control the heating process. Rapid heating Is attributed to Increasing electrical loss with temperature, and It Is found that the presence of small Impurities significantly affect the loss factor. He reviews a number of popular applicators, Including the ring resonantor, the meander type traveling wave applicator as well as single mode cavities. The use of a mode stirrer is Important In reducing temperature extrema produced by standing waves. Boslslo, et al, [12] address the problem of thermal Instabilities In the heating of Debye materials (ethanol). They discover that measurement of the loss factor becomes difficult above a certain power level and that localized 11 breakdown of tha liquid is tha causa. Thay do not discuss tha problea mathematically, but point out that localized breakdown la enhanced by the low thermal conductivity of the material which tends to sustain localized teaperature increases. An interesting approach to the runaway problem is found in [13], where rapid heating was observed in nylon monofilaments. The loss factor of nylon Increases with teaperature causing acceleration in heat generation. This change in loss factor produces an Impedance change in the applicator, or load, resulting in a change in the power delivered to the sample. If the applicator is properly tuned, the system becomes self-adjusting. That is, as the saaple heats up, the dielectric loss Increases, coupling decreases, and the rate of heat generation is reduced. A stable operating point is established at the desired teaperature by controlling the input power and the tuning of the load. Roussy, et a l , [14,15] use a numerical approach to the thermal runaway problem. cylinder. The geometry is an infinite The heat equation is solved using convective heat transfer at the boundary. They solve the steady state heat equation analytically for the case of constant electrical conductivity. When conductivity Is allowed to vary quadratlcally with teaperature, a numerical solution Is 12 needed. The reeult Is a series of curves defining regions of stable and unstable heating for the case of teaperature dependent conductivity. The meaning of stability Is not well defined and the result depends on the convective heat transfer constant and the polynomial coefficients of the conductivity. Radiation effects are not Included. They use their results to develop an analytical analogue for a control system to monitor the heating of a rubber material. Sekerka, et al, [16,17] runaway are interested In thermal as It affects crystal growth. They analyze the RF heating of an Infinite cylinder using a shooting method. They use an activation temperature model for the electrical conductivity, o ■ o0exp(-TA/T) . Their results show regions of stable and unstable heating. For silicon, a sigmoidal curve Is derived, Indicating a region of lower temperature stability and an upper branch of stable heating, separated by a region of unstable heating. The upper branch for silicon has been observed experimentally. More complex curves are used to describe the melting process and the solld-llquld Interface. To summarize, these papers Indicate that 1) microwave sintering is feasible, offering decided advantages over conventional sintering; 2) alumina is almost universally used as the model material; 3) thermal runaway is attributed to electrical conductivity that increases rapidly with 13 temperature; and 4) runaway aay ba controllad by faadback tachnlquaa If tha ayataa la atable. Our work builds on prior studlas at tha Mlcrowava Tharaal Processing Laboratory at Northwastarn. [2] Aranata, at al, [18,19] hava davalopad an lapadanca analysis for rods In ractangular wavagulda, and have verified tha slnterablllty of a variety of aaterlals Including a and 0alualna, N10 and ZnO. Recant work has Included studlas In single aoda applicators Including, tha slab applicator, the ractangular applicator, tha TEItl [3] circular cavity applicator and tha T M ^ circular cavity applicator. The circular cavities feature alrrored walls to reflect radiated heat back to tha sample, resulting In high sintering temperatures and reduced thermal gradients. Tlan, et al, [1,20] have succeeded in producing very fine grain structure In alumina and have sintered alumlna-TlC. In the investigation to follow, we will perform a theoretical analysis of microwave heating for both electrically thick and electrically thin bodies to describe temperature profiles during heating and to better understand the runaway phenomena. choice for a model material. Alumina is a logical CHAPTER III THEORETICAL ANALYSIS In this chapter the theoretical aapecta of nlcrowave sintering are explored. We describe the general sintering process which Includes a discussion of Maxwell's equations, the heat equation and sintering dynamics. These equations are then applied to some simple geometries to learn more about electromagnetic heating. We first consider the seml-lnfInlte space, considering the cases of constant electrical conductivity and temperature dependent conductivity. A discussion of the planar slab geometry reveals the possibility of temperature Instability, and a relationship Is developed between steady state temperature and applied power. Cylindrical and spherical geometries, and the "hot wall" applicator problem are also considered. A. Qeneral Theory The basic Ideas Involved In microwave sintering can be pictured In the following way. Consider a sample of arbitrary shape Irradiated by microwaves. Loss mechanisms within the sample (such as electrical conductivity) cause the microwave energy to be converted to heat producing a temperature rise In the sample. 14 This generated heat Is redistributed via thermal conduction, and lost to the surroundings via convection and radiation. As the temperature rises, the saaple undergoes physical changes including denslflcation, thermal expansion, etc. In a controlled sintering environment, the temperature will rise rapidly to a stable "sintering temperature" where it remains until sintered, then it is allowed to cool. A typical sintering run is pictured in Pig. 1 [1]. It is Important to note the temperature and temporal dependence of the material properties during sintering. As temperature rises dielectric constant, electrical conductivity, thermal conductivity, etc. change. With time, the density and microstructure of the material change altering the electrical and thermal properties as well. 1. Electromagnetic Aspects The electromagnetic problem reduces to the questions: what are the fields everywhere within the applicator and more specifically within the sample, and what is the total power absorbed by the sample? Por an arbitrary sample, lnhomogeneous and isotropic, Maxwell's Equations are (for a source free region): [30] 7 * H ■ J ♦ (la) 16 (* C ) 3300 PHCHCAT1NC STACC 3000 I STAfiLC SINTTKlNC STAGE transition ST ACE TEMPERATURE I 1300 1000 t t / r ; 'I 300 I I ^ i i i i M TIME H g u r * 1. I. l r a A - l l l l* l^lq 11- kU,*^ 2 1 1 1 • j ilx J (MINUTE) Tsnpsrstur# history o * aluaina aaapl* during •Intaring. Adapted froa [1]. 17 7 x E ■ (lb) 7 •D - 0 (lc) 7 •B - 0 (Id) where D - £E , B ■ uH and £ Is permittivity, ,J ■ oE, u Is permeability and o la conductivity. Since electrical properties change with time as the sample Is heated, these equations cannot In general be simplified. However, temperature change to If we assume the time scale for be much slower than themicrowave period, the electromagnetic problem can be reduced to the sinusoidal steady state over a finite time Interval. (This Is reasonable because electrical effects are measured In nanoseconds, while thermal effects are measured In milliseconds.) In this case, Maxwell's equations become 7 x H - (o +ju£)E (2a) 7 x E ■- jwuH (2b) 7 • (€E) - 0 (2c) 7 • (UH) - 0 (2d) where cj Is the angular frequency. 18 Heating of the sample is Induced by material losses, namely conductive, dielectric and magnetic. In terms of observables, dielectric loss (due to dipole action) and conductive loss (due to collisions) are Indistinguishable. We write the loss terms as follows: (3a) (3b) where £0, the permittivity of free space, Is 8.854 x 10~‘zF/m and n0, the permeability of free space, Is 4ir x 10‘7H/m. In general, these losses vary with temperature. To solve for the fields In the sample, we must solve the non-homogeneous wave equation. [31] 7ZH ♦ <i>2t V H + 7^H • (7 x H) ] x (7t‘) - 0 (4a) For homogeneous samples, this reduces to the familiar wave 19 equation. 7*H ♦ w V u ’H ■ 0 (5a) 72E + w 2€*u'E ■ 0 (5b) Note that materials which are homogeneous at room temperature can become lnhomogeneous astemperature due to the temperature dependence rises of materialparameters and the presence of thermal gradients. In this Investigation, ceramic materials are of particular Interest. They are typically non-magnetlc, and their dielectric properties change with temperature. In Fig. 2, Westphal [32] shows the variation of permittivity and conductivity as a function of temperature for alumina. The dielectric constant Increases slowly with temperature, while the conductivity (proportional to the loss tangent) Increases dramatically with temperature. Ho [33] attributes the Increase in dielectric constant to the Increase In volume, and thus polarlzablllty, due to thermal expansion. In high grade ceramics, loss Is attributed to low quantities of Impurities. The power absorbed by the sample can be computed from the Poynting Theorem, and Is given as a time average of the absorbed power. [34] 20 0.010 0.000 oooo Temperoture (C) <*4o odd 666 " i5 Temperature (C) 71gure 2 a) loss tangent and b) dielectric constant of ainalna ae a function of teaperature at 3.6 QHz. Adapted froa [32]. 21 P.>. “ * / o | E | * d v ♦ ^ V (6) { Ur" IH 12dv *V In an actual system, incident power will be partially absorbed and partially reflected. If we consider a slab sample backed by a short-circuit, the reflection coefficient Is [35]: (1 - jc7)e™ + (1 + J67)e-y« (7) (1 ♦ J O e y‘ ♦ (1 - £7)e"v<’ where d Is the thickness of the slab and Y Is the propagation constant in the sample. The absorbed power, In terms of the Incident power and the reflection coefficient, p, is (8 ) P.*. - (1 - lPl‘)P,Be Figure 3 shows percent absorbed power as a function of conductivity. For low conductivities, there is low absorption since there Is no means of generating heat; the energy is transmitted through the sample. For high conductivities, there Is low absorption since there is high 22 1.00 Power A bsorbed 0.80 0.60 0.40 0.20 0.00 10 '* 10"* 10 Conductivity Figure 3. 1 1 (S /cm ) Percent absorbed power vs. conductivity for s planar slab of thickness d ■ 0.95cs, f - 2.45 GHz, €r ■ 10. 23 reflection from the front eurface and weak field penetration. The conductivity of maximum absorption depends on the thickness of the sample. Note that transverse variations in the electric field (such as occur in rectangular waveguide) have been neglected. 2. Thermal Aspects The heating of a sample is governed by the heat equation. The general equation is [36] 7 • (k 7T) (9) where k Is the thermal conductivity, c, is the heat capacity and n is the density. When thermal conductivity is constant this becomes (1 0 ) Heat is generated from the absorbed microwave power and at a point in the sample it is (ID Heat energy is lost to the surroundings through convection and radiation [37]. The boundary condition is 24 where T0 Is the ambient temperature, h Is the convection constant, s Is the Stefan-Boltzmann constant and £> is the emlsslvlty. The first term on the right hand side Is due to convective heat transfer and the second Is attributed to radiation. Initial conditions require a uniform temperature and a homogeneous sample. Thermal parameters vary with temperature [38]. The thermal conductivity for many ceramics Is shown to decrease with Increasing temperature, Fig. 4. Figure 5 shows specific heat as a function of temperature. At high temperatures it approaches the 3Nkb limit, or 5.96 cal/ga'tom°C. [39] 3. Sintering Dynamics Figure 6 shows spherical particles In contact.[40] In the region of contact, mass transport occurs causing the formation of a neck at the boundary between grains. Mass transport occurs via several mechanisms: lattice diffusion, grain boundary diffusion, surface diffusion and plastic flow. Those processes which produce denslflcatlon are 25 004 M E U u E W 003 Smgl«cry$t»l TiO. II to r axit I $ T * V Potycry»t»llmt Al2() o 17m grain tin o 9m grain tin £ 002 1 1 001 Polvcryttallina TiO. 200 Figure 4. 400 800 1000 1200 Thermal conductivity v s . taaparatura for varloua caraalca. Adapted froa [38]. 26 (o. UJ0|» 1/1*}) li.}*<J«} )**H sc MrO Al . 0 0 Temperature (*C) Figure 5. Heat capacity va. taaparatura for various caraalca. Adapted froa [38], 27 Cram boundary Figure 6. Ceramic partlclee In contact, allowing a grain boundary. Adapted from [40]. 28 lattice diffusion, grain boundary diffusion and plastic flow. Coarsening Is produced by surface diffusion. Conventional sintering favors surface diffusion at low temperatures because of Its low activation energy. Rapid sintering techniques have a shorter dwell time at the lower temperatures, and tt\us tend to inhibit excessive grain growth. [41] In a typical sintering operation, a green sample will have an Initial density of 30-70% theoretical density. Particle size Is on the order of a micron or less. Sintered samples show 95% theoretical density or better. [1] He note that as density Increases, shrinkage occurs, causing a change In surface area, which affects heat transfer. A change in density affects both electrical and thermal parameters, and should result In a higher electrical conductivity, dielectric constant and thermal conductivity. 4. Case Study In the theoretical Investigation to follow, results are applied to particular examples. material. We use alumina as a model The physical constants listed below will be used throughout the next section, unless otherwise noted. [42] 29 f - 2.45 GHz €r - 10 o0 * .002 S/m (C « 20 W/m*K cp - 1000 J/kg-K P ■ 3970 kg/m? h - 10 W/m2*K a ■ 5.67x10"’ W/m2-K’ („ « 0.6 where f Is operating frequency, €r Is dielectric constant, o0 Is electrical conductivity, k Is thermal conductivity, cp Is heat capacity, p Is density, h Is convection constant, s Is Stefan-Boltzmann constant and Is emlsslvlty. The melting point of alumina Is approximately 2050°C, but this condition will be relaxed In the theoretical analysis to follow. Temperature dependent electrical conductivity for sintered alumina Is determined from published data. [32,43] 30 B. Thtorttlctl H a t i n g of a S— 1-Inflnlte Space We consider the simplest geometry, a semi-infinite space. The goal is to determine temperature profiles as a function of incident power and material parameters. The analysis is divided into two parts; 1) constant electrical conductivity and 2) temperature dependent electrical conductivity. Both steady state and transient solutions are developed. The geometry of the system is shown in Fig. 7. A uniform plane wave is Incident from the left upon a planar surface. Part of the signal is reflected, and the remainder penetrates the space. The transmitted wave heats up the material and a temperature distribution with time and position is the result. Lt Constant Conductivity The temperature profiles are determined by solving the heat equation with appropriate boundary conditions. The time dependent heat equation in one dimension is: X > 0 Heat is lost to the surroundings by convection and (13a) UNIFORM PLANS WAVS RADIATION rigur* 7. I CERAMIC SAMPLE Gto M t r y of planar half spaca. 32 radiation at the aurface. icfj - h(T - T..,) ♦ a€K(T4 - T ^ ) . x - 0 (13b) The flret tera on the right Is due to convection and the second to radiation. In the U n i t of Infinite depth, C m T ■ T„ < oo . (13c) The space Is Initially at a reference tenperature of 300K. Heat Is generated electronagnetlcally, and Is proportional to the absorbed power, P(x) - Jo(x) | E(x) | *. (14) The total absorbed power Is the Incident power less the reflected power. P0 - (1 - |p|*)P1Be (15) If constant conductivity Is assumed, the electric field within the saaple Is: [44] E(x) - E0e y" (16a) 33 where v - (16 b) end (16c) | B(x) | * - E(x)-E*(x) - E^e'2" where a * $ 4 r [ i ♦ ] sln( ^ tan'1^ ) <16d> The generated heat le [40] Q, - no0P0e-*" - no0(l - |PI*)Plnee‘*“" (17) The reflection coefficient from a planar homogeneous surface Is: [46] l - rf~t P - ---- 1 i ♦ £7 • (18) He consider the variation of transmission as a function of conductivity In Pig. 8 where the percent absorbed power Is plotted as a function of conductivity for a fixed dielectric constant of 10. For low conductivities, the absorbed power Is determined by the dielectric constant. For high conductivities, above 2 S/m, the material becomes more reflective, approaching the limit of total reflection by a perfect conductor. 34 POWER 1.00 0.80 ABSORBED 0.60 0.40 0.20 0.00 1 -3 10 ■* 10- 1 10 10 CONDUCTIVITY ( S / M ) Flour* 8. Parcant powar abaorbad v*. conductivity for a planar half spaca, f ■ 2.45 OH*, €r - 10. 35 a. TriMltnt Solution The previous aquation la aolvad nuaarlcally to dataralna taaparatura aa a function of tlaa and position, ualng an axpllclt aathod [47]. Wa represent the derivatives by finite differences: all -irJL - 2T 3x* -t T (dx)* - T dt (19a,b) whers ths subscripts represent Increments In position and the superscripts, Increments In time. The temperature at time t ■ j+1 can be expressed In terms of temperatures at time t ■ J: T f i s') j Q„dt - r T + T ♦ (1 - 2r)T + • V ‘“ I IM / 1 P* where (20) r ■ — — =c^P (dx) The boundary conditions at the eurface are linearized and recomputed at each time Increment. where hr ■ h ♦ e€xT* The grid le extended one point beyond the boundary. (22a) (22b) (22c) The Infinite region Is approximated by a finite grid. It la necesaary to determine boundary conditions In the Interior to minimize error propagation from the rear surface where the grid la truncated. The temperature gradient Is extrapolated one point beyond the grid. (23) The grid spacing Is constant. A multipoint method for predicting the end value could further Improve the accuracy. 37 [48] The •xplicit method Is not unconditionally stable. an Interior node, we auet choose r < 0.8. For At the surface, the stability criteria Is: [49] h Ax —ljj— ) < 0 . 5 r( 1 (24) Since hr can becone large due to radiation we choose r < 0.25. Before examining the transient temperature profiles, we first consider the steady state solution as a function of electrical conductivity, Incident power and thermal conductivity. b, Steady State Solution As time Increases to Infinity, the heat equation becomes (25) The same boundary conditions apply. The solution of (25) yields temperature as a function of position. 38 Q T - - •"*••] ♦ D (26) D Is found froa the solution of a quartlc aquation. D ' - [ssfe:+ + ♦ T-.*] - 0 Tha taaparatura at tha surfaca, x ■ 0, la D. i” ) Tha taaparatura daap within tha aatarlal la T- - D ♦ (28) and la navar lass than tha surfaca taaparatura. Wa consldar two Halting cases, low and high electrical conductivity. In the first case, o -* 0, and In this Halt, . _Q2_ 24T % (29a,b) 2€IP 'm (4^7 ♦ 1 )*S4k - 4€ *P WKCJC ♦ 1J (29c,d) and ws sea that steady state surfaca taaparatura Is Independent of electrical and thsraal conductivity, and Is a 39 strong function of Incident power, heat transfer at the boundary, and dielectric constant. In the Interior, the asymptotic value of temperature Increases Inversely with electrical and thermal conductivity. In the second case, o -» oo, and we have (30a,b) (30c.d) Note that the surface temperature Is Independent of electrical conductivity and dielectric constant, and that the Internal temperature limit approaches the surface temperature as the electrical conductivity becomes large. The thermal gradient Is strongly related to the electrical skin depth and the thermal conductivity. c. steady Stmts Results He evaluate these equations using a model material, which approximates the material parameters of aluminum oxlds. The physical constants are listed In Section A.4 of this chaptsr. He use different values of electrical conductivity to understand the role of skin depth in the constant conductivity model. The external temperature Is » 40 equal to the initial taaparatura of tha aatarial and is 300K. Tabla I shows how tha surfaca and daep tanperaturaa vary as a function of physical paraaatars. Tha first two coluans raprasant tha alactrlcal and tharaal conductivities. Tha next* coluans list tha assuaed incident and absorbed power densities. Tha fifth and sixth coluans are tha surfaca and deep taaparatures. Tha final coluan is a ■assure of tha rata of rise of interior taaparatura as raprasantad by tha depth at which the taaparatura rises froa tha surfaca taaparatura to 90% of tha total rise. Taaparatures above tha aalting point of aluaina (2000°C) are theoretical only. At a vary low electrical conductivity, Line A, there is a large rise in taaparatura as one proceeds a considerable distance into tha aatarial. Increasing conductivity, B, decreases the interior taaparatura and shortens the length of tha taaparatura gradient. Surfaca taaparatura and power absorption are unchanged since reflection is doalnatad by tha dielectric constant. A further increase in conductivity, C, further decreases tha interior taaparatura and shortens tha gradient. Decreasing tharaal conductivity, D, only increases interior taaparatura. Increasing alactrlcal conductivity, at tha original tharaal conductivity, E, 41 o (S/a) K(W/aK) P lM <*/«») *P.H T0(C) T.(C) .9T_(a) A. 0.002 20 200 73.0 1899 98900 9.70 B. 0.02 20 200 73.0 1899 11600 0.97 C. 0.2 20 200 72.6 1898 2869 0.097 D. 0.2 10 200 72.6 1898 3840 0.097 E. 2.0 20 200 56.8 1857 1962 0.0115 P. 20.0 20 200 21.4 1805 1827 0.0027 0. 20.0 20 100 21.4 1505 1516 0.0027 H. 20.0 20 900 21.4 2283 2339 0.0027 Tabla 1. Taaparatura profllaa in a aaal-lnflnlta aolld. 42 decreases absorbed power (due to Increased reflection), lowers the surface temperature slightly, reduces temperature lnhoaogenelty and shrinks the gradient. A further Increase, F, shortens the gradient and further laproves the hoaogenelty. Changee In applied power, Lines G and H affect the temperature and not the gradient. d. Tranelent Results Transient teaperature profiles are determined by numerical calculation. The computer program le given In Appendix A. The same material parameters are used. Again, the external temperature Is 300K. Figure 9 shows teaperature within a half-space as a function of depth. The electrical conductivity Is 0.2 S/m and the Incident power Is 50 W/cm*. 1mm and the time step Is 0.04 eec. The grid spacing Is Figure 9(a) shows the temperature profile after the 500 sec. We notice that heat Is transferred to the surface where It Is lost to the surroundings by convection and radiation. Heat Is transferred to the Interior via conduction, and after 1500 sec, Fig. 9(b), the teaperature has penetrated more deeply Into the material. The maximum temperature Is In the interior of the material. Given sufficient time, Ignoring the melting point, the temperature profile will approach the 43 steady state value, Fig. 9(c). In the figures to follow we will truncate the grid near the surface and observe only In the near Interior heating effects. The following sequence of figures, Figs. 10-13, show the transient teaperature profiles at constant conductivities varying from .002 S/m to 2 S/m. extended 200 mm Into the material. The grid Is This has been shown to be sufficient for an observation area of 100 mm by comparing these results with a grid extending 300 mm Into the material. These results become less accurate as time progresses. A typical transient profile Is shown In Fig. 10 where the conductivity Is 0.002 S/m and the Incident power Is 5000 W/cm:. Reflections reduce the power by 21% to 3650 W/cm:. This power level Is not practical In a laboratory, but serves to Illustrate the heating phenomena. The transient profile after 100 s shows pratlcally uniform heating. After 400 s, there Is a temperature gradient near the surface due to heat transfer to the surroundings. Heat Is also conducted to the Interior as evidenced by the shallow gradient on the right hand side of the figure. All transient profiles have these characteristics. The next figure, Fig. 11, Is for a conductivity of 0.02 S/m and an Incident power of 5000 W/cnr . At 100 s and 400 s the profiles are almost Identical to Fig. 10. This 44 TEMPERATURE (C) 5000 40001 3000 2000 melting point - ■ 10001 0 100 200 300 400 500 DISTANCE (MM) Figure 9. Transient teaperature profiles In a planar half space; o ■ .2 S/a, P - 50 tf/ca*, a)500 sec, b)1500 sec, c) steady state. 45 (C) 3000 -] m _ii TEMPERATURE 2000 1•** 1000H 0 20 40 60 80 100 DISTANCE (MM) Figure 10. Transient taaparatura profiles In a planar half space; a ■ .002 S/a, P - 5000 W/ca*. a) 100 sac b)400 sac. (C) 46 TEMPERATURE 2000 . ... ■ •M in i p * >nt 1000 20 40 60 80 100 DISTANCE (MM) Plgura 11. Transient taaparatura profllaa In a planar half apaca; o ■ .02 S/a, P ■ 500 W/ca*, a)100 aac b)400 aac. »1 y' tr * K. . X% fV ^ ***• ’**0° *'»»c »w # v •* ' *• * - 4'I40° mC < : «.; -'i0° «*e* * -*c • 48 (C) 3000n TEMPERATURE 2000 1000 - 20 30 40 50 DISTANCE (MM) Plgure 13. Transient taaparatura profiles In a planar half space; o ■ 2 S/a, P • 80 W/ca*, a)100 aac b)200 aac. c)292sec, d)1256 sac. 49 demonstrates that hsat is being generated at the same rate In both cases. It would take 4000 s for the previous example to reach these temperatures If the incident power was 500 W/cm2. Since condutlvlty has been Increased 10- fold , power must be reduced proportionally to maintain the same heating rate. Figures 12 and 13 are for conductivities of .2 and 2 S/m respectively. figures. Incident power is 100 W/cm2 in both These plots show the reduced field penetration by the decreased heating of the interior. Reflection from the surface has Increased. Figure 14 shows the effect of changes in power, observation time, thermal conductivity and skin effect upon temperature non-uniformity, for an electrical conductivity of 0.2 S/m. The solid curve at 100 W/cm2 is observed after 400 seconds and exhibits a small gradient near the surface. At a higher power, 200 W/cm2, the dashed curve, we achieve similar temperatures in a shorter time, 200 s. near the surface has been Increased. The gradient If we decrease thermal conductivity for the 100 W/cm2 - 400 s curve, we have the dotted curve. Note the poorer uniformity arising from reduced heat flow into the interior. Note also the relative values of surface versus interior temperature. For the solid and dotted curves, which differ only in thermal conductivity, the surface 50 TEMPERATURE (C) 3000 n 2000 1000 20 40 80 100 DISTANCE (MM) Figure 14. Transient teaperature profiles In a planar half space; a)solId, o - .2 S/a, k • 20 W/a-K, P ■ 100 W/ca*, t ■ 400 sec, b)dash, o » .2 S/a, k - 20 W/a-K, P - 200 W/ca*, t - 200 sec, c)dot, o - .2 S/a, k - 10 W/a-K, P - 100 W/ca*, t 400 sec. 51 temperatures are practically the same whereas the internal peak temperatures are significantly different. Upon comparing the solid and dashed curves, with only the power Increased, we see the expected Increase In surface teaperature as a result of the Increase In absorbed power. The dotted curve coincides with the dashed curve In the deep interior. ** It should be noted that steady state temperature distributions are well off the graph. In normal sintering practice high power Is applied to heat the sample to the sintering temperature, then power Is reduced to maintain a constant teaperature. 2. Temperature Dependent Conductivity The time dependent heat equation In conjunction with the time harmonic wave equation are used In the temperature dependent conductivity problem. In one dimension they are , x > 0 “ h <T - T.«<> ♦ *x(T« - O , x - 0 (31a) (31b) (31c) fjj ♦ -r„E - 2 % . x - o Old) 52 where 0 ■ an<* vo *he free epace propagation conetant. Since the nonuniformity le In the direction of propagation, and TE waves are assumed, dielectric gradient terms arising from (4) can be neglected. The electrical conductivity Is a function of temperature a - o0f(T) (32) The function f Is defined to be unity at the background temperature, i.e. f(T.„t) ■ 1. conductivity case f(T) Is 1. For the constant To complete the mathematical statement of the model problem, conditions are required on T and E as x approaches Infinity. |E| - 0 (33a) T - T„ < oa (33b) where T_ Is a bounded constant. Transient Solution The transient solution to the heat equation remains the same as before. However, it Is necessary to modify the solution to the wave equation. We note that 53 e„ ♦ r*E - 0 . (^ ♦ in fa s (34) ■ i r >B - 0 In the constant conductivity case there Is no backward tranvsling wave In the saterlal. In the teaperature dependent conductivity case, we aust Include forward and backward traveling waves as part of the solution. We do this In discretized fora, allowing each region of the grid to be coaprlsed of both coaponents, representing aultlple reflections In that grid region. We assuae that at soae point deep within the saaple, there Is no reflected wave. By proceeding backwards along the grid, the electric field can be deteralned at each point in the saaple. [50] where (35b) To reduce computation time, we can assume that at low temperatures the conductivity is nearly constant. Reflections between layers can be Ignored and the simple attenuation equation of the constant conductivity case can be applied, using the new conductivity of each grjld element. At higher temperatures, where conductivity gradients are significant, the full model must be used. A computer program using this algorithm Is given In Appendix B. We apply the transient analysis to our model material, using an exponential conductivity law, where o ■ .002exp(.0026T), which is a good fit to the conductivity vs. temperature data for alumina (Honeywell A-203, 95*). [32] Figure 15 shows the the temperature profile vs. depth for a power input of 500 W/cmz. After 1000 seconds, the material Is at a uniform temperature of 400 C to this depth. After another 500 seconds, the teaperature has doubled, and continues to accelerate due to the exponential conductivity variation. The surface teaperature tends toward a limiting value, while heat Is conducted Into the Interior. Again, 30CC r* — 1 2000 1800 sec 1700 sec 60 Figure 15. 80 Transient taaparatura prof1las in a planar half spaca with taaparatura dependent conductivity, o • .002 exp(.0026T), P - 500 H/ca*, a)1000 sac, b)1500 sac, c)1700 sac, d)1800 sac, a)1900 sac. 56 the M l ting point is ignored. This is bsttsr illustrated in Fig. 16, where the surface teaperature is plotted vs. tlae. regions of heating. He see three Region I is initial heating, the teaperature rises slowly as the conductivity is only weakly dependent on teaperature. Region II is rapid heating, where the heat generation rate accelerates exponetlally; and Region III, teaperature saturation, where the skin effect has H a l t e d the field penetration into this high conductivity aaterlal and surface reflection has Increased, reducing the total absorbed power. Figure 17 shows the conductivity as a function of depth in this Halting region. At low teaperatures, the theraal gradients are saall, and consequently, there is little variation in conductivity. In the teaperature saturation region, however, a conductivity gradient is evident. This reinforces the need for the aore exact solution to the wave equation. In the constant conductivity section, we discussed truncation errors — that is, a finite grid must be used to approxlaate the infinite solid. result froa the grid spacing. length be s m Inaccuracies can also It is laportant that the grid II enough to resolve detail in both the theraal and electrical regiaes. The theraal length is defined as a ratio between conduction and surface transfer constants: 67 (C ) 4000 TEMPERATURE 3000 2000 000 500 iME Flgur* 16. 1000 1500 (S E C ) Surface taaparatura vs. tlao for Trsnslsnt hosting of s plsnsr half spscs with taaparatura dspsndsnt conductivity, o ■ .002sxp(.0026T), P ■ 1000 W/ca*. 58 10 1 uo >- o 3 o z: o o 1 0 DEPTH Flour* 17. 40 20 (MM) Tranaiant conductivity £ » * “ • tor planar half gpicc aftar 1900 **c« with o ■ .0 0 2 axp(.0026T), P ■ 1000 W/ca*. 99 At zero degrees, the theraal length le 2 aetere. Ae teaperature rleee, the theraal length decreaees. At 600*C, lt - 1 a; at 1500*C, 1, le .1 a; at 3800“C,« lt Is .Ola; and at 8200°C, lx la .001 a. Since the grid apaclng la .001a, the explicit aethod le unatable above thla teaperature. The electrical length la referenced to the akin depth, which la If we uae the aaae exponential law for conductivity, we notice that at 0°C, 1. la .2a; at 23904C, o Is .1 S/a and 1, le .Ola; at 4161°C, o le 100 and la Is .001, which la on the order of the grid apaclng. So we eee that In the teaperature dependent conductivity caee, the electrical length la auch leae than the theraal length at high teaperaturee. To resolve greater detail In the field distribution, a saaller grid spacing Is needed. Unfortunately, to aalntaln stability, the tlae scale aust be reduced In proportion to the square of the length, greatly Increasing processing tlae. A variable grid aethod aay be valuable In this case, and is outlined in [51]. 60 fe, Steady 3t>ti Solution In this section we summarize the results of technical report [23] and a publication [24] which show that a steady state solution does exist for the sesl-lnflnlte space, and furthermore, the relationship between incident power and steady state temperature Is monotonlc, and unconditionally stable. The solution of the mathematical problem posed by these two coupled differential equations is Impossible to find analytically, since the temperature and heat flux are related non-llnearly. However, we can obtain an approximate solution by exploiting the smallness of thres dlmenalonless paramsters. =. * KT • c« ‘ * zzfc (37»-c) Ths constant c, Is the square of the ratio of the skin depth to wavelength In the slab at t-0. Ths constants c, and cz are the ratios of thermal lengths to wavelengths in the slab at t-0, and are difficult to physically lntsrpret in the seml-lnflnlte solid. Thess paramsters are derived from a non-dimensional analysis of the heat and wave equations In Appendix C. The non-dimensional forms of the these equations are: 61 B„ ♦ (1 - Jc,f(u)))E ■ 0 , x > 0 (38a) -JT *. ♦ J« - 2j , x - o (38b) u„ - - Xf(u) I E(x) I (38c) , X > 0 U, - C, U ♦ CjUu-fl)4 - 1), X - 0 where X - (38d) , u - ^ ^ Tfl In [23] a formal asymptotic analysis was performed In the limit as c, -* 0 and bounded. with c,/c, and c,/c, fixed In this case the conductivity was assumed exponential, I.e. f(u) - exp(u). An approximate solution was obtained In the fora of a power series In ct, T ~ T0 + c,T, ♦ (c,)*!, + ... (39a) E 'v E0 ♦ c,E, ♦ (ci)tE2 ♦ ... (39b) where explicit formulae wereobtained forT0,T,, E0 and E, . In particular, the surface teaperature T0(0)was found function of the incident microwave power, P0. as a However, for sufficiently high power, the series became invalid because the correction terms c,T, and c,E, became larger than T0 and Eq , respectively. This nonunlforalty was removed by seeking a different asymptotic expansion of T and E which Involved a generalized power series, using the method of multiple scales [52]. (A similar effect Is demonstrated in Appendix 62 D.) By using ths method of matched asymptotic expansions [53] a composite result was obtained which yielded a uniform approximation for T and E for any power level, P0. The functional relationship between the surface teaperature T, and the Incident power P0 Is stated below. P - M T . - Tq) ♦ »€k (t / 1 - where 1 - |p|* ■ V) | p|* ____________ ( 1 ♦ J2(r(x+l) ♦(rx)<j2(x-l) + c. Por low Incident powers, where c, << l, this reduces to h(T - T0) ♦ st (T * - T04) P„ - - L-J----- ^ — .-5^* 1----- ^ 4^7 <i ♦ JC)2 (40b) Por high powers, where c, >> 1, we have P0 - ^ c ~ 3 s€kT.\ (40c) Plgure 18 shows surface temperature vs. Incident power where the material parameters are the same as ths constant conductivity case, except the electrical conductivity Is 63 3000 j (C) J 1 -l H Temper at ur e -4 2000 i 1 i i / J / J Surface 1ooc -I 1/ -II HI f I I II I I I I I f I I I I II I II |'l I I I I I I T | I I I I I I I I I I I M I'IT T I II 0 200 ncident Figure 16. *00 600 800 1000 Power ( W / c m 2 ) Steady state surface teaperature vs. Incident power (W/ca*) for planar half space with teaperature dependent conductivity, o ■ .002exp(.0026T). 64 .002exp(.0026T) S/m. Notice that lt raqulras ralatlvaly •■all power to achlava modest temperatures, but that a •light Increase In teaperature at the high end requires a large Increase In power. We note that the solution Is monotonic, a one-to-one correspondence exists between Incident power and steady state surface temperature. The stability of this solution Is proved In [23] Indicating that this system is unconditionally stable, every temperature can be achieved by simply adjusting the Incident power level. Theoretical Heating of a Planar Slab The next geometry we consider Is the planar slab, which Is shown In 71g. 19. sample from the left. A uniform plane wave Impinges on the A portion of the signal is reflected from the sample and a portion Is transmitted through the •ample. The remainder of the signal Is absorbed by the sample as heat. The analysis, however, Is based on the fields In the sample and Is not related to the Incident signal. In the discussion to follow we consider both transient and steady state solutions to the heat equation, and derive a model to predict the onset of temperature Instability. Por the steady state analysis, we assume that the sample Is electrically thin, that Is, the fields are uniform. We will show In section D of this chapter that the 65 UNXrOftH PLANE MAVE Flgur# 19. Qaoastry of planar slab. CEMNXC SAMPLE 66 solution for s thin cyllndsr only dlffsrs from ths solution for s slab by a constant. Therefore, tha slab analysis, which Is slaplsr aathaaatlcally, Is prafarrad. Tha taaparatura distribution Is found by solving tha hast aquation subjact to approprlata boundary conditions. In tha slab gaoaetry, tha haat aquation Is - Q , -d/2 < x < d/2 (41a) The boundary conditions are: -K§| + h(T - T0) ♦ e€K(T4 - T0) - 0 , x - -d/2 + h(T - T0) * “ T o> " 0 ' x " d/2 (41b) <41c> Haat Is lost to tha surroundings through convection and radiation. We assuae that tha aablant taaparatura Is 300K and tha theraal conductivity Is constant. Tha ganaratad haat Is Q “ *0P 1B€o ( x ) ( l - IP I* - |T|*)Ig ^ * “ inc (42) 67 Lx Corntant Conductivity For tha constant conductivity casa, tha aatarlal la homogeneous. Tha raflactlon and transmission coafflclents can ba axprasaad axactly. If the sample la Illuminated by TE waves, and Is terminated by a matched load (l.*e. free space), the reflection and transmission coefficients are: [54] p m T - (1 - 6r*)slnh Vd f (€r* + l)slnh yd ♦ 2 cosh yd ----------- --------------- = (€r ♦ l)slnh yd ♦ ------- : (43a) (43b) ------------------------ cosh yd If the sample Is terminated by an Ideal short circuit, the net transmission Is zero and the reflection coefficient Is given by (7). TraMient Solution We use the sane technique described In the semllnflnlte space presentation to numerically determine the transient temperature profiles In the planar slab. modification Is that there are two boundaries. The only The right hand boundary Is given by (22b) and the left hand boundary Is 68 hrT J - (hT f 0 *"* ♦ sc^T ' ) - -^fl * *1 n ♦I - T ‘ 1 n-l J (44) For computational purposes, ths grid extends fros 0 to n, where ndx “ d, the thickness of the slab. Appendix 7 lists e progres for an explicit numerical solution for transient heating of a planar slab. sample la In free space. We assume that the We now use the transient analysis to study the steady state temperature profiles In a slab with physical properties outlined in Section A.4 of this chaptsr. Figure 20 shows ths stsady state teaperature distribution for conductivities of .002, .02 and .2 S/m with constant power Input of 900 W/ca2. While more heat Is gensratsd for ths lossler samples, ths dlffsrence between surface and Interior temperatures Is small. thermal conductivity. This Is dus to As the thickness of ths slab Is reduced, for fixed power, thermal gradients will decrease. 71gure 21 shows the steady state temperature distribution for conductivities of 2, 20 and 200 S/a and power of 500 W/ca2. symmetry Is reduced. Am the conductivity Increases, ths 7or Infinite conductivity, we have surface heating, with a linear decrease In temperature towards ths other boundary. Note that as ths conductivity 69 2500 i .2 S /r temperature (C) i 200C " 1500 .02 S /m 1000 .0 0 2 500 ^'rn—r~r- n 0.00 S /m -I--T— r~r~i i ' i > i ' > r 2.00 position Fiaura 20. t t v rT T-n ~ T- n ~i ■; *00 (mm) 6.00 traiiMrif t Stwady atata taaparatura P*0*1*** c\ 2 planar •lab. with a).002 S/a. b).02 S/a and c).2 S/a. 70 (C) 3500- 300C temperature 20 S /m 2500 2000 200 S / 1500 i i ? » i 0.00 t—i—n —|—i—i—r i i—i i i i—r i i i i "i—r—i—i—r - | 2.00 position Figure 21. 4.00 6.00 (mm) Steady atata taaparatura profIlea for a 6aa planar alab, with a)2 S/a, b)20 S/a and c)200 S/a. 71 rises, absorbed power diminishes because of Increased reflection. Me conclude fros these figures that the heating of a planar slab with constant conductivity Is quite uniform. The maximum teaperature is In the Interior, and the difference between the central maximum and the surface teaperature Is determined by the theraal constants for conduction, convection and radiation. In the heating of constant conductivity materials, measurement of surface temperature Is sufficient to determine the average temperature of the material. Stmmdv Stmts Solution Por thin samples, we can assume that the field distribution within the sample Is uniform. The steady state heat equation Is £ ■ - I If the electric field Is constant, the solution must be symmetric about the center of the sample, T(-d/2) ■ T(d/2) and the boundary condition at zero is sufficient, (41b). Solving (49), the steady state temperature as a function of position Is 72 (46) where B la a solution to ths quartlc (47) The temperature at the surface Is T(d/2) ■ T(-d/2) - B; the temperature at the center Is (48) In the absence of radiation, the ratio of the central teaperature to the surface teaperature Is (49) which Is small for thin samples. The assumption of symmetry Is Invalid for samples with high conductivity, since the field distribution Is not uniform. A 16th order equation must be solved, because each boundary must be treated Independently. Lt *— perature Dependent Conductivity In most ceramics, the electrical conductivity Increases 73 rapidly with taaparatura. We apply thla crltarla to tha slab geoaetry consldarlng both tha transiant and staady atata solution. L Tr«nyi«n<; Solution Tha nuaarlcal analysis for tha taaparatura dependant conductivity case Is Identical to that for tha constant conductivity planar slab, except that tha field distribution aust be recalculated as taaparatura rises. Wa use tha prograa listed In Appendix F to obtain transient taaparatura profiles in a planar slab. Figure 22 shows surface taaparatura as a function of tlae for an elactrlcal conductivity of .002 exp(.0026T) S/a, and various power levels. Whan tha Incident power Is low, tha transiant solution converges to a steady state, (a) and (b). Whan tha Incident power Is high, (c), runaway occurs. This Is conslstsnt with experlasntal observations. Figure 23 is a plot of surface taaparatura vs. tlae for a sintered alualna saapla of .5 ca dlaaeter. Tha aeasureaents ware taken with a pyroaeter using ths rectangular applicator. Tha power was turned on to tha aaxlaua level (approxlaataly 700 W) and tha saapla was allowed to haat up. In Fig. 23, wa sea slow Initial heating, followed by a region of rapid rise. Tha power aust be dlalnlshed to 74 2000 (C) 200 W surface temperature 1500 150 W 1000100 W 500 1000 time Figurs 22. 2000 3000 (sec) Surfscs traptratur* vs. tlas for trsnsisnt hsstlng of plsnsr slab with tsapsrstur* dspsndsnt conductivity, a ■ .002sxp(.0026T), s) 100 W/ca*, b) 150 W/ca* c)200 W/ca*. 9*1 M CJ rt SC • • ■ 9 92 2 <o rt (S O 9 < M. • 9 *1 *% rt • r t ** 6 s M* s c *1 Ml » n SORTACE TEMPERATURE (C) prevent destruction of the sample. steady state heating Is achieved. When power Is reduced, The periodic variation In the data la due to a slight misalignment of the rotating sample relative to the pyrometer. Steady state can be maintained up to about 1500-1800°C, above which, heating appears to be uncontrollable. In the steady state analysis to follow, we consider the question of Instability In greater detail. 77 b. >ttidv « i t i Solution and the Onset of Inata bllltv In this section, we demonstrate that Instability can sxlst In tha heating of a planar slab.[25] Ha first raprasant tha haat aquation In non-dlmenslonal fora, then wa find a solution to tha steady state haat aquation using a numerical "shooting" method. An approximate analytical solution using regular perturbation theory Is seen to match tha numerical result whan Important constants aresmall. Tha result Is that whan conductivity Increases rapidly enough with temperature, there Is a critical power level above which steady state cannot be attained. In a later section, He study the control of the stable/unstable regions by varying the temperature dependence of conductivity and thermal constants. He assume that the sample Is thin enough so that the electric field Is uniform. The heat equation can be expressed In non-dlmenslonal fora as shown In Appendix C. The length Is scaled to half the thlclcneas of the slab. u„ - ut ■ -\f(u) , -1 < x < 1 (50a) -u. (50b) c,u ♦ c, ((u+1)4 - 1) ■ 0 , x ■ -1 uB ♦ c,u ♦ ct((u+l)4 - 1) * 0 where , x ■ 1 (50c) 78 and x - 2X/d. Tha paraaeter c, la tha Blot nuaber, [49] which and coaparaa tha affacta of convactlon and conduction. Whan c, << 1, tha raalstanca to haat conduction within tha aaapla la auch laaa than tha raalatance to convection across the boundary, and tha taaparatura within tha saapla Is nearly unlfora. Tha constant ct can be viewed as tha radiative equivalent of tha Blot nuabar, and tha saaa rule applies whan lt Is saall. For thin saaples and low power levels (and hence low teaperatures), both c, and c, are saall; typical values are .002 and .00018, respectively. 1> Shooting Hethod Solution Tha solution to tha steady state aquationIs found ut ■ 0. whan Because of Its nonlinear nature, a nuaerlcal solution aust be found. Tha goal Is to express tha non- dlaenelonal taaparatura, u, as a function oftha Incident power, X., with ct and c. as paraaatars. Me use a nuaerlcal "shooting" aethod to deteralne this solution. [55] Tha approach Is to convert tha steady state boundary value problea Into an Initial value problea and solve using tha Runge-Kutta aethod. 79 * ” +\f(«) -0 4»(-1) ■ a (51a) ifc'(-l) ■ b • c,a ♦c2((a+l)4 - 1) g(X) - U»'(1) ♦ c.wd) ♦ c2((*(1)+1)4 whan g(X) (51b,c) - 1) (51d) ■0, than tit ■ u Ha can aolva for X using Newton's aathod [47] V. - V whara <»2> la found from solving K" + U»J-1) - 0 - -*(*) (53a) vD/(-l) - 0 gf - U»x*(1) + c,^(l) If X << 1 then a - (53b,c) ♦ 4ct[U»(l) ♦ 11 30/k <1) (53d) , which constitutes a good first guess for tha numerical solution. Tha aquations are solved numerically using tha Runge-Kutta method. [47] Tha two second order differential equations are separated to form a system of four first order differential equations.Appendix G contains a computer program that Implements this solution. Ut" ♦ Xf(tlt) - 0 (54a) 80 (54b,C) (54d,e) (55a) ^ - S - *(#) 0 (55b,c) (55d,e ) * 0 Figure 24 shows the solution fdr f(v) ■ exp(.78v), where normalized power vs. normalized temperature are the axes. It Is evident that for low power levels, there Is a corresponding steady state temperature. There Is a critical power level, above which, steady state cannot be maintained. Power levels above this critical value will produce sample melting. Points on the upper branch are not physically realizable. 11, Approximate Analytical Solution We note that In many applications the constants c. and cz are small, e.g. ct - .002, cz - .00018. We shall exploit the smallness of these values to derive an approximate analytical solution using regular perturbation theory [52]. The approximate solution compares favorably with results In the prior section, when the thermal 81 nor mal i zed sst at e temp 10.00 -3 8.00 6.00 4.00 2.00 0.00 ■ 0.000 0.001 0.002 0.003 normalized Figure 24. 0 .0 0 4 0.005 0.006 power Normalized steady stats surface teaperature vs. normalized applied power for planar slab, with f(v) ■ exp(.78v), c, ■ .002, ct • .00018, using a numerical "shooting msthod." 82 a parameter, we redefine the other constants. a - ct/cl , 0 ■ X/c, (56) where a and 0 are at most order one quantities. We find that the steady state temeperature distribution, v (x ;cj) , satisfies the nonlinear boundary value problem. v„. " " c,0f(v) (57a) -V„(-1) + ctv(-l) + «c(((v(-l)-H)4 - 1) ■ 0 (57b) v, (1) + c,v (1) + ac, ((v (1) +1)4 - 1) - 0 (57c) We shall now determine an asymptotic approximation of v in the limit as c, -* 0 with a and 0 held fixed. First we assume that v has the representation (58) v - 2 vn(x )Ci" Inserting this into (50), expanding the nonlinear terms using Taylor series, and equating to zero the coefficients of the powers of c, yields an infinite set of equations. These sequentially determine the vn(x). We shall just list the first few which are sufficient to deduce the leading order behavior of v. They are: 83 a) leading ordar approxlaation, v ■ v0 v0‘ ■ 0 , -l < x < l (59a) - v 0 ■ 0 , x - -1 (59b) v0 - 0 , x - 1 (59c) b) first ordar approxlaation, v ■ v0 ♦ c,v, v,* - -flf(v0) , -1 < x < 1 (60a) ♦ v0 ♦ <*((v0+l)4 - (60b) -V ,- v, 1) - 0 ,x - -1 ♦ v0 ♦ or( (v0+l )4 - 1) - 0 , Naxt we solve (59) for v0 x- 1 (60c) andfind that It Is a constant, which Is unknown for the moment. Then we solve (60a) and find that v, » A, ♦ B,x - J x*f(v0) (61) Inserting this result Into the boundary conditions (60b) and (60c), and eliminating the constant Bt we obtain v0 ♦ <x((v0+l)4 - 1) - |f(v0) (62) This gives v0 lapllcltlly as a function of X, the scaled power parameter, and hence the first term of the expansion (58) Is determined. 84 In suaaary, we have found that v - v0 4- 0(C,) C j V 4- C j ( ( V4-1 ) * - x " (63a) 1) (63b) r w i --------- Thus, to leading order, the teaperature within tfre thin slab la unifora and Its value la given lapllcltly as a function of power by (63b). This can be re-expressed In teras of physical paraaeters by reversing the normllatlon procedure. For an 8ma alualna slab, this Is accoapllshed by a) aultlplylng the normalized teaperature, v, by 300 to obtain temperature In degrees Celsius; and b)multlplylng the normalized power, X, by 50,000 to obtain power In Watts per square centlmenter. The revised equation Is: p ■ 5 3 5 ^ ? r f h <1 - T«» + - V ) ] (64) Figure 25 shows the steady state temperature distribution within the slab and Is derived from (61). The applied power level, \, Is .005, f(v) - exp(.78v), ct * .002 and ct ■ .00018. We note that Interior teaperature variation Is saall, and that aaxlaua teaperature occurs In the Interior. Figure 26 displays the relationship between power and 86 nor ml ai z ed temperature 3.60 -i 3 .5 5 - 3 .5 0 3 .4 5 - 3 .4 0 - 1.00 - 0 .5 0 0.00 normalized Pigure 25. 0 .5 0 1.0 0 distance Normalized steady state teaperature profile In a planar slab, X ■ .005, f(v) ■ exp(.70v), ct * .002, and c, ■ .00010. 86 nor mal i zed temperature 10.00 unstable 8.00 m elting point 6.00 H 4.00 d 2.00 H stable o.oo o.ooo 0.002 normalized Figure 26. 0.004 power 0.006 cri tical Normalized steady etate surface teaperature vs. normalized applied power for planar slab, with f(v) ■ exp(.?8v), c, - .002, c, - .00018, using an approximate analytical solution. 87 teaperature for a slab with tha saae alactrlcal propartlas aa Fig. 25. We nota that v0 aa a function of power la aultlvalued. Wa ahall show In Sactlon 111 that tha lower branch la atabla whereas tha uppar branch la unatabla. In tha praaant aodal wa obaarva that ralalng tha powar abova Xa raaulta In uncontrollad haatlng. Plgura 27 la ldantlcal to Flgura 26, axcapt tha unnoraallzed axpraaalon, (64) was usad to coaputa tha curva. It shows that for this aodal, tha aaxlaua stabla teaperature for alualna Is approxlaataly 1250°C. This valua Is lower than experlaental observations, Indicating that batter conductivity data as a function of teaperature la needed. Plgura 28 coapares tha results of tha "shooting" aathod solution (a) with tha approxlaate analytical aolutlon (b). It Is seen that for low powar levels and low taaparaturas tha results are nearly ldantlcal. A higher ordar approxlaatlon Is needed to aatch tha solution for high taaparaturas. We digress hare for a aoaent to sake an laportant observation. Tha reeults given In (63b) allow us to deduce f(v) froa powar and teaperature aeasureaents. valuable In characterizing slab aaterlals. This Is Proa pyroaatar aeasureaents of tha surface teaperature, for a given Incident powar, wa can obtain a functional relationship between v0 and X, Independent of (63b), v0 ■ 0(X) . 88 3000 temperature (c) unstable 2000 J 1000 melting point - stable 100 power ( W / c m 2) Figure 27. 300 200 c r it ic a l Steady etate eurface teaperature v s . applied power; un-noraallzed from Fig. 25. 89 nor mal i zed sstate temp 10.00 8.00 6.0C -3 4.0C 2.00 0.000 0.001 0.002 0.003 normalized Figure 28. 0.004 0.005 0.006 oower Normalized ataady atata aurfaca taaparatura va. normalized applied powar for planar elab, with f(v) ■ exp(.78v), c, ■ .002, cg - .00018, comparing a)aolld, approximate analytical aolutlon and b)daah, numerical "ehooting" method. 90 Inserting this into (12b) we find c.Q(X) ♦ c, ((Q{\)+l)* - l — ----F(X) - f (v0(X )) - -1 -- ---- Y which gives conductivity as a function of X. (65) Pros these parametric descriptions, 7(X), v0(X), we can deduce f[v0(X)] as a function of v0 ili-t Stability Analysis In the previous section we have computed an approximation to the steady state teaperature distribution and now we will study its stability to small perturbations. If thess perturbations decay to zero as t -» oo, the solution is stabls. If thess perturbations grow without bound as t -* ®, then the solution is unstable and will never be realized in the laboratory. Accordingly, we add a small time dspsndent perturbation to the steady state u(x,t) - v(x;c,) ♦ 6e"*“9(x) (66) where 6 is a measure of the size of the perturbation and & << ctz. Inserting this into (50), using Taylor ssrles and 91 (32), we find after omitting nonlinear tarns In e. ♦«. ♦ [u ♦ X f (v)]d - 0 , -1 < x < 1 (67a) “♦« * (ci + 4c1(v+l)3)e - 0 (67b) ♦ (CI ♦ 4Ct(V+l )*)• ■ 0 , x - -1 (67c) , X • 1 This Is a classical Stura-Llouvllle eigenvalue problem.[56] We shall now shorn that the smallest eigenvalue, ut, Is positive (negative) when > 0 < oj. That Is, when the response curve la monotonlc, the steady state Is stable because the perturbation will decay to zero. Similarly, If the response curve has a branch where ^ < 0, then this steady state Is unsta'ble. To prove this assertion we fora the "variational" equations corresponding to (50) by taking the derivatives of these equations with respect to X. Denoting by , we have ♦ Xf'(v)K» - -f(v) , -1 < x < 1 (68a) -Hi. ♦ (c, + 4cI(v+l)3)U) ■ 0 , x - -1 (68b) ♦ (c, ♦ 4ct(v+1 )3)Ui ■ 0 . x - 1 (68c) Note that (67) and (68) have the same boundary conditions. Next, we multiply (67a) by «, and (68a) by e, take 92 their difference, end Integrate over the length of the eleb. *[♦„ ♦ <u ♦ Xf'(v) )e] - e[u»„ ♦ xf (v)«i» ♦ f(v)] - / * / -1 UU»6dx ■ -1 f o ef(v)dx (69a) (69b) -1 * where j (*♦.. - 6*„)dx ■ |' - 0 (69c) and we have U, f 6,il>dx ■ * -i f f (v)e,dx (70) ■* -■ Since f(v) la always positive, and the lowest eigenfunction of a regular Stura-Llouvllle problem, e,, Is always positive, the right hand side of (69) le also positive. From this we deduce that ut Is positive (negative) when & > ° ( &<<>) • Proa this result we see, for example, that the lower branch ehown In Fig. 26 is stable whereae the upper ie unstable. When ut ■ 0, we have ■ 0 and this defines the "critical power level," \e. A etudy of the effect of variation In thermal and electrical parametera is performed In Section Q. 93 3. 8 f dv 8tit» Solution for an Insulated Planar Slab In son* cases, sicrowsvs heating Is perforaed on Insulated samples. In this section, the approxlaate analytic aethod Is applied to the Insulated planar slab to detsralna Its hsatlng behavior. The result Is similar to the uninsulated case except that the critical power Is lower. The sample configuration Is shown In Figure 29. A thin sample, Region II, Is bounded on both sides by an thin layer of Insulation, Region I. Region II Is subject to microwave heating, while Region I Is a window material, Inert to microwave heating. Region II extends. In normalized units from -1 to 1. Region I has thickness a, on the same order as Region II. For simplicity, symmstrlcal hsatlng will be assumed. We write the heat equation In each region In nondlmenslonal form as developed In Appendix C: Region I There Is no source of heat In Region I, so the heat equation Is: I: INSULATION II: CERAMIC SAMPLE Plgura 29. OtOHtry of lnsulatad planar alab. 95 V,„ - 0 , -a-1 < v , - ^v, x < -1 , 1 < X < ♦ ct((v,-fl)4 -1) ,x "vi.« " civi ♦ cf((v,-fl)4 - 1) , where c, - ^ 1-fa (7la) - -a-1 (71b) x (71c) - a-f1 , c, - Tha conatanta c, and cs ara acalad with raapact to the thlckneee of region II. conaldar Because of syaaetry, wa only need -a-1 < x < -1. Tha aolutlon to (71a) la alaply v, - Ax ♦ B (72) Ualng (72), wa aaa that at tha boundaries v,(-l) - B-A ; v, ,(-1) - A (73a,b) v, (-a-1) - B-aA-A ; v, „(-a-l) - A (73c,d) Substituting (73c,d) Into (71b) wa have A - c,(B - A - aA) ♦ c, ((B - A - a A + l ) 4 - l ) (74) 96 Region II Heat Is generated In Region II and transferred across the boundary to Region I. The boundary conditions require the continuity of teaperature and energy flow. The appropriate equations are: vi:.«« " “ ^ fvri) (75a) V ,,(-l) (75b) - V ,(-l) (75c) To solve the heat equation In Region II, we perform the regular perturbation analysis discussed earlier. He use only the first two terms of the expansion: (76) We can rewrite the heat equation by substituting (76) Into (74a), v, - -8f(v0) (77) and the solution Is: v» " ' ff<v0)(l - x») ♦ B, (78a) vt,. ■ " 8*(v0)x (78b) 97 where K, la a constant of Integration. Froa (75b) and (74) we gat A - c,(v0 - aA) ♦ c4((v0 - aA ♦ 1)4 - 1) „ (79) A „ °.v. - c,((v,+.)’ - 1 ) 1 ♦ c,a ♦ 4ac,(v0+l) o Froa (75c) and (73a), wo obtain an expression for A: K..\f(v) A - -STE-C — k7 (81) Equating (80) and (81) yields Ate, k 7JT(v T _ ic, c ,v ♦ c2( (v+1)4 - 1) K,,f(v) j + c a + 4ac,(v-fl)3 (82) When a - 0, this reduces to (63b), the solution for the uninsulated planar slab. When the thickness of the Insulating layer Is on the order of the saaple thickness, the aaxlaua stable teaperature does not chango, however, less power Is needed to achieve the sane teaperature. Figure 30 shows a noraallzed steady state teaperature profile In the Insulated slab. Ths distribution Is linear In the insulated region, where no source Is present and quadratic In the Interior, where electroaagnetlc heating 98 3.005 <D 2.995 CL ~D • - 2.985 2.975 - 0 .9 - 0 .4 normalized Plgura 30. 0.1 0.6 1.1 thickness Staady atata taaparatura profila for on lnaulatad slab. 99 occurs. Vlgurs 31 compares ths uninsulated and Insulated samples, assuming that the thermal conductivity of region I Is half that of region II. Ths dashed curve represents ths Insulated case, showing that Instability still occurs above a critical power level. While the critical temperature remains largely unchanged, the power necessary to achieve this operating point Is reduced. As the thermal conductivity ratio between the two regions Is diminished, the critical power level Is also decreased. This demonstrates ths value of an insulated sample In reducing ths energy consumption of the system. Lx Stsady Stats Solution for a Thick Slab The approximate analytical solution Is based on the assumption that the electric field Is uniform In ths sample. This Is valid when sample thickness Is much less than a wavelength and much less than the skin depth. In this section we develop an Improved solution when sample thickness Is on ths order of a wavelength, but still much less than ths skin depth. Ths field Inside a homogeneous sample, backed by a short-circuit one-quarter wavelength away is derived from (35) : nor mal i zed temperature 100 8.00 6.0C 2.00 -4 0.00 0.000 0.001 0.002 0.003 normalized Figure 31. 0.00* 0.0 0 5 0.006 oower Normalized steady state surface teaperature ve. normalized applied potter for Insulated planar elab, ttlth f(v) - exp(.78v)f c, ■ .002, ct .00018 and a ■ 0.1; ajaolld, uninsulated elab, b)dash, Insulated elab. 101 . (£7 - 1 ).-'“- ,x’ ‘ [ST'. »>••* * (J<7 ♦ iK ''*"’ ♦ (J*7 - 0.-‘'4 and tha reflection coefficient le given by (7). ' ’ The heat equation becoaea v.. - -\f(v)(1 - |p |*) | E(x) | * - -Xf (v)g(x) (84) Uelng the perturbation analysla, we write v l>>B - -6f(v0)g(x) (85) Integrating (85) once, we obtain v, , - -aftvj f g(T)dT (8 6 ) • _■ Subetltutlng (86) Into the boundary conditions (60b,c) we find that e.v ♦ c ((v+1 )* - l) *-------------, f(v) J g(x)dx » . -Ii \ « — (B7) 102 where the expression in the denominator represents an average value for the fields. It should be emphasized that the analysis of the planar slab Is valid only when the field distribution In the sample le Independent of temperature. When skin effects, produced by teaperature dependent conductivity, reduce field penetration, the heat and wave equations cannot be de coupled. simple. The question of stability (formally), Is no longer The prediction Is that the planar slab will re- stablllze at high temperatures. Whsn skin depth Is sufficiently small, the slab will resemble the seal-lnflnlte space, which Is always stable. 103 IL Theoretical Heating of an Infinite Cylinder The next geoaetry of Interest Is the thin, Infinite cylinder pictured In Tig. 32. A unifora redial wave laplnges on the saaple, and a fraction la abeorbed. We will consider the constant conductivity case, a aethod for transient analysis, and steady state heating for the teaperature dependent conductivity reglae. We consider the special cases of a tubular saaple and a thick cylinder. In general, the results for the cylinder are slallar to the results for the slab. The heat equation In cylindrical coordinates Is: K (Trr ♦ t t,> - C,PT* ■ - £ o0 | B(r) | * , 0 < r < a (88a) The boundary conditions are: ICTr ♦ h(T - T0) ♦ s£k(T4 - T04) - 0 , r - a (88b) and we require the teaperature to be bounded at the origin; T(0) Is finite. UNIFDRA ■> PLANE WAVE Plgura 32. CERAAIC SAAPLE A / Qaoaatry of lnflnlta cyllndar. 105 1. Twntltnt Solution The explicit sethod used to obteln teeperature profiles for the half-space and the planar slab aust be modified In the case of the Infinite cylinder. If we fora a difference equation based on (88), we have a singularity at the origin. At the origin, we average adjacent points, yielding [57]: A prograa to aodel the transient solution for the cylinder has not been iapleaented. 2. Steady State Solution for Constant Conductivity To determine the steady state solution to the constant conductivity case, we restate the heat equation: (90a) which has for a solution: -5) ♦„ (90b) 106 where D la a root of tha quartlc, D' + & D - [ $ ; ♦ £ to ♦ v ] - o ( 91 ) Thla reault la alallar to (47) and (48), tha aolutlon for tha planar alab. 3. Steady State Solution for Temperature Dependent Conductivity We conaldar tha atablllty of the cylinder when the conductivity lncreaaea with temperature. Tha regular perturbation analyale la applied to the ateady atate heat equation aa daacrlbed In Section II.C. The non-dlaenalonal heat equation, from Appendix C, la: r3rlr SrJ m 'Xf<v > . 0 < r < 1 (92a) vr ♦ c,v ♦ ct((v+l)* - 1) * 0 , r ■ 1 (92b) where c, - ^ , c. - — -g8 - , X - Baaed on the aaauaptlon that ct and c, are eaall, we fora the aolutlon ualng an aayaptotlc expanalon (76). Equation (92a) becoaea 107 (93) and the solution la (94) The constant A must be zero, since the temperature Is finite at the origin. Applying the solution at r * 1 gives v(l) - v0 (95a,b) Substituting for the boundary conditions, (92b), we have This Is virtually Identical to (63b), the solution for the planar slab. The difference Is In a scaling constant. Heating a cylinder to a desired temperature requires twice the power per unit volume as a slab of equal thickness. Inverse solution, The (63b) applies to the cylinder as well, Indicating that cylindrical samples could be used to characterize temperature dependent conductivity using steady state temperature and power measurements. loe L, S t M d Y S t t f Solution for a Tubs He extend the discussion of the thin cylinder to Include the case of e tubular geometry. He flret assume that there Is no convection in the tube and as a consequence there Is no heat flux across ths Inner boundary. Then we consider the effect of forced convection In the center of the tube. He use the scaled heat equation (92a,b), with an additional boundary condition at r ■ b < 1: vr - 0 , r - b (97) Hhen c, and ct are small, the solution to the heat equation, (94), still applies. In this case, however, A Is not zero. In fact, (98) Inserting (94) and (98) Into (92b), we have x - 2 (c,v ♦ Cj ((v+1) - 1) f(v)(l - b‘) (99) Hhen b - 0, this reduces to (96). If b is nearly one, X becomes large, and the analysis Is not valid. If we allow air flow through the center of the tube, 109 (97) b«COM« • vr - c',v , r * b (100) . h.a where c , * The convection constant Mill bs greater in the interior when there is forced convection. (92b), get 2(c.v ♦ fbc .v ♦ c, ((v+1)4 - 1) 2— --- :-------------------------------------------f(v)(l - b*) - - 1 - i ----- 1---- i - \ ms Using (94) and (100) in This increases the heat lost to the surroundings, increasing the power required to heat the sample to a desired temperature. h Interior thermal gradients are also Increased. Steady etate Solution for a Thick Cylinder As in the planar slab, we have assumed that the electric field within the cylinder Is uniform. When the cylinder is on the order of a wavelength in diameter, this assumption is no longer valid. For low conductivities, the field distribution looks like Ja, the Bessel function of the first kind, order zero [58]. The approximate analytical solution to the steady state solution applies, using an 110 X » a (c1v » * (V) (v+1) - 1) f0'g(r)rdr C,( (1 0 2 ) As conductivity rlsss with temperature, ws have the sase difficulty as we encountered in the slab; naaely that the sanple becomes lnhoaogeneoua, and the heat equation must be coupled with the wave equation for an accurate solution. When the fields within the sample do not vary with temperature, we can consider three types of conductivities: a) low conductivity, field distribution dominated by J0; b) Intermediate conductivity, Bessel functions of complex arguments are needed; and c) high conductivity, where J0 can be represented by ber and bel functions [58]. a) E - (103a) b) E - J0([fl - Jo]r) (103b) where v ■ a ♦ Jfl (103c) c) 7or large enough conductivities, the fields decay exponentially as thsy do In planar geometry. We can conclude from this analysis that the heating behavior of the cylinder closely resembles the heating behavior of the slab, and therefore analysis using a planar Ill •lab geometry Is sufficient to pradlct heating effects in a cyllndar. L Theoretical Heating of a Sohara The last class of geometries we consider is the spherical sample. He discuss the steady state solution to the heat equation based on the perturbation study, we consider the Inverse problem of characterization, and we describe the fields In a sphere as a function of conductivity. Li Steady State Solution The steady state heat equation, assuming uniform fields, and using the non-dimensional analysis in Appendix C , Is : (104) The boundary conditions are the same as the Infinite cylinder, (92b), where the scaling Is with respect to the radius of the sphere. Again, the temperature at the center Is bounded; u(0) Is finite. Applying the series expansion (76) to ths solution, and assuming c, and cz are small, we 112 obtain: flf l v I r 1 v, ----'-f1— 4- A/r ♦ B where A must ba zaro. (105) Subatltutlng (105) Into (92b) tha following ralatlonahlp batwaan powar and taaparatura la obtalnad: 3 (c,v ♦ Cj ((v+1)4 - 1) >- - (106) Thla dlffara froa (96), tha aolutlon for tha cyllndar, by a conatant, Implying that tha eaaa stability phanoaana la oboarvad In tha cyllndar. Ha can compare tha powar taaparatura relatlonehlps for planar, cylindrical and spherical geoaetrlea. C.V + C. ( ( V4-1 )* - SLAB: X - -»-------CYLINDER: X - They are: 2( C . V 1 (107a) 4- C. ( ( V 4 - 1 ) 4 - 1) :----- - 3 ( C . V 4- C. ( ( V4- 1 ) 4 - (107b) 1) SPHERE: X - — L-*---- -------------- (107c) For the cyllndar and sphere, c, and cz are unchanged. Note that tha same functional ralatlonahlp Is seen In all three gaoaatrlas. This Indicates that tha basic stability relationship Illustrated by Plgure 28 Is still true. Tha constant by which (107a,b,c) differ is simply the 113 ratio of normalized surface area to volume of the sample. Heat transfer to the surroundings Is Influenced by this ratio, and hence more power per unit volume Is required to heat the sphere to the same temperature as a cylinder of equal thickness, and more power per unit volume Is required to heat the cylinder to the same temperature as a slab. In terms of un-normal1zed parameters, the surface area to volume ratio will depend on the size of the bodies. Important Information about the conductivity of the sample can be obtained from the Inverse solution of these equations, as given by (64). Experimentally, this requirement of nearly uniform field distribution and uniform temperature limits the application of (64). A planar slab, Inserted in a rectangular waveguide will have a transverse variation In the electric field proportional to sln^. Visual observations Indicate that a cylindrical rod in a rectangular waveguide can have a substantial axial temperature gradient. The sphere, due to Its compact nature, Is likely to have a uniform temperature profile and thus is the preferred geometry for characterization experiments. To determine the temperature dependence of conductivity using a sphere, it Is necessary to know the power absorbed by the sphere and the surface temperature. The power absorbed can be calculated by subtracting the reflected 114 power fro* the Incident power, •■•using no wall loss**. Is Independent of applicator gaosatry. It Bacauaa tha taaparatura distribution la naarly uniform, whan c, and cz arc asall, [49] tha aurfaca taaparatura la a good aaasura of tha avaraga taaparatura In tha aphara, and can ba obtalnad by a pyroaatar. Additionally, knowledge of tha heat transfer coefficients la Important In determining tha constants c, and ct. This characterization method la dlacuaaad In graatar detail In Chapter V. 2, Fields in a Sphere As In tha slab and tha cyllndar, as the thickness Increases, tha assumption of uniform flelda la no longer valid. Tha fields In a thick slab are given In (83), and for a cylinder In (103). The fields In a sphere are described using spherical Bessel functions. The electric field distribution, for the fundamental TM mode ls:[S9] (104a) (104b) Note that this mode yields fields which are a maximum at the origin, which Is consistent with the field distribution In ths slab and cylindrical gsomstrles. For higher 115 conductivities, in ths hoaogeneous sphsrs, spherical bessel functions with coaplex arguaents aust be used. Hhen conductivity Is very high, functions analogous to the ber and bel functions (103c) are derived and strong attenuation Is observed. Hhen the fields In the saaple are described by (104a,b), the heat equation aust be aodlfled to include angular variation: The solution to this equation Is not sought here. Li generalized Stability Toraula for Thin Electrical Bodies A general relationship between Incident power and steady state teaperature based on the geoaetry of the systea has been discovered. He notice that the power necessary to achieve a desired teaperature varies with geoaetry. The aaxlaua stable teaperature does not vary with geoaetry. He know that at steady state, for soae arbitrary solid, the heat generated In the aaterlal aust equal the heat leaving the aaterlal. This can be expressed as [60]: 116 $o0t (T) 18 |*dv - <f> K(7T-n)do (106) (Applying the divergence theorea to the left hand side yields the faalllar heat equation.) rron the boundary conditions we describe the heat leaving the aaterlal by heat transfer conditions. (107a) (107b) We constrain this arbitrary solid In order to nondlaenslonallze the equations. First, It Is assuaed that the solid Is convex, that Is, all tangent lines or planes touch the surface at a single point. It Is sufficient for the heat transfer conditions at the boundary to assuae that the noraal vector at every point on the surface, S, does not re-lntersect the surface, so that the saaple transfers heat to a uni fora background. Furtheraore, we assuae a polar coordinate systea for two-dlaenslonal solids and a spherical coordinate systea for three-dlaenslonal solids so that the length nay be scaled to soae aean radius. Non-dlaenslonal power can now be expressed In teras of non-dlaenslonal steady state teaperature. 117 \ X f (108) f(v)|l|*dv V Por electrically eaall bodies, the fields are uniform In the solid and we can make a leading order approximation, assuming that c, and cz are small. (109) V The normalized surface area to volume ratio provides a geometric correction factor. It Is 1, 2 and 3 for the slab, cylinder and sphere respectively. Consequently, the power required to achieve a desired steady state temperature is dependent on geometry. We determine the maximum stable steady state teaperature by evaluating the derivative of the power with respect to temperature and setting It to zero. Por the leading order approximation, where the field distribution does not change with temperature, this becomes 0 - [ct ♦ 4cf (v+1 )3]f (v) - [ctv ♦ ct((v+l)* - 1) jfv (v ) which Is Independent of geoaetry. We conclude that the (111) 118 maximum taaparatura la not a function of geoaetry, eventhough tha critical powar la gaoaatry aansltlva. A stability analysis Is naadad to foraally prova this assartIon. 0. Parameter Study In this section we will axaalna how tha critical power, \e, and tha corresponding taaparatura, v, change as functions of tha electrical conductivity and thermal constants. Tha critical powar Is found by taking tha derivative of (63b) and setting It to zero. This yields (ct ♦ 4clv*)f(v) - (ctv ♦ c,v4)gj - 0. (112) Tha root of this aquation Is tha maximum stable taaparatura, v.If tha electrical conductivity Is aonotonlcally decrsaslng with taaparatura, (112) indicates that heating is always stable since there Is no positive solution. However, In aost caraalcs, [40] tha conductivity Increases with temperaturs, which allows ths possibility of Instabilities. We shall now axaalna some specific cases within ths present theoretical framework. 119 l-i Constant Conductivity Sine* the electrical conductivity has been scaled with respect to 9s$*0, we take without loss of generality, f(v) - 1 ' SJ " (113> Substituting (113) Into (112), we obtain v3 ♦ ^ - 0. (114) Since the teaperature Is always positive, we conclude that the heating of a planar slab with constant conductivity follows a aonotonic law. The solution Is stable. 2. Linear Conductivity The slaplest aodel Is a linearly Increasing conductivity, and f (v ) - l + kv . - l (115) Substituting (115) Into (112), we have 3c,kv* ♦ 4c2v* ♦ c, ■ 0 (116) 120 Again, thara la no poaltlva aolutlon, tharafora tha haatlng of a planar alab with llnaarly Increasing conductivity la unconditionally atabla. In tha absence of radiation, c, - 0, the solution Is still stable, since c, Is positive. 3_! gmdrfltjc Conductivity It Is convenient to aodel soaa aaterlals by using a quadratic conductivity law: [14] f(v ) - 1 + 2k,v * k,v* , - k, ♦ 2kjV (117) Substituting (117) Into (112), we find that v satisfies (c, ♦ 4Cj (v+i )3)(1 ♦ k,v + kjV2) - (c,v ♦ Cj( (v+1)4 - 1) )(k, + 2k,v) - 0 (118) Since a positive solutlon(s) can exist, Instability Is possible. Suppose the power level Is low enough to neglect radiation effects, that Is, c2 - 0. Equation (118) yields (119) 121 This yields ths aaxlaua stable teaperature, vB, with corresponding critical power X- ’ *, * l2 Q Z ' ,12°» Stable heating Is possible only when \ < X,, and cannot be achieved above this power. If, on the other hand, the radiative effects doalnate the convective ones, then c, ~ 0, Instead, and (118) becoaes In this H a l t 2k,v* ♦ (3kt ♦ 4k, )v4 ♦ 4 (2k, + ljv3 2(6 ♦ 3k, - 2k, )v* ♦ 12v ♦ 4 + 2k, (121) For high teaperatures, (v > 1), there Is no soltulon to (121), and stability Is therefore assured. These two cases suggest that radiative effects will restablllze the systea at high enough teaperatures. general, roots. In (118) can be shown to have 0, 1 or 2 positive He shall only consider the case when k, ■ 0, which Is Illustrated in Fig. 33. Here teaperature, v, Is plotted as a function of \ for different values of ot (where we recall that a Is c,/c,). If there Is no Intersection, / or,, the solution Is aonotonlc and stable. If there Is one lnteresection, <*,, there Is an Inflection point, and again the solution Is stable. For saall enough a, there 122 normalized tem perature 10.00 8.00 6.00 4 .0 0 2.00 0.00 0.000 0.001 0.002 0 .0 0 3 normalized Plgure 3 3 . 0 .0 0 4 0 .0 0 5 power Normalized steady state surface teaperature vs. normalized applied power for planar slab, with f ( v ) - 1 ♦ .3 v * , a)a - . 0 0 2 , b)a - . 0 0 3 , c)a ™ . 0 0 4 . 123 are two stable branch**, oi, saparatad by an unatabla ona. In thla casa tha radlativ* affacta hava rastablllzad tha systaa at high taaparatura*. 4. Ixponantlal Conductivity An axaalnatlon of conductivity data In [32,43,61] shows that an axponantlal raprasantatIon Is more appropriate for ceramic materials than a quadratic law. f(v) - exp(kv) , ££ - k exp(kv) (123) Substituting (123) Into (112) wa find that the critical taaparatura la a root of c,(kv - 1) ♦ c,[ (v+l)*(k(v+l) - 4)] ■ 0 (124) Thera la a positive solution to this aquation, Indicating that there Is a range of steady state temperatures that cannot be realized. Furthermore, when cz Is 0, this range Includes all teaperatures greater than 1/k. If ct la 0 Instead, stability Is maintained until vB Is somewhat less than 4/k. In the presence of both convection and radiation, the maximum stable teaperature Is between these two values. Figure 34 shows the power vs. teaperature curves for several values of c, and ct. The 124 nor mal i zed temperature 10.00 8.00 6.00 4.00 2.00 0.00 {Til111111111 t i rr>n fr11 m 11»i; n n 11111 11 ■1111111 |m 111 11111 0.000 0.001 0.002 0.003 normalized Flgura 34. 0.004 0.005 0.006 power Normallzad ataady atata aurfaca taaparatura va. normallzad appllad powar for planar alab, with f(v) • axp(.?8v), a)c, • 0, ct ■ .00018, b)c, - .002, c, ■ .00018, c)c, ■ .002, c, • 0. 129 corresponding critical powar levels, \e, ars also shown. By adjusting thermal parameters, the maximum stable teaperature can be controlled up to a maximum value determined by the c, ■ 0 curve. Turther control of the maximum teaperature la possible by adjusting the teaperature dependent conductivity through doping. In Fig. 35', the power vs. teaperature curves for sevsral valuss of k ars shown. The lower the value of k, the higher the maximum teaperature. This Is consistent with our experiments with A1203-T1C. [1] Ths conductivity of alumina Is low at room teaperature, but It Increases rapidly with teaperature. Thermal Instability has been observed at high teaperatures. On the other hand, TIC has a aodsrats loss at room temperature which Increase slowly with teaperature. Thermal runaway has not besn observed In TIC. Mixtures of AljOj-TIC with 10% TIC or lsss show thermal runaway. Mixtures of greater than 30% TIC sinter stably. 9. Other Exponential Forms While the exponential law has besn sufflclsnt for our preliminary work, there are more complex relations which bstter fit the experimental data. We examine two such sxprssslons below. 126 nor mal i zed tem perature 10.00 -I 8.00 6.00 0.68 .78 4.00 0.88 2.00 0.002 0.004 0.006 normalized rigur* 35. 0.008 0.010 power Normallzsd st«ady stats surfacs taaparatura vs. normallzsd appllsd powar for planar slab, c, ■ .002, C| - .00016, a)k - 0.68, b)k - 0.78, c)k - 0 .88 . 127 f(v) - 1 4 k0« x p ( ^ 1) , exp(^) (125) Substituting (125) into (112), yields (c, ♦ *c2(v*l)’)[ 1 ♦ k o s x p ( ^ ) ] - (c,v ♦ c2((v+l)4 - •xp(‘^ r O ] ■ 0 (126) Again the possibility for instability exists, but in this case, for large values of v, stability is regained. Figure 36 is the power teaperature curve where k,, is 600 and k, is 13.3. The constants wsre derived froa a curve fit to the data in [32]. The curve shows a stable low teaperature region, an lnteraedlate range of instability, and high teaperature stability. Another useful fora for expressing the teaperature dependence of conductivity is: f(v) - 1 ♦ ^ - e x p ( ^ ) , ^ •xp(^) [ v - l] (127) 128 normali zed tem perature 10.00 8.00 6.00 4.00 2.00 0.00 -fr0.000 0.005 0.010 normalized Flgura 36. 0.015 0.020 power Normallzad ataady atata aurfaca taaparatura va. normallzad appllad powar for planar alab, with f (v) - 1 ♦ 600 axp(-13.3/v), c, - .002, ct ■ .00018. 129 Substituting (127) into (112), w* obtain (c, ♦ 4c, (v-fl )*) [ i ♦ V« x p( - rr O ] - (c,v ♦ c,((v+l)4 - 1 ) [ ^ « x p ( ^ ) “ l]] - 0 (128) As in ths prior csss, high tsmpsrsture stability sxlsts, yet an intermediate region of instability can be found. Figure 37 shows the power teaperature curve modeled after ft-alualna. [62] k, is 378.S and k, is 4.67. a «»— This section shows the importance of teaperature dependent electrical conductivity on the stability of the systea. The functional fora for the conductivity will vary with the material. Ideally, a curve fit to actual data points could bs used to establish the stability of a particular systea. Suitable dopants might be added to alter the conductivity teaperature curve to Improve system stability. Control of the maximum stable temperature is also possible by adjusting heat transfer constants. Ths results of this section are consistent with [16,17], where a numerical study of Instabilities in the RF heating of ceramics is performed. A coll surrounds a thin 130 nor mal i zed tem perature 6 .0 0 -i 4.00 .00 - 0.00 0.000 0.002 0.004 0.006 normalized Vlgur# 37. 0.008 0.010 power Norullzsd steady stats surface tnptratur* vs. noraallzsd sppllsd power for planar slab, with f(v) - 1 ♦ 378.8/v sxp(-4.67/v), ct ■ .002, ct * .00018. 131 cylindrical saapla. Inducing RF currant In tha saapla, generating haat. An activation taaparatura aodal la uaad for tha alactrlcal conductivity, o0axp(-Ta/T) , and tha raault la a algaoldal curva ovar power and taaparatura with atabla and unatabla branchaa. Experimental atudlaa on alllcon achlava atabla oparatlon on tha uppar branch, aa wall aa low temperature atablllty. "Hot Wall" Applicator From our analyala of tha half>space, wa notad that thara ara thraa dlatlnct raglona of haatlng. Thaaa ragIona ara a raault of tha taaparatura depandanca of tha alactrlcal conductivity. At low tamparaturaa, haatlng la alow, dua to tha low conductivity of tha aatarlal. At lntaraadlata taaparaturaa, tha accalaratad haatlng rata producaa a rapid taaparatura rlaa; at high taaparaturaa, aurfaca raflaction, dacraaaad field penetration and lncraaaad radiation contribute to a taaparatura aaturatlon or Halt. In thla aactlon wa ara concerned with tha low taaparatura region, where taaparatura rlsea slowly. If tha background taaparatura ware fixed at a taaparatura above that of tha saapla, wa could rely on convective haatlng as wall as alcrowava haatlng, to "wara up" tha saapla. Tha use of "hot-wall" applicators Is not unknown to tha 132 microwave haatlng literature [7,10,63]. The general approach la to place a eaaple In an Insulating container In a microwave oven. sample. A susceptor may be In contact with the The microwaves heat up the susceptor, and the susceptor In turn heats the sample. We examine the "hot-wall" problem from a theoretical perspective, comparing the case of a background temperature of 1000*C to a background temperature of 0*C. cases, the sample Is Initially at 0°C. In both The numerics are the same as before except that when the sample Is colder than ambient, heat flows Into the sample. The programs In Appendix A,B and 7 are used with the appropriate background temperature. Lt B f U - f P tf f f Figure 38 shows transient conventional heating of the half-space, where the temperature as a function of position Is plotted at selected time Increments. Incident microwave energy. There Is no Notice that the maximum temperature Is at the surface, and that the heating rate Is quite slow. An analytic expression for the transient profile has been described In [64] 133 — temp (C) 100 5 0 -- 0 10 20 depth Figure 38. 30 40 50 (mm) Transient hot wall haatlng of planar half space with no alcrowave radiation, T.., ■ 1300K, a)100 sec, b)400 sec. 134 - .xp( ^ ) .rfc( ^ ^ ) (129) where a - Figure 39 ehowe the coablned effect of alcrowave and conventional heating, where an exponential conductivity law hae been asauaed, f(T) - exp(.0026T). The surface teaperature la plotted as a function of tlae. the low teaperature heating rate Is Increased. Observe that Interaedlate and high teaperature heating Is still doalnated by alcrowave effects. cases. The saae aaxlaua teaperature Is achieved In both A decrease In saaple voluae should laprove the conventional heating rate. Ls Pl«iwr Slab solution Figure 40 shows the heating of a planar slab, surface teaperature vs. tlae, for different power levels. background teaperature Is fixed at 1000*0. The Steady state is achieved at 1000°C, when there Is no Incident alcrowave. When alcrowave power is applied, we know that Instability 136 s urface temp (C) 3000-- 2000 1000 - - - - 0 400 200 time Flgurs 39. 600 (sec) Surfacs traptratur* vs. tias for hot wall hosting of planar half spaca, P ■ 500 W/ca*, o ■ .002 sxp(.0026T), a)T ■ 300K, b)T..t - 1300K. (C) 136 Surface Temp 1000 500 200 Time Plgurs 40. 400 600 (sec) Surfacs tsapsratur* vs. tla* for hot wall hsstlng of planar slab, T#>t ■ 1300K, o ■ .002 sxp(.0026T) a)P - 0, b)P - 100 W/ca*. c)P - 500 W/ca*. 137 can occur If this lsvsl Is too high. Flgurs 41 shows ths cross-sectional variation In temperature for ths saae slab, at various tlass. As long as ths teaperature of the saaple Is less than the background, the aaxlaua teaperature Is at the surface, since there Is a net flow Qf heat Into the saaple. Microwave heating Is voluaetrlc and yields a central teaperature aaxlaua. The hot wall applicator Bay prove particularly useful in sintering aaterlals whose low teaperature conductivity Is quite saall. As ths teaperature rises, through "hot wall” heating, the conductivity rises. Above a certain value, alcrowave heating will becoae possible at reasonable power levels. 138 (C) 1500 Temp 1000- Surface 2000 500- Ml t Distance(mm) Plguro 41. • l(k kMllkiry T Tooporoturo profiles In hot wall hosting of on 8oo plonor olob, o ■ .002 oxp(.0026T), K m 5 W/n-K, P - 500 W/co*, T„, - 1300K, 0)40 ooc, b)140 ooc, c)200 ooc, d)240 ooc. CHAPTER IV APPLICATOR SELECT IOil Microwave sintering la studied using a variety of applicator techniques at the Microwave Thereal Processing Laboratory, at Northwestern.[65] In this chapter we discuss the experlaental apparatus. This Includes the slab applicator, the rectangular applicator, the TEtll circular cavity applicator and the TM,,,,, circular cavity applicator. Por the rectangular applicator we convert the existing lapedance analysis Into a cavity equivalent circuit and exaalne theraal stability as a function of cavity paraaeters. Li Experalaental Apparatus The equipaent used In our alcrowave sintering experlaents Is shown In the slapllfled block diagram, Pig. 42. A aagnetron operating at 2.49 GHz, with a aaxlaua output power of 2000 H Illuminates a saaple within an applicator. Between the source and the applicator Is a circulator and a slotted line. The purpose of the circulator la to protect the alcrowave source froa large reflections. The power reflected froa the applicator Is directed to a calorimeter which measures the magnitude of 139 140 f*Mkuk ----------TEMPERATURE *-y PLOTTER STRIP-CMART RECOVER CONTROLLER A -i CALORIMETER POUR-PROBE STSTBI PYROMETER MICROWAVE T GENERATOR ♦ _L -H CIRCULATOR Flgurs 42. 3LOTT® WAVEGUIDE APPLICATOR Block dlagraa of aicrowava sintering apparatus. 141 the reflected power. The four-probe eyetea conelete of four dlodee, epeced at eighth-wavelength Intervale, that extend Into the waveguide. The combined output of the theee detectore yields the real and Imaginary parts of the reflection coefficient, which can be plotted on a Smith Chart. They reveal changes In applicator Impedance while sintering. A variety of applicators Is available. Typically a rectangular or circular resonant cavity Is used In order to couple energy to the saaple most efficiently. Saaple surface teaperature Is monitored by a pyrometer. The output of the pyrometer Is fed back to the magnetron In order to regulate the sintering temperature. 1. Slab Applicator The simplest applicator Is the slab applicator In TE,0 rectangular waveguide, and Is pictured In Pig. 43. This design places a slab sample of thickness "d" a quarter-wavelength In front of a short circuit where the electric field Is a maximum. The Input Impedance of the applicator la: follows: [19] - vufcd] where v ■ >(S) <” 0> - w*u( ♦ Jwiio 142 V V d r * slob rigur* 43. Slab applicator gaoaatry. \ i short 143 and "a" is tha width of ths wavsguids. Knowledge of the input lapedance is useful in deteraining the total power absorbed by the saaple. It does not, however, give the distribution of this power within the saaple. For the slab geoaetry this can be laportant, since the electric field distribution for the TEt0 aode in rectangular waveguide is proportional to slnf^P), it is zero at the waveguide walls and aaxlaua at the center. Because of this uneven field distribution, nonunlfora heating is observed. The principle advantage of this applicator is its simplicity, both in analysis and design (tuning not required). Its disadvantages are uneven field distribution and large saaple size if the operating frequency is 2450 MHz. h Kactmiular Applicator The next applicator we will discuss is the rectangular applicator, shown in Pig. 44. Here, the slab has been replaced by a rod so that the entire saaple is in the region of aaxlaua electric field. [4] The short circuit is now adjustable, and an adjustable iris (inductive) has been added to control coupling into the structure.[66] In 144 K T O C CUTOFF T\JBT tC N -O M T A C T I» e M F T Flgura 44. Ractangular applicator gaoaatry. 145 ganaral, tha applicator can ba rapraaantad by an aqulvalant Input lapadanca, whara tha rod la aodalad aa a T aqulvalant natwork. For low loas aaaplaa wa hava a TEIOn cavity. Wa praaant tha lapadanca daacrlptlon of tha applicator aa daacrlbad by Aranata [19] and latar show that for low loaa aaaplaa a parallal RLC circuit la an accaptabla alapllf1catIon. A variational mathod, daacrlbad by Schwlngar [67] waa uaad by Marcuvltz [68] to aodal wavagulda obataclaa, Including thin dlalactrlc roda. Aranata axtandad tha Marcuvltz approxlaatlon ualng hlghar ordar taraa, hla raaulta ara valid for thick roda and roda at raaonance. Tha rod la modalad aa a T aqulvalant natwork, Pig. 45, whara tha lapadancea ara In ganaral coaplax. Tha aquatlona that ralata tha circuit paramatera to tha cantarad dlalactrlc poat ara atatad balow. A flrat ordar (* x 1) approximation haa been uaad. [19] f 3 , ODD ► (131a) 146 Z 11 - Z 12 Z - Z 11 12 cz::.3—i—i---- v 12 rigura 45. T aqulvalant natwork for cylindrical rod. 147 g 11 It ' k 9 , ODD I - ![(¥)■ ♦ ( & ) ’ * W ] , fiJJflJYJa) - *JjB)YJa) ~[ TRT L «3,(®)J0(a) - ai0(fl)Jt («} J «*k*r (131b) where T - .5772 , a - kR , fl* - 6rV , k - w/c and R Is the radius of tha rod. When this rod Is placsd In the applicator, we aust Include the Iris and the short In ths equivalent circuit, as shown In Fig. 46. When the rod Is placed a sultlple of a quarter wavelength froa the iris, and ths short is ideal, the Input adalttance of the applicator Is <Z„ - Z^MZ,, ♦ Z„) ♦ J Z , , t a n f ^ O (132) Z„ * * - where Y, Is the adalttance of the Iris and lt Is ths distance between the rod and the short. The adalttance of 148 Z 11 - Z Z 12 11 - Z 12 IN XftXS o Plgure 46. *•> <---- Equivalent circuit for rectangular applicator. 149 •n inductlve iris, with no loos la derived by Marcuvltz. [68] For aaall aparturaa, w, this Is Ths rectangular applicator has sssn widest uss In our alcrowavs slntsrlng sfforts. 1* Circular Cavltv Applicators Ms consider two types of cylindrical cavity applicators. Ths TEtll and the TM0I0 cavities. The purpose of the cylindrical geoaetry Is to Increase the aaxlaua teaperature of ths saaple by using a theraally reflecting wall. [3] Ths aaxlaua saaple teaperature Is prlaarlly H alted by heat radiated froa the saaple which Increases as teaperature to the fourth power. By silver- plating ths walls of a cylindrical cavity, this radiated energy can be reflected back onto ths saaple reducing heat loss and thus Increasing saaple teaperature. The TEllt cavity Is preferred because of ease in tuning— ths end caps are adjusted to bring ths systsa Into resonance. However, there are soae disadvantages. One disadvantage of the TE,,, cavity Is field non-unlforalty. The electric field varies axially ( s ln ( ^) , where ISO d la tha cavity height) reaching a aaxlaua at tha cantar and la noraal to tha rod. Soaa unuaual axparlaantal raaulta hava baan obtalnad with tha TE,,, cavity. Whan haatlng SIC, tha hot zona algrataa to tha anda of tha cavity, rather than raaldlng In tha cantar Ilka S13W4 or A1Z0,. Thla aay Indicate that tha conductivity of SIC la high enough that haatlng occura at aagnatlc field maxima, reeultlng In IZR loaaaa, rather than at electric field maxima (oE2 loaaaa). Thla ahlft In hot zona doaa not appear In tha other appllcatora alnca aamplaa ara always placed In an electric field maximum. Tha TE(II cavity la relatively lnaeneltlve to the preaanca of email aaaplaa. A perturbation analyela [69] of dlelectrlcally-loaded cylindrical waveguide shows little change In tha propagation constant for thin samples, Fig. 47. In tha TM^ q cavity, tha electric field la z-dlrected, with radial variation. The electric field aaxlaua la at the cantar of tha cavity, analogous to tha TEI0 rectangular waveguide. Tha resonant frequency la a function of radius and tha cavity la tuned by partially Inserting sapphire tubas (low loss) to change tha cavity volume. Tha )TMo|o cavity la currently under development. The field distribution for both these cavities, 151 t-o I 1-0 4 Plgura 47. Propagation constant for sanpla-fIliad T*llt circular wavagulda, whara b/a la tha ratio of tha dlalactrlc radius to tha cavity radius. Adaptad froa [69] 152 neglecting the coupling aperture, can ba dataralnad by solving a straightforward boundary value problea — Appendix H. (Neglecting tha coupling aparture la Justified by tha close agraaaant between theoretical and exparlaental values of Q and resonant frequency.) We discuss the THolo cavity first, because Its solution Is slapler. In the eapty T ^ (0 cavity, we have an axial electric field and an azlauthal Magnetic field. They are given as follows: Et(r) - E0 J0(kr) (134a) H*(r) - JE0n J, (kr) (134b) where When the rod Is Inserted Into the cavity the field coaponents reaaln the saae, but their values must change to satisfy the additional boundary conditions at the rod/alr Interface. The fields In the two regions are given as follows: (135a) E.| - E0 J0(k,r) E„ - E# (J„(k,r) H., - J B , , ^ J.llt.r) Y0(k,r) (135b) (135c) 1S3 (13Bd) The propagation conatants k, and k2 ara In ganaral complex. Tha TEI(1 cavity, rig. 48, can ba traatad In a almllar aannar. For thin roda a parturbatlonal approach may ba uaad to datarmlna tha propagation conatant of tha partlally-fIliad cavity. A mora ganaral approach la the hybrid mode analyals. [31] It racognlzaa that In tha empty cavity wa have radial and azimuthal E-flelda and axial Hflelda; but whan tha aampla la praaant, tha boundary condltlona require an axial E-fleld aa wall. Although In moat caaaa thla component la weak, tha mode excited In tha aample-fIliad cavity la neither tranaverae electric nor tranaveraa magnetic. Appendix H. A complete derivation la given In In tha fundamental mode, tha electric field lnalde tha aampla la: J,<*„,) J c o m ■»» " « l ny 1J, (k..p) ]alne eln^f (136a) (136b) (136c) 154 iT U t •X • M tT M M IT COM » Vlgura 48. TBU1 applicator gaoaatry. *11 155 where A, and B, art given by (H-14) and (H-18) raspactlvaly. Hlghar sintering taapaaturea hava baan obtalnad axparlaantally In tha TBII( cavity than In tha rectangular applicator. [1] As expected, samples whose electrical conductivity Increases rapidly with frequency are still difficult to control. ft; [2 0 ] Thermal Stability In a Rectangular Cavltv Applicator In this section we use the expression relating Incident power to steady state teaperature In the Infinite cylinder (63) to study the thermal stability of a cylindrical rod In the rectangular applicator. It Is found that stability can be controlled to some degree by proper Iris selection. He first derive a cavity equivalent circuit and calculate the coupling coefficient at resonance to determine the absorbed power as a function of sample conductivity. This relationship Is then assimilated Into (63) to fora a modified expression for Incident power and steady state temeperature. It Is assumed that the cavity Is lossless, that is, all available power Is either reflected by the cavity or absorbed by the sample. Por thin rods, (128a and b) reduce to the equations developed by Mareuvltz, [6 8 ). The equivalent circuit Is re expressed in terms of normalized reactances, which may be 196 lbssy, and la shown In rig. 49. Tha approprlata ralatlonahlpa ara: r tjir 6l3J«)3l(6) - «J0U)J,(c«) (137a) z. - iz. • 5?: [ ( n* ” 2 2 ) " /I ■a] n-*.ODD irR (137b) a'JAb) JTta- y ^ r r J T T E r ^ W T T T T ^ y • 2 Figures 50-53 show the real and laaglnary parts of the complex lapsdancs as a function of conductivity, using (131), where Z. ■ R. ♦ JX. and Z* ■ R* + JX^. These figures show that for low conductivities, X, dominates, so we can neglect Z» and view the circuit as a shunt reactance with a series resistance, Fig. 54. For very high conductivities, the approximate solution for thin metallic posts given by Mareuvltz [6 8 ] Is obtained. The empty rectangular applicator is modeled as a resonant cavity using a parallel RLC circuit, while the Iris Is modeled as a transformer. This circuit Is shown in Fig. 157 JXb JXb -JX Figure 49. m Equivalent circuit of cylindrical rod using Mareuvltz nodal. 158 ’° 1 1 10 3 t 10 » M iiim '* i iiittbi~'i rrn wi 10 ’4 10 ■* 10 i 1 1 hi t * m u m ~ i r i m n 10 conductivity Figura 50. 1 10 i m i n n ~ i t mttoi 10* (S /m ) Variation of rod lapadanca, R., with conductivity. 10 3 159 5.0C 3 -1 o 4.0C i 3.0C 1 2.00 X - 3 3 j .oc ■3 3 T H 3 0.00 J 3 H 3 3 3 1.00 — I'TTIIIW i ’0 -» r» -« iiiiirnj 'i 11iirai TTTTTWr -a » 1 10 “* 1 10 “*m 10 111 inn conductivity Figura 51. iiiim i mini i i mint 10 . 10 2 10 (S /m ) Variation of rod lapodanco, X , with conductivity. 160 10 _Q -sj -7 -s -4 10 '3 10 10 ” conductivity Flgura 52. 1 10 (S /m ) Variation of rod lapadanca, R*, with conductivity. 161 0.005 -n 3 o.ooo 4 - 0.00 _Q X -0.015 - 0.020 4 10 '* 10 ' 4. 10 "* 10 ‘ * 10 ” 1 10 conductivity (S/m) Flgura 53. Variation of rod lapadanca, conductivity. with 10 * 10 5 162 Flgurs 54. Simplified circuit for low loss, thin rods. 163 95. The rod la lncludad by adding an additional shunt raactanca with sarlas loss. Tha ssrlss loss can bs convsrtsd to an squlvalsnt shunt loss whan tha quality factor, Q, of tha circuit la high. According to [70], tha new raslstanca la: R. X* " TT (138) Ha can now study tha cavity In teras of lapadanca: R, L and C, or by tha cavity paraaatars: rasonant frequency, Q and coupling coefficient. Tha reflection coefficient can be deteralned froa tha Input adalttance of tha cavity. (139) Tha circuit eleaents are related to tha cavity paraaatars as follows: z o (140a) (140b) (140c) where Qo represents tha quality factor for tha unloaded cavity with saaple present. Figure 55. Cavity equivalent circuit. 165 The input admittance of tha cavity axpraaaad in teraa of cavity paraaetere la: [71] and the reflection coefficient is [5] IP I* - 1° (0 - »)* ♦ (142) + !)■ ♦ ( 2 Q„Sgy Y - Y where p ■ y8 4. y'a 0 1n The normalized absorbed power is * »n« - 1 - ♦6 IPI1 (# «■ IT (143) ( * .« ) Maximum absorption occurs at resonance when the coupling coefficient is unity (critical coupling). Note that this model is valid for cylindrical cavities as well, since it is a general expression for resonance. Using (136a-c), the absorbed power is expressed in terms of circuit parameters: 166 4n»Va 1 - |p |* (144) - When the cavity la tunad to lta rasonant fraquancy, wa hava 1 - |p|* (145) - Since tha raalatanca la lnvaraaly proportional to tha conductivity of tha saaple wa hava (146a) where f(v) la tha noraallzed teaparatura dependent conductivity function and £ la an arbitrary conatant. Then, Coupling to the cavity can be controlled by changing the conatant £. Since f(0) - 1 , a value of £ ■ 1 defines critical coupling at v ■ 0. If f(0) la aonotonlcally increasing with teaparatura, the cavity will always be over 167 coupled. If we chooee 5 leee then one, the cevlty will becoee critically coupled when the eaaple reachee a certain teaparatura defined by f(v) - 1 /5 . Below thle teaparatura the cavity la undercoupled, and above thle teaparatura It le overcoupled. The effect of cavity coupling on theraal stability can be seen by aodlfylng (63). 2( c,v ♦ Cj ((v+ 1)4 - 1 )) He have 2(c,v ♦ Cj ((v+ 1 )* - 1)) f(v)[l - 1* 1-1 (1 , ~ Sf(v))z ,147) This expression Is valid when the following conditions are satisfied: 1 ) fields Inside the saaple are unlfora, there la no theraally induced skin effect; 2 )the saaple Is thin enough to slapllfy (131); 3) cavity Q Is greater than 10, allowing for the circuit transforaatlon; 4) the cavity Is aalntalned at resonance. Figure 56(a) Is a plot of noraallzed teaperature vs. power when the cavity Is critically coupled at rooa teaperature, where f(v) ■ exp(.78v), c, ■ .002 and cz ■ .00018. Figure 56(b) Is the power vs. teaperature plot for the cylinder In free space. Is aonotonlc. He note that heating In 53(a) However, It requires extreae power levels to Increase teaperature In the high teaperature region, because coupling Is weak. These results aatch experlaental 168 8.00 -| tem perature 6.00 4.00 - 2.00 ■ 0.00 0.000 0.010 0.020 0.030 0.040 0.050 power Figure 96. Normalized steady state surface teaperature vs. noraallzed applied power for cylindrical rod In rectangular applicator with f(v) * exp(.78v), c, ■ .0 0 2 , c, ■ .00016, a) critical coupling at 300K, b) free apace cylinder. 169 observations, when a small Iris Is used to begin heating a low loss alumina sample. The sample cannot be raised to the sintering temperature with this fixed iris, because the available power is limited. Figure 97(a) is a plot of the other extreme, where the coupling coefficient is chosen for critical coupling at 2000*C, somewhat above the sintering teaperature. Figure 57(b) is the reference plot of the free space cylinder. As sxpscted, low tsmpsrature hsatlng requires large power levels. As tsmpsrature rises, coupling Improves, accelsratlng the instability. In tha frse space cylinder, instability occurs nsar a noraallzed temperature of 4, while for the undercoupled cavity instability occurs much sarller, nsar a noraallzed teaperature of 1.4. observation confirms this effect. Experimental When a large iris is chosen, high power is rsqulred for initial heating and high temperatures are impossible to control. An intermediate iris may be selected, as shown in Fig. 58(a), where critical coupling occurs at 1000°C. Figure 58(b) is the free space cylinder as defined earlier. The compromise iris shows soms difficulty in initial hsatlng, but aonotonlc behavior at high temperatures. Intsrasdlats tsapsraturss ars difficult, but not impossible to control. Experimental observations using alumina rods showed that thsrs is an optimum iris size. It must be 170 8 .0 0 -i tem perature 6.00 - 4.00 2.00 0 .0 0 - f - r 0.000 0.020 0.040 0.060 power Flgura 57. Noraallzad staady stats surftea taaparatura vs. noraallzad appllsd powar for cylindrical rod In rsctangular applicator with f(v) ■ sxp(.78v), c, ■ .0 0 2 , ct ■ .00018, a) critical coupling at 2300K, b) frss apaca cyllndar. 171 8 .0 0 -i tem perature 6.00 - 4.00 - 2.00 0.00 -Hr0 .000 0.010 0.020 0.030 0.040 power Figure 58. Noraallzed steady stats surfacs tsapsraturs vs. noraallzad applied power for cylindrical rod In rectangular applicator with f(v) ■ exp(.78v), c, ■ .0 0 2 , c, ■ .00018, a) critical coupling at 1300K, b) free epace cylinder. 172 large enough to peralt high teaperature heating at reaeonable power levels, but saall enough to provide high teaperature stability. A good guess would place the critical coupling aldway between rooa teaperature and the sintering teaperature. Initial heating Is enhanced by Increasing coupling, while theraal runaway at high teaperatures Is controlled by decreasing coupling. An adjustable lrle Is valuable In dynaalcally controlling coupling. However, for low loss saaples, like alualna, a fixed circular Iris Is necessary to reduce arcing. Figure 59(a-c) shows the coupling as a function of teaperature for the three Irises used In Figs. 56-58. Calculations show that the Q of the cavity Is 1000when the conductivity Is l.E-5 S/a, for a .5 ca dlaaeter rod. The drops with teaperature as the conductivity rises. corresponds to a conductivity of .001 S/a. Q A Q of 10 Using the exponential law, this corresponds to a noraallzed teaperature of 5.9, or 2070 K. Experlaental observations show that tuning Is required during low teaeperature heating of low loss saaples As the teaperature rises, the resonant frequency decreases. At high teaperatures, the Q Is sufficiently dlalnlshed to spread the resonant spectrua, thus cavity tuning Is not required. This chapter shows the laportance of lrle eelectlon on etable heating. It aay be that the Iris which aaxlalzes 173 power 1.00 0.80 absorbed 0.60 0.20 0.00 0.00 2.00 4.00 6.00 8.00 10.00 tem perature Figure 59. Percent power absorbed ve. noraallzed teaparatura for different coupling arrageaente, a)solld, 300K, b )daeh, 1300K, c)dot, 2300K. 174 efficient energy transfer say not be the beet choice for stable heating. 41 CHAPTER V MATERIAL CHARACTERIZATION Knowledge of the teaperature dependence of electrical conductivity la important in determining the thermal atablllty of the syetem. In this chapter we dlacuas methods of material characterization. First we consider low temperature measurements (0 - 200°C) using the Automatic Material Characterization Equipment (AMCE). A discussion of measurement range and sources of error is included. We next consider high temperature characterization methods. Of particular interest is an in situ characterization method derived from Chapter III, where measurements of applied power and surface temperature are used to determine conductivity. A. Automatic Material Characterization Equipment A scalar network analyzer has been developed to measure permittivity and conductivity over a wide frequency range (2-lSGHz) with temperature control from (0 - 200°C) . [27,28] It is useful in determlng initial values for materials to be sintered. We describe the apparatus and discuss measurement limitations arising from Maxwell's Equations, poorly fitting saaples and component imperfections. 178 176 i_i The Apptrtttt A simplified block diagram la presented In Fig. 60. A alcrowsve source, HP8390B Sweep Oscillator and accesorles, sampled by the Incident alcrowave bridge, Narda Model 5082, excites a saaple-f1 1 led segaent of coaxial line enclosed In a temperature-controlled chamber. A second bridge Is used to sample the reflected signal, while the transmitted signal Is measured directly. The saaple Is padded to minimize undeslred multiple relfectlons. The use of an Internally terminated alcrowave swlth enables the computer to selectively measure the transmitted or reflected signals. The source Is modulated at 1 kHz and the detector diodes fsed directly to narrow-band amplifiers. The computer measures the signals at each of the detectors and controls frequency and temperature. The saaple holder la an ordinary APC-7 connector which can been modified by undercutting the Inner and outer conductors to support the samples. shown In Fig. 61. It Is Care Is needed to prepare tightly fitting saaples with parallel front and back surfaces. Tsmpsrature Is controlled using a two thermocouple method. The oven controller uses an lntsrnal set point (set by the computer) and a direct thermocouple feedback to achieve a steady state oven temperature. A second thermocouple Is ussd by ths coaputsr to monitor the oven teaperature. The computer can alter the set point to m ic ro w a v e SOURCE (U M t SO WAVE) M l M l ■**/“ NARROW IAND AM RUFIERS (IkNtl OVM Flour# 60. Block diagram of Automatic Hatarial Charactarization Equlpaant (AHCE) 178 ALL DIMENSIONS Figure 61. ARE IN MILLIMETERS AMCK Sample holder, ehowlng a) end view and b) croea-eectlonal view of the modified APC-7 connector. 179 enhance the heating rate. L Theory The reflection and transmission coefficients are derived fros a total wave analysis [54] and are given In terse of complex permittivity P m .... (1 - £r i*) sinhYd . (€r* 1)sinhYd ♦ 2j€^Tcoshvd (£,* ♦ l)sinhYd ♦ 2ji7TcoshYd (148ft) (148b) where free space permeability is assumed. An Impedance change in the coaxial line is produced by undercutting the sample holder, thus modifying (148a,b). The corrected expressions are - e-~) mlnhyA (€r* ♦ kMsinhYd ------- 2k,|i/coshYd 2kJ?7 ((,* ♦ k*)sinhYd ♦ 2kJ*7Tco*hYd where (149b) iso k r H " The numerator la tha Impedance of the undercut eectlon. In the ecalar network analyzer, we measure the magnitudes of the reflection and transmission coefficients. Consequently, the complex permittivity must be derived numerically. Initially we utilized the Newton-Raphaon method [47] to analyze the data at each frequency. However, small measurement errors can produce wild variations In the calculated results. It Is preferred therefore to use the large frequency range of the Instrument to smooth the data. Flrat, the measured reflection and transmission coefficients are plotted as a function of frequency. He then calculate the reflection and transmission of an Ideal material with assumed complex permlttltlvlty and superimpose this data onto the plot of the measured data. A good fit Is determined visually. b. Results He present data obtained from measurements of teflon, Stycast H1K, and silicon at room temperature from 2-18 GHz. These results demonstrate the ability of the Instrument to measure the constitutive parameters of low and high 181 permittivity, and low to medium conductivity eamples In the coaxial line frequency band. Typical published data for teflon Indicates that at microwave frequencies, the dielectric constant la 2.08 and the conductivity la .0004 S/cm [72]. Data taken on a 4.56 mm teflon sample showed an average dielectric constant of 2.11 over the range 4-14 (3Hz. The transmission coefficient was very large, on the order of 0.5dB, Indicating low loss. The haIf-wavelength resonance of this sample occurs at 22.9 GHz which Is beyond the range of the Instrument. The quarter-wavelength anti-resonance occurs at 11.45 GHz and corresponds to the region of high uncertainty In dielectric constant. Next we consider a high permittivity, low loss sample. He used Stycast H1K, with a nominal permittivity of 16. The curved surfaces of the sample were coated with Heraeus 5450 conducting paste (a silver-filled polymer) to fill gap regions between the sample and the walls of the holder, as substantial error can be Introduced by poorly fitting samples. Figure 62 shows the measured reflection and transmission coefficient of a 4.43 mm sample In dB as a function of frsqusncy. Two resonance points are observed, the first at 8.48 GHz and the second at 16.93 GHz representing the half-wavelength and one-wavelength resonance points, respectively. Calculated from the half- 182 •• •• o/ -10 0 1 i o I -15 20 20 FREQUENCY* GH* figura 62. Raflactlon and transmission v s . fraquancy for Styeaat H1K, o - la raflactlon data, □ - la transmission data. 163 wavelength reeonant point, the dielectric conetant of the Material la 16.0. Using the foraula for the Q at reeonance, the conductivity of the saaple la .005 S/ca. The eaooth curve euperlapoeed on the data polnte le calculated froa a eaaple of dielectric conetant of 16 and conductivity of .005 S/ca. Good agreeaent le eeen over the entire frequency band. As a final evaluation we perforaed aeasureaents on silicon. used. A p-type, (111) oriented single crystal saaple was DC aeasureaents were aade by the four point probe aethod, and a DC conductivity of .017 S/ca was Measured. Silicon has a peralttlvlty [73] of 11.8. The contacting surfaces of the saaple were coated with conducting paste. Figure 63 shows reflection and transalsslon coefficients In dB plotted versus frequency; thickness is 4.50 aa. this case Is noticeably low. The Q In At the point of resonance, the data Indicates a peralttlvlty of 11.8 and a conductivity of .02 S/ca. However, the curve for a peralttlvlty of 11.8 and conductivity of .017 S/ca seeas to fit the overall data better, and Is superlaposed on the graph. It should be noted that due to the low Q, the calculated conductivity based on resonance Is only an estlaate of the true conductivity. 184 *• Og oo - 46 - -10 -15 •20 -20 2 4 C I 10 12 14 FREQUENCY - CHi Vlgura 63. Raflactlon and transmission vs. frsqusncy for silicon, o - 0.017 S/ca. 0 - la rsflsctlon data, □ - la transalsalon data. 18S 2. Measurement Range and Sou r c e of Error Three factors contribute to measurement Inaccuracies. First, there Is a fundamental limitation Imposed by Maxwell's Equations — when conduction or displacement current dominates, measurement of the other quantity Is obscured. Second, small gaps between the sample and walls of the coaxial line produce errors In high permittivity and high conductivity materials. He describe a theoretical model for the gap and determine that gaps must be eliminated to acquire useful data. Third, Imperfect components Introduce multiple reflections and degrade the measurements. A signal flow graph analysis la used to describe these errors. ib Llmltatlonm Eased on Maxwell'a Equations In this section we determine the physical limits on accurate measurements and provide criteria for the selection of a suitable saaple thickness. He use Maxwell's Equations to develop the theoretical boundaries on the measurement of conductivity and permittivity. know that In the steady state, From these equations, we 186 7XH - 0 E ♦ Jw(,(0E (150a) We note that when the displacement current dominates there la high uncertainty In the conduction current, and conversely, when the conduction current doalnates, the displacement current Is uncertain. By applying this concept to the present systea, the physical limitations on the range of measurable material parametere may be discerned. Theee conditions guide selection of proper sample thickness. In terms of complex permittivity, (150a) becomes: 7XH - Jw£0tr'E - (150b) Assuming fixed dielectric constant, the prior Inequalities can be expressed In terms of two limiting cases: *nd -» 0 “* 00 • The first limit is obtained by either decreasing the conductivity or Increasing the frequency. Since the maximum frequency Is restricted to the microwave frequency range, only the diminishing conductivity limit will be considered. From (148) we see that: 187 to* \p\ V 1 *ln * * - --------------B-------------- , to* Ir I - £ (13la,b) where D - 1 ♦ ' S ; / 1' ^JT) and thus, |P|2 ♦ |t|* - 1 (152) The second Halting condition Is aet by either decreasing the frequency or by increasing the conductivity. We consider each case separately. In the low frequency llalt the following results are obtained. ton p - ------------------------------------------------ (153a) #-•0 Om T - t ^ ,153b) and thus, p + t ■ 1. (154) 188 Keeping the frequency fixed end taking the H a l t as the epnductlvlty goes to Infinity, we obtain, for large conductivity: Ir | - NI T • - N-lT (155a) |t | - 0, (155b,c) In the Unit, |p| - l, £*t and thus, iPl ♦ |r| - 1 (156) Equation (156) can be used to describe the relationship between reflection and transmission In either the low frequency or the high conductivity case. Equations (152) and (156)describe the boundaries that restrict the measurenent range of the Instrument. He note that (151)-(152) describe a region of high uncertainty In conductivity and (153)-(156) describe regions of high uncertainty in dielectric constant. These limits are shown In Fig. 64, where (152) and (156) are plotted as a function 189 1.0 04 1.0 ♦ €,• 12 6 • 0.3mm 0.5 10’* 10*' CONDUCTIVITY - S/cm Figure 64. Su b of aagnitudas and s u b of squares of Bagnltudes of reflection and tranoBlaalon va. conductivity. 190 of conductivity for a fixed frequency. In the low conductivity region, the sue of the equaree approachee unity, (192). In order to eatlafy thle Halt, the reflection and transalsslon euat be Independent of conductivity, (191), and thle lepllee that dlaplaceeent current dominates the conduction current and there la uncertainty In the conductivity. conductivity region, the (196). bub Llkewlae, In the high of aagnltudes approachee unity, Thle U n i t le net only when the reflection and tranemlealon are Independent of dielectric constant, (193) and (199), and lnplles that conduction current donlnates the dlsplacenent current and thus high uncertainty appears in the dielectric constant. Therefore we conclude that accurate meaaureaents of both dielectric constant and conductivity are confined to a Halted range of conductivities. These results are then used to deteralne the optlaua thickness. He view the saaple In three reglaes: the aultlply reflecting saaple, (191)-(192), the high ineertlon loss saaple, (199)-(196) and the electrically thin saaple, (193)-(194), In each region, we will see that there Is a different criteria for selecting the proper saaple thlckneee. In the aultlply reflecting caee, where the conductivity Is sufficiently saall, and the saaple is sufficiently long, 191 aultlple reflections between the front and back surface of the saaple produce resonant and antl-resonant behavior. This resonant behavior can be predicted fros the sinusoidal nature of (148) when the peralttlvlty Is purely real. At resonance (the half wavelength saaple) the reflection Is a alnlaua and the transalsslon reaches a aaxlaua. Anti- resonance Is observed when the reflection Is a aaxlaua and the transalsslon Is a alnlaua (the quarter wavelength saaple). The haIf-wavelength region Is ths aost sensitive to changes In dielectric constant, whlls the quarter wavelength region Is least sensitive. The saaple thickness required to observe resonance at a particular frequency Is: d - j fjf. (157) and we note that an Increase of the dielectric constant of the aaterlal, results In a lower resonant frequency. In Fig. 65, where reflection and transalsslon coefficients are plotted as a function of dielectric constant at a fixed frequency, conductivity and thickness, the sensitivity of the resonant and antl-resonant regions Is deaonstrated. To aeasure dielectric constant aost accuratsly, the halfwavelength thickness should be chosen. As ths dsslred frequsncy Is decreased, this thickness would be Increased, Halted only by the physical length of the eaaple holder. Note that the resonant behavior of the aaterlal REFLECTION COEFFICIENT MAGNITUDE - \ p\ 192 DIELECTRIC CONSTANT Vlgura 6 8 . Raflactlon coafflclant vs. dlslsctric constant for flxad fraqusncy and conductivity. 193 provides an altarnata way of measuring tha alactrlcal propartlas of tha saapla. Tha aaapla can now ba vlawad aa a tranaalaslon cavity whara tha raaonant fraquancy la ralatad to the dielectric constant and tha bandwidth or Q, la related to tha conductivity (Appendix I). Tha unloaded Q for any homogeneous cavity, accounting for dielectric loasas only, la [74]: Q - € (158) This Independent eolutlon can ba uaad to give tha first guess for tha nuaerlcal analysis. As tha conductivity Increases, resonant behavior diminishes. Whan tha conductivity is sufficiently large, resonance disappears entirely and this defines the onset of the high Insertion loss regime. Here, the determination of conductivity is very dependent upon the thickness of the sample. The maximum measurable conductivity Is determined by the Insertion loss of the sample, the power available from the source, and the noise level of the detector. This Is Illustrated In Fig. 6 6 , where the magnitudes of the transmission coefficients of a thick saaple (a) and a thin sample (b) are shown as a function of conductivity. For the thick sample (a), at a critical conductivity, 0c* the transmitted signal would not be observed because It Is below the noise level of the detector. Thinning the sample, (b) Flgur* TRANSMISSION COEFFICIENT MAGNITUDE 66. |t|- 46 Transmission 8 N o o vs. conductivity o o z o c o oI M •s. 194 for s) d • 4.56ss, O. 199 brings ths signal Into obssrvatlon. However, for a saapls of lower conductivity, still In ths high lnssrtlon loss rsglae, ths thicker saapls Is a bsttsr cholcs, bscauss ths transalsslon cosfflclsnt Is aors ssnsltlvs to changss In conductivity. This Is sasn by coaparlng ths slopss of ths transalsslon cosfflclsnts at conductlvltlss below oe. This Is also sssn In (155) where ms nots that for high conductlvltlss: log |t | ~ -oid (159) Therefore, the thinner saaple Is aore useful for aeasurlng high conductivities, as long as we are above the noise threshold, while the thicker saaple Is better for aeasurlng low conductivities. A saaple Is teraed to be In the thin saaple region when Its thickness Is less than a quarter-wavelength. As the frequency Is lowered, or as the thickness of the saaple becoaes a saaller fraction of a wavelength, we approach a Halting condition— (153) and (154), where the reflection and transalsslon are deteralned solely by the conductivity. This result Is Illustrated In Figs. 67(a) and (b) where the reflection and transalsslon of saaples of given thickness and peralttlvlty but different conductivities, ars plotted versus frequsncy. At very low frequencies, the reflection $4 o %4 > o c • s » S TRANSMISSION COEFFICIENT MAGNITUDE-lT l O® 2 1 10 TRANSMISSION COEFFICIENT M A G N lT U O E -|r| u %* m > ■ O e s o 01 • H • « ■H ■ • a £ o >o z w 3 o m m a ■o ■ e o 3s M c oH •H o *• «/> b» O \d • u • • H %* o • M • m O I- aonilNOVN IN3QliJ3O0 NOI133U3U O n O l^l-aonilNOVN 1N3ICMJJ300 NOU03U3U « b and transalsslon bscoas constant In both flgurss. To asasurs peralttlvlty, however, the aultlply reflecting region of Fig 67(a) can be aoved to the left by Increasing the thickness of the saaple The desired thickness of the saaple can be estlaated by (157). However, If the conductivity Is so high that no aultlply reflecting region can be observed, Fig. 67(b), Increasing the thickness would only weakly laprove the aeasureaent of peralttlvlty. The highest frequency at which aeasureaents can be aade Is deteralned by the onset of higher order nodes. Higher order nodes can propagate In a aaaple-f1 1 led region of line at lower frequencies than In an enpty line. However, we have found no evidence of excitation of higher order aodes as long as the saaple front and back surfaces are sufficiently plane parallel. Therefore It Is satisfactory to deternlns ths upper bound on aeasureaent frequency fron the cutoff frequency of the enpty line. To suaaarlze our findings on optlaua saaple thickness, we outline the following procedure. Prepare a saaple of nodest thickness and take a aeasureaent. If weak resonance or anti-resonance Is observed, an approxlaatlon of the dielectric constant can be aade. If the transalsslon drops below the noise level of the detector at the higher frequencies, a thinner saaple aust be used to deteralne the conductivity. If the reflection and transalsslon are 198 constant at the lower frequencies, the saaple is too thin. These results can then be used to prepare a new saaple whose thickness Is aore appropriate for characterization. In general, the dielectric constant Is aost accurately found by considering a saaple whose thickness produces resonance at the desired frequency. Accurate conductivity Is found by choosing thicknesses bounded by aaxlaua aeasurable insertion loss on the high end and Insensitivity on the low end. Note that the analysis of optlaua saaple thickness Is based on physical principles and thus can be applied to a vector lnstruaent as well. k:__ Q»p Effects Another source of Inaccuracies In aeasurlng aaterlal paraaeters In a coaxial line systea is the presence of narrow gaps between the saaple and the aetal walls.[29] We have found large discrepancies bstween neaaured and known values for aaterlals that should be within aeasureaent range. We first review prior work on this topic, presenting the capacitance aodel. We develop a aore rigorous analysis using the transverse resonance nethod, and finally we consider the effect of filling the gap with a lossy aatsrlal. The results show aarked laprovsaent over a calculable range. 199 The contact problem has been coneldered in the literature, for various waveguide geometries [79,76,77,78,79] and In coaxial line at fixed frequency [80.81]. There are three basic approaches to this problem. The first reduces the gap by applying pressure to the saaple hoder, forcing the walls Into contact with the saaple [75.80.81] A second technique Is to use a TE0l-mode circular waveguide, where the electric field Is small near the waveguide walls.[77] The final method Is to derive a correction formula for the gap so that aeasureaents may be adjusted to yield the proper result. The aost elementary approximation Is a perturbation analysis.[82] Next Is the capacitance model, [83] and the aost advanced uses an expansion of modes near the Interface.[84] An Illustration of the gap effect Is shown In Pig. 6 8 , where observed peralttlvlty is plotted vs. gap thicknesses. true peralttlvlty for different The consensus Is that a correction formula Is only valid for very small gaps. A large gap obscures the measurement and must be eliminated. A pressure method Is Impractical In a coaxial geometry, since It Is difficult to apply outward pressure on the Inner conductor. The second method, using a ?E01 circular waveguide, Is not viable at low frequencies because the physical dimensions are large. The third approach, a correction formula, is considered in more detail. Observed dielectric c o n s ta n t 200 100.00 0.01 S/ cm 0.0 80.00 60.00 40.00 20.00 i.OO True d ie le c tric c o n s ta n t rigur* 6 8 . Obssrvsd dialactrlc constant vs. trus dlslsctrlc constant for an air gap of a).l ua, b)l ua, c) 10ua, d )100 ua 1. Capacitance Modal The distributed cepecltence model (DCM), first proposed by Westphal [83], views the section of line containing the saaple as an equivalent lnhoaogeneous coaxial transalsslon line, Fig. 69a, where r, and r„ are the radii of the Inner and outer conductors, and r,. and r„ are the Inner and outer surfaces of the sample, Fig. 69b. In the absence of a gap, the Inductance and capacitance per unit length of a saaple filled line Is: (160a) L ■ oi ■ L ■ & * ($ ? ) 2nf * When the effect of the gap Is Included, the measured capacitance Is: (160b) (160c) 202 COAXIAL LINE Figur* «*. O«o»«try of tho g»P- 203 «nd L, - *»(??) . L, - * t ( ^ ) . L, - L, - L,. The equivalent clrculte corresponding to Figs. 69a and 69b are shown In Figs. 70s and 70b. The aeasured peralttlvlty, €., can be expressed In teras of the saaple peralttlvlty, 6., and the gap peralttlvlty, €,: €t% - re rt J (1«2 ) L,€ ♦ L,€ 1 r« 1 r| Separating (162) Into real and laaglnary parts, we have £r. - £r £ L, ' € L,(l + tan*6 ) ♦ €r L {1 + tan*6 ) — ----- ;--- =-----“ v --------- H -- Z ♦ L2£rttan*6f) £r.L tanfi (1 ♦ tan26.) ♦ er.L_tan6 (1 +• tan26.) tanfi ------------------ H ------*—1-----;--- *— " « , . M l ♦ tan26.) ♦ £,,1^(1 «■ tan26t) where tan6 ■ (163a) (163b) ' ' 204 1• Figure 70. Equivalent circuit for tha gap ualng a distributed capacitance model. 205 Note that for a lossless saspls, DCM is lndspsndant of frequency. In Pig. 68 the seasured permittivity Is plotted as a function of sasple permittivity for different gap thicknesses. The presence of a gap places an upper limit on the observed permittivity. This limiting value Is: - €r, ^ (164) In Fig. 71, the measured conductivity Is plotted as a function of sample conductivity. 0.01 Note that even a gap of microns can cause erroneous measurements for samples with conductivities greater than 100 S/cm. Using (15a) and (15b), the observed permittivity and dielectric constant and conductivity are plotted as a function of frequency In Figs. 72 and 73 respectively. The permittivity of the sample Is 1 2 , and the conductivity Is .01 S/cm. The gap does not Influence the measurement above 1 QHz when It la less than 10 microns, since the loss tangent decreases with frequency and the Influence of the gap Is diminished. At this time there Is no criteria to establish the size of the gap. In general, for tests over a large temperature range, gaps increase with Increasing temperature. ( S /c m ) 206 12.0 0.0 0.1 gap Observed conductivity 2 GHz . ....W . ■ ....Ml 10 ■* ■ . ...uJ . 1 0 * i p True c o n d u c tiv ity ( S /c m ) Figure 71. Oba.rved conductivity vs. true conductivity for an air gap of a ).01 im, b).l iia, c)l iia, d )10 Hi, a )100 iia. 3 •t w Observed d ie le c tric c o n s ta n t O O’ • • ►* H S *1 TTfl < <Q O. m •a a o • Hi M • • O — r* •1 O HH O a o P o 5■ ° o ;s ~•-*srt 55 o* -Ull s i 207 Figure 73. TW Observed Observed c o n d u c tiv ity ( S /c m ) conductivity «P v s . frequency 208 for an sir 209 T r rtirr avw N to n w c t Model The accuracy of DCM for aaaplea with high loaa tangents aay be questionable. In fact, In the H a l t of Infinite conductivity, the samples aay be viewed as a noncontacting short circuit. „The noncontacting short has been extensively Investigated by Huggins In [8 8 ], where the gap regions are aodeled as narrow transmission lines. A model which Is valid for both high and low conductivity samples Is needed. The transverse resonance method (TRM) Is used to derive such a model.[8 6 ] The cross-section of the partially-fllled coaxial line Is viewed as a radial transmission line. TRM Is used to determine the propagation constant for the dominant mode In the lnhomogeneous coaxial line. For simplicity, we consider only a single gap, between the sample and the outer conductor. From a reference plane at the boundary between the sample and the gap we establish an equivalent network of two short-circuited radial transmission lines of different characteristic admittances, Y k and Y2. The geometry and equivalent network are shown In Fig. 74. Resonance in the transverse plane Is determined by equating the two admittances at the gap sample boundary. equal, the system will propagate. [8 6 ] When they are 210 k «1 kca Yi(r ) Ya (r ) EQUIVALENT NETWORK Figure 74. Equivalent network for Traneveree Reaonance Model (TRM). 211 Ytct(k„r„,kelr() - Ylct(k.ir...kelr0) where £ ' - ^ * £r| , ke, k..£ ** r I (168) , keI «* The redial cotangent function ct(x,y) la defined aa * ct(x,Y) " j i (*>y q (Y> - Yi(*) Ja(y) J0 (x)Y0 (y) - Y0 (x)J0 (y) • (166) When the eaaple Is lossy, kel, ke2 and the radial cotangent are conplex. The usual separation relations apply. k* - y* « S*^£ *.i , it-1• - / _< el C r* «* - 5* _• C r* * (167a,b) a The propagation constant, y, Is > - Solving for the Measured permittivity, £/, using (17,19 and 2 0 ) we have (168) 212 (169) Proa this general expression, two special cases are exaalned, low and high conductivity. When sample conductivity Is low, and the gap Is narrow, the approximations for Bessel functions of small arguments are valid. As a consequence, (169) becomes (170a) where A ■ (170b) When the gaps are sufficiently narrow, or the frequency Is sufficiently low, A ■ 1. When A - 1, (170a) reduces to DCM dsscrlbed In ths previous section. The second limiting case of TRM Is highly conducting 213 saaples. that art insulated froa the walls of the coaxial line by a narrow gap. It is necessary to first aodlfy the equivalent network that represents the radial transalssion line systea. He note that if the conductivity of the saaple is high enough, reflections froa a radial short circuit will be unobservable because ofthe high saaple a consequence, the saaple, attenuation. As viewed froa the inner gap, represents an unbounded, highly attenuating radial guide. The saaple, viewed froa the outer gap,coapletely fills inner region, that is, the the presence of the inner conductor does not effect the calculation. The result of this aodlflcatlon is that the radial cotangent functions appear in a aodlfled fora. The corrected expression for an inner gap is (171a) and for an outer gap ct (k^r.. ,k^r, ) ■ (171b) The result is that wave propagation occurs in the gap region, while the signal is attenuated in the saaple. If the saaple conductivity is high enough, the gap regions can 214 be modeled as TSM llnss. problem. This Is ths non-contacting short It Is lntsrsstlng also to nots that for largs radii and narrow gaps, ths sxprssslon for ths propagation constant is equivalent to that of a parallel plans guide [87], since radial tangents becoae tangents in the H a l t of e large radii. We compare the two aodels, TRM and DCM in Pigs. 75 and 76. Figure 75 shows observed conductivity vs. true conductivity as a function of frequency, where (a) is TRM and (b) is DCM. value of 12. The peralttlvty is assuned to be a constant There are two branches to the transverse resonance solution, one of which is spurious. The other is in close agreeaent with DCM. Figure 76 plots observed peralttlvlty vs. true peralttlvlty with (a) and (b) as described above. The conductivity is assumed to be a constant value of 0.01 S/cm. agrees with DCM. Again, the correct branch A computer program for TRM analysis is contained in Appendix I . TRM describes the effect of the gap regardless of saaple conductivity. In the low conductivity limit, and when the gaps are narrow, DCM is valid. ilLx Tschniquas for Removing Gaps We now discuss means of overcoming ths effect of the gap, using DCM. Accordingly, we note that in order to 215 in «, - 0.1 12.0 9, m 0.0 •i - 1-0 , outer gop “ SO f • 2 GHz . i •5 o 3 •d tS o a •d a> 0.01 0.001 r & eapocitonce ^ model \ v O o.oooi ~ ■ 0.0001 Figure 75. ■ *■■■■«•! 0.001 0.01 0.1 ■ ■■■!■< 1 ■ . I 10 \..«1 • 100 True c o n d u c tiv ity ( S /c m ) Oboorvod conductivity v « . truo conductivity for • ) TRM nodol, b)cop«cit*nco nodol. Observed conductivity ( S /c m ) 216 0.1 0.01 copocttonc* modal 0.001 o, - 0.01 S /c m - 12.0 0.0001 o. - 0.0 S ■ 1-° outer gap 0.01 Figure 76. — SO pm illll 0.1 1 10 fre q u e n c y (GHz) 100 Obttrvvd conductivity ve. frequency for a)TRM nodel, b) capacitance aodal. reduce the effect of the gap, the gap capacitance auet be auch larger than the saaple capacitance, or In teras of reactance, (172) This can be accoapllshed In two ways: the first aethod seeks to reduce the gap width to zero, and hence 1/C# ■ 0. In the coaxial systeas, this option Is open only to samples such as liquids, powders or "soft" solids, where the saaple can be presssed Into the saaple holder. An alterantlve approach Is filling the gap with a known aaterlal. We assuae that the gap Is saall, that is, (173) If we assuae that the gap Is coaposed of a material whose conductivity Is greater than that of the saaple, then (163a) and (163b) are Cf.L. L. r• (174a) tsn6a - tan6. ♦ tanfif (174b) for tan 6t >> 1 , tanim - tan&. (174c) 218 So we ••• that whan tha conductivity of tha gap la graatar than tha saaple aaaaurad valuaa thaoratlcally match trua valuaa. Two tachnlquas ara usad for filling tha gap. First, a conducting paata la palntad on tha curvad aurfacas of tha aaapla. Tha aaapla la lnaartad In the aaapla holder and tha paata la allowed to dry. Tha hlghaat maasurabla aaapla conductivity la H al t ed by tha conductivity of tha paste. Tha second technique for achieving a higher conductivity gap entailed electroplating tha saaple. Tachnlquas for metallizing plastics [8 8 ] are usad. A thin layer of silver Is deposited on the saaple surface using an electroless method, than tha sample Is electroplated with copper. The details of the process are described In Appendix K. After electroplating Is complete, the saaple Is sanded to remove excess metal froa the flat surfaces and to smooth the curved surfaces for a snug fit Into the sample holder. If the saaple dimensions are the same as the standard 7aa coaxial line, and the electroplating Is thick enough to fit the undercut saaple holder, we can eliminate the Impedance step caused by the undercut. In more accurate measurements. This should result 219 l£i intrlHntt 1 We M i au r «d the characteristics of thrse known saaplss to deteralne study ths gap offsets. Measureaents wers aads on Stycast H1K, and on two silicon saaplss of asdlua and high conductivity. Ws consider results for both an air gap and a filled gap. The Stycast H1K aaterlal was 4.43aa thick with a nonInal peralttlvlty of 16. Figure 77 Is a graph with three different plots of the aagnltude of the reflection coefficient In dB as a function of frequency: (a) the theoretical results (solid curve), (b) the experlaental results for the filled gap (circled points), and (c) the experlaental results with an air gap (dashed curve). The theoretical curve Is based on a dielectric constant of 16 and a conductivity of .005 S/ca, and the gap filling aaterlal Is Heraeus 5450 conducting paste. Figure 77(a,b) Is a reproduction of Fig. 62. In coaparlson, the plot of the reflection coefficient with the air gap Is characterized by a coaparatively large aaount of Irregularities, and the resonance Is shifted In frequency. The half-wavelength calculation using (157) yields a false peralttlvlty of 10.4. The capacitance aodel indicates that an air gap of 10 alcrons would produce this shift In dielectric constant. Figure 78(a) and (b) Is a reproduction of Fig. 63, and Fig. 78(c) Is the erroneous data acquired In the presence 220 m a g n itu d e(p ) (dB) 0.00 -3.00 - 10.00 Stycast HiK a » 0.003 S /c m 00 fre q u e n c y (GHz) Plgurs 77. Ths sffsct of air gaps on tha aaasuraaant of Stycaat HIK; a) dash, aaasuraaant with gap, b) solid, aaasuraaant with flllad gap. 221 m a g n itu d e (p ) (dB) o.co •V 20.00 silicon l.OO fre q u e n c y (GHz) Vlgura 78. Tha affsct of air gaps on tha aaasuraaant of .017 S/ca silicon; a)dash, aaasuraaant with gap, b) solid, aaasuraaant with fIliad gap. 222 of a gap. Tha faulty data la again charactarlzad by a shift in rasonanca to a highar fraquancy and by graatar irragularitias in tha rasponaa. For high conductivity samples, tha conductive pasta cannot ba usad, because its porosity H al t s its conductivity. Instead, we auat electroplate tha saaple. Meaaureaents ware perforaed on p-type silicon. DC aeasureaents of tha 3.6aa saaple showed a conductivity of .28 S/ca. A aetalllzed saaple was prepared and tha results are shown in Fig. 79. Tha circled points show tha aeasured transmission coefficient vs. fraquancy. Tha solid curves show theoretical transmission given a dielectric constant of 11.8 and conductlvtles of .2, .28 and .3 S/ca. Tha reflection coefficient was too large to aeasure accurately. Notice that the conductivity is sufficiently high to suppress resonance behavior. For still higher conductivities, loss tangent greater than 1000, Fig. 71 shows that even an oxide layer on the sample surface could Influence the measurement. In this case, a metallizing layer aust wet the surface of the saaple. The gap effect can be corrected by using a suitable filling aaterlal. Theoretical models accurately predict the effect of the gap but are unable to provide data correction because the exact gap width is unknown. 223 -2 5 .0 0 silicon OQ - 3 0 .0 0 - 3 3 .0 0 0 .2 S /e m •• •-* - 4 0 .0 0 0 .2 9 S /c m 0.2 S /c m - 4 5 .0 0 f r e q u e n c y (GHz) Figura 79. Transaisslon vs. frsquancy for high loss (oDC " *28 S/ca) silicon using an •lsctroplatsd saapls. Plgurs shows thsorstlcal curvss for .2, .25 and .3 S/ca. 224 c. Component Imperfections The other major source of error originates In the microwave systea. It arises froa laperfectlons In the coaxial line coaponents. Me found that the Initial version of the Instrument, without appropriate attenuators, produced erratic results. Through a slgnal-flow graph [89] analysis, It was possible to Isolate the source of error to aultlple reflections between the bridge and the saaple, and aultlple reflections between the saaple and the diode that aeasures the transaltted signal. Additionally, It Is possible to Identify alternate paths to the detector diodes that limit the sensitivity of the experiment. Other aeasureaent errors, produced by diode non-linearity or an unleveled source, can be curtailed by proper calibration. Figure 80 Is a signal flow graph model of the system. The microwave bridges are modeled as three port devices and are assumed to be Identical for the simplified analysis to follow; where Stl Is the Insertion loss, S3I Is the coupling loss, and S3, Is the directivity. Slt, S„ and S33 represent the return loss of the bridge at each corresponding port. Both the attenuators and the sample are modeled as two ports with the terms for Insertion loss and return lose being clearly displayed in Fig. 80. coaponents are all assumed to be reciprocal. 1,2 These The diodes, and 3 are represented as one ports loading the proper 225 9dB 3dB BRIDGE I BRIDGE 2 ATTEN I «» a SAMPLE ATTEN 2 '•T u ’7 7 5*9 Plgura 00. Signal flow graph rapraaantatlon of AMCE. OS 226 portions of ths graph. In reality, dlodss 2 and 3 rsprsssnt a slngls dlods that Is controlled by ths switch. (Fig. 60) When ths transmission Is bslng saspled, ths reflection port Is terminated In a matched load. Conversely, when the reflection Is being measured, the transmission port Is matched. The Important first order loops are: ^33^*01 * ^33^01' ®77^03 ' ®«7^D3 (175) These loops represent multiple reflections between the sample and the left attenuator, the sample and the reflection bridge through the attenuator, the Incident bridge and Its detsctor diode, the reflection bridge and Its detector diode, the right attenuator and the transmission detector diode, the sample and the right attenuator, and, the sample and the transmission detector diode through the attenuator, respescltlvely. The signal at a point in the graph can be computed by the relationship [90]: b, ST " P,[l - 2 L,(l) ♦ •••] ♦ P,[l - S 1,(1) +•••] 1 - 2 M l ) ♦ Z L(2) - Z L(3) ♦ - (176) 227 Where P„ represents the poeslble paths to point b, and Ln represents non-touching loops with respect to path Pn. The denominator Is the sua of all loops. Por the system, we first calculate the reflected signal produced by a short circuit as normalized by the Incident signal. (177) The error ln the reference signal, br can be reduced by comparing It with the signal produced by an open circuit. The aultlple reflections produced by the open circuit are 180o out of phase with the signal produced by the short circuit. The effect of these aultlple reflections can be reduced by forming the reference signal from an average of the short-circuit reference and the open-clrcult reference. 228 where the remaining error terms arise froa aultlple reflections between the detector and the bridge, and are Independent of aultlple reflections between the termination and the bridge. The signal reflected froa the saaple can then be measured with respect to this reference. The approximate expression for the reflection coefficient Is: s,,s4g} ♦ s„r 02 ♦ s 33r0l J C^ * ^ ♦ c" « <179> The teras ln the expression are explained as follows: The first tera contains the reflection from the sample and errors due to aultlple reflections between the sample and the attenuator, bridge and diode. The remaining teras represent alternate paths, the first Is reflection froa the front attenuator and the second Is due to the directivity of the microwave bridge. The secondary paths set the sensitivity H a l t of the reflected signal, and the aultlple reflections produce fine structure ln the measured signal. If the attenuator between the bridge and the saaple Is oaltted, a different approximate expression for the reflection coefficient Is obtained: 229 (180) The positive effect of the attenuator Is seen ln reducing the aultlple reflections between the saaple and the bridge. The negative aspect of the Insertion of the attenuator Is that provides an alternate signal path to the detector and that It Increases the effect of the directivity of the bridge on the aeasured signal. The transaltted signal can be considered ln a similar way. In this case, the reference signal Is obtained by noraallzlng the transalsslon through an eapty holder to the Incident signal. The approxlaate expression for the transalsslon coefficient Is given as: r ■ Note that all of the error teras are due to aultlple reflections. There are no alternate paths to the 230 transalsslon detector. The effect of the ettenuetore ie eeen by reaovlng both ettenuetore froa the circuit, the treneeleelon coefficient le now given ee: r ■ (182) So the left ettenuetor is useful in reducing reflections between the bridge end the saaple, while the right ettenuetor deeps reflections between the seaple end the trensaleslon detector. The deleterious effect of the ettenuetor on the trensaltted slgnel Is to decreese sensitivity. While the Insertion of epproprlete attenuation reduces the Interaction between reflective alcrowave components, the aeesured values of the reflection and trensaleslon coefficient aegnltudee still appear with soae residual quasi-perlodlc saell variation due to reaelnlng multiple reflections end secondary paths In the systea. Their effect could be reduced further by aore sophisticated data analysis. The signal flow graph analysis has been useful in Identifying sources of error caused by reflective alcrowave coaponents. Multiple reflections between the saaple and the bridge were reduced by Interposing a 3 dB attenuator. 231 Reflections between the saaple and the transalsslon detector were reduced by a 9 dB attenuator placed between the two' coaponents. The particular value of attenuation was chosen to reduce Interactions between coaponents without degrading the sensitivity of the experlaent too auch. 232 ». High T— pgratiure Characterization In addition to rooa taaparatura aaaauraaants, It la Important to know aaapla propartlas as taaparatura lncraasas. We saak a aystea to aeasure coaplax peralttlvltles froa rooa taaparatura to sintering teaperatures, noalnally 2000°C. The aysten should provide data at 2430 MHz and use actual saaples for sintering. In this section we consider existing aethods of high taaparatura characterization, and propose a new technique baaed on theoretical findings In Chapter III. High Teaperature AMCE The aost logical alternative would be to adapt AMCE for high taaparatura use. AMCE is currently Halted to operation below 200*C for the following reasons: 1)teflon spacer In saaple holder chars; 2 )copper conductors becone Increasingly lossy 3) differential expansion nay cause gap effects to re-energe. To alleviate these difficulties we aust a) replace the teflon spacer with a refractory window naterlal such as beryl11a or alualna. A lower dielectric constant Is preferred since the spacer will add nultlple reflections to the systea. The saaple holder should be evacuable to avoid high taaparatura oxidation effects, b) the saaple holder 233 should bs fabricated out of aolybdanua or a slallar aatal. c) tha saaple should ba nickel plated. Electroless deposition of nickel is described in [8 8 ]. Differential expansion aay either cause the saaple to fracture or cause gaps to reappear around the saaple. A correction foraula for gaps produced through differential, expansion Is developed using a aodal analysis In [91]. A correction foraula Is now possible since the gaps produced by differential expansion can be calculated. These changes should lncrsase the operating range to 1000°C. 2.! Other High Teaperature Methods A survey of the literature reveals several Interesting characterization aethods. Westphal [92] coapares several cavity aethods for dielectric constant and loss tangent aeasureaent up to 1200°C: a) A TMo,0 re-entrant cavity with a centrally aounted cylindrical saaple Is used near 1 GHz, with aeasurable loss tangents ranging froa 9 x 10'* to .02 b) A coaplstely filled TEltl cavity at 4 GHz aeasures loss tangents between 4 x 10'* and .009. transversely aounted saaple In a TEllt re-entrant c) A cavity at 8 GHz aeasures loss tangents between .002 and 1. In all cases silver or platlnua foil Is used to Bake contact 234 between the staple end the wells of the cevlty. Neesureble lose tengente ere lower then those obtslnsble by AMCE et roos tespereture. High tespereture charecterlzetlon et sllllseter wevelengths Is performed using quasl-optlcal techniques. [93,94] Results compere well with lower frequency (alcroweve) measurements [99]. These aethods would require huge samples et 2450 MHz. A method using e TE0I cavity composed of graphite Is used to measure silicon end aluminum oxides et 10 GHz, froa 0 to 2000*C [96]. The sample Is 4.1 cm In diameter. The conductivity of alumina et 2000°C Is given as .1 S/a. 3. In Situ Characterization While high temperature characterization may be of use In determining the trend of complex permittivity with temperature, the response during actual sintering would be desirable. Araneta describes an In situ characterization method using the rectangular applicator. From the Impedance analysis, we can deduce complex permittivity froa reflection coefficient data.[18] This asthod assumes that the saaple Is heated uniformly, and works best when losses In ths iris and shortcircuit are kept to a minimum. Ultimately, accuracy depends on the ability to measure the reflection coefficient and 235 this is llaltsd by ths purity of the aagnetron sourcs. Ms proposs an laproveaent to this charactsrlzatlon asasursasnt using two alcrowave sources. A possible systea configuration Is shown In Fig. 81. A aagnetron source Is used to heat the saaple as before. Since the aagnetron Is only transalttlng 50% of the tlae (120 Hz), a precision low power source can be used to sweep the cavity during the off cycle of the aagnetron. The low power source Is protected by a directional coupler and a suitable nuaber of Isolators. The high voltage froa the aagnetron Is used to switch a transalt/receive (T/R) tube to protect the detector froa high power levels. Acquired data could be stored In coaputer aeaory and analyzed at the coapletlon of a run. Lx A Hew Method for High Teaperature Characterization We describe a high teaperature characterization aethod using the analysis of Chapter III on a spherical saaple placed In a alcrowave applicator. This aethod provides a aeans of deteralnlng teaperature dependent electrical conductivity by aeasurlng surface teaperature, Incident power and reflected power, along with Icnowledge of heat transfer constants. The aeasureaent Is Independent of applicator geoaetry. Equation (63) relates power to steady state surface teaperature for a thin planar slab. A slallar expression 236 Figure 81. Block diagram of In situ characterization aathod using a tranaait/rscsivs tubs. 237 has bssn dsrlvsd for an Infinite, thin cyllndar (1 1 2 ) and a snail sphsrs (152). Ths dlffsrsncs in ths sxprssslons Is a constant dstsrnlnsd by ths surfacs arsa to voluns ratio of ths sanpls. To dstsrnlns ths conductivity froa sxpsrlasntal data an lnvsrss rslatlon, (67), must bs ussd. Accurats rssults rsqulrs that ths tsapsraturs bs unlforn within ths saapls, othsrwlss ths asasursasnt will ylsld sobs average conductivity. In ths slab geometry, a saapls aountsd In rsctangular wavsgulds has a non-unifora slsctrlc field, producing centralized heating. Observations indicate that for cylindrical rods, non-unlforalty Increases with temperature and is aost pronounced In low theraal conductivity naterlals. Figure 82 Is the axial teaperature variation of an alualna rod. Figure 83 shows the axial teaperature variation for an alualna rdd with 30% titanium carbide. The profile Is aore uniform In Fig. 83, the higher electrical and theraal conductivity aaterlal. These figures show the trend In axial teaperature variation and are consistent with visual observations— the pure alualna saaple has a shifted aaxlaa and Is highly non-unlfora. Note that absolute temperatures may be in error due to occlusion of the pyrometer. (The pyrometer looks at the saaple through a beyond cutoff tube. If the cone of observation intersects 238 t e mp e r a t u r e (C) 1800-< H 1 1 r> 60 u 1400 -1 i t_ * i i neignt Plgura 82. 0 1 f 9 \ (cm) Axial taaparatura profllaa In a alntarad alualna saapla for varloua powar lavala. 239 Temper at ure (C) 1600-1 1400J * ♦ 1200 - 1 0 0 0 ? i i i i i i i i r | i i i i i » r r tT i i i » i r l r i |~i i » i » » r t -> i - 2 - 1 0 Height Plgura 83. 1 (cm) 2 top Axial taaparatura proflias In a slntarad alualna-TlC (30%) saapls for various powsr 1 avals. 240 the beyond cutoff tube, a aeaeureeent error Mill Introduced.) In addition to axial non-uniforalty, cylindrical aaaplee are preferentially heated on the side facing the source. The sphere is aore likely to have a uniform field variation due to its aore coapact geometry. (Theoretically a saaple can be Bade saall enough to eliminate any nonuniformity. ) Phenomenological experiments with a pill- shaped (diameter ■ height) alualna saaple suspended in a quartz tube confirm this hypothesis. We observe no axial variation, and heating appears to be uniform around the circumference of the sample. Therefore, it is proposed that spherical saaples be used in characterization studies. To aake an accurate measurement of the sample requires that the absorbed power be known. This requires a measurement of incident power and reflected power. It is not necessary to acquire reflection coefficient data. The location of the sphere in the cavity does not affect the calculation, since we assume that all of the power dissipated in the cavity is dissipated in the sample. However, low loss samples should be placed in a region of maximum electric field to improve heating efficiency. With a knowledge of theraal properties, saaple diameter and surface teaperature, the electrical conductivity can be deduced. 241 An uncertainty analysis rsvsals ths relative Importance of certain measured values. The error Is found froa (63) and Is Error ■ 1 (183) where two supposedly Identical measurements are compared. A 5* error In measured power or sample geometry translates directly to a 5k error In conductivity. A 5k uncertainty In ck has a large effect on conductivity at low temperatures. Pig. 84. It Is plotted as a function of temperature In The result is anticipated, since radiation dominates at high temperatures. Plgure 85 plots the effect on conductivity of a 5k error In cz as a function of temperature. As expected, cz has less effect at low temperatures, where convection dominates. A 5k error In surface temperature can produce up to a 20k error In conductivity as shown In Fig. 8 6 . This is a result of the fourth power variation In temperature. The uncertainty analysis reveals that the surface temperature measurement can produce the largest errors In conductivity. In a typical experiment, we know the temperature within 50°C at 1500°C. This would translate 242 A 00 - T r\ r* «*/ . sJW ir 2.00 \ *^ (* I n n r\r* _» I ..Jw -Ji -J 0 0.00 2.00 4.00 6.00 8.00 10.00 tem oeroture Plgurs 84. Porcont orror In conductivity B M s u r m n t vs. normalized taaparatura for a 5% error In c,. 243 5.00 - error 4.00 -i 1.00 0.00 2.00 4.00 6.00 temperature — ... 8 oo 1 0 .0 0 244 20.00 --i 16.00 -s 3 / -t *^ ^ ^ ^ o •4 . .VS-/ J -1 3 -< o & 8 •nW nw J ^ / 3 -i t / 2/ A .00 3 3 n 3 -t 0 . 0 0 I n ' i i r r 'r i 0.00 rr i i 2.00 n i i i i i i i i i i *00 11 i i i i i i t ) i i i i i i i i 6.00 8.00 i 10.00 tennDeroture Flgur* 8 6 . Psrcsnt srror in conductivity M u u r n t n t vs. normalized tsmpsraturs for • 5% srror In v. 245 into an error of a little aore than 10% In conductivity. While thle aethod aay not provide high accuracy reaulte, it ehould provide ueeful conductivity data for alcrowave alnterlng applications. It has several advantages over the aethods described earlier: A saaple does not need to contact the cavity, so the "gap effect" is not a concern; phasor detection of reflection coefficient is unnecessary, only aagnltude is laportant, so source quality la not critical; and saaples can be studied in an "real" sintering envlronaent. C H A P T M VI C0WCL08I0W8 AMD RICOHHgyDATIOHS FOR f U T U W WOWC This study has provldsd sddsd Insight Into ths alcrowave hsatlng of aatsrlals. Specifically, wc havs dsaonstrated the laportance of teaperature dependent electrical conductivity In deteralnlng the heating behavior of the saaple. Matheaatlcal analysis Indicates that when electrical conductivity rises too rapidly, for electrically thin saaples, the heating process Is uncontrollable above a critical power level, and that there Is a aaxlnua attainable steady state teaperature associated with this power level. We have considered the heating process In a variety of geoaetrles Including a seal-infinite space, a planar slab, an Infinite cylinder and a sphere. The result Is that heating of a seal-infinite space is always aonotonlc because the "skin effect" H a l t s field penetration at high conductivities (teaperatures). theraal Instability. Thin saaples can exhibit In fact, the aaxlaua teaperature Is the saae for the slab, cylinder and sphere. This "runaway" behavior is consistent with laboratory observations. To predict stability/instability In the heating of a saaple the teaperature dependent electrical conductivity 246 247 aust be known. Characterization aathods and their llaltatlons are discussed for both rooa and high teaperatures. He find that to aeasure high loss tangent aaterlals the saaple surfaces which contact the walls of the saaple holder aust be electroplated to reaove any gaps. A high teaperature technique based on the relationship between steady state teaperature and absorbed power Is also described. This study has answered a nuaber of questions concerning the nature of anoaalous behavior observed In experlaent. Core aeltlng Is attributed to theraal gradients due to low theraal conductivity In which the surface teaperature Is cooler than the Interior. "Theraal runaway" can occur above a critical power level when electrical conductivity Is a strong function of teaperature. Control of cavity coupling can In soae cases restablllze the system. Puture work Is needed to further our understanding of the sintering process. In the theoretical area, a coaplete analysis of the slab Is needed. He have developed a solution for the case where the electric field distribution Is nearly unifora and does not depend on teaperature. At high teaperatures, the skin effect reduces field penetration, coapllcatlng the analysis. For very high conductivities, the saaple reseables the seal-lnflnlte space. Hork Is also needed In developing aore sophisticated aodels for the radiant wall applicator and the "hot wall" 248 applicator. Tha cavity analysis should bs sxpandsd to lncludsd wall lossss. Studios should bs sxpandsd to consldsr teaperature gradlsnts In aultlpls dimensions, as wall as sintering dynamics. High tsapsraturs charactsrlzatlon squlpasnt Is badly nssdsd. As ws slntsr unlqusly prsparsd aatsrlals such as alualna-TIC, at various concsntratlons of TIC, a knowlodge of tsapsraturs dspsndsnt conductivity Is essential. Such squlpasnt aay help answer still unresolved problems, such as: why do unslntsrsd saaples heat aore quickly than sintered saaples? Unslntsrsd saaples should have a lower conductivity due to their greater porosity. It Is recommended that a characterization study be implemented to classify aatsrlals as good/bad candidates for alcrowave sintering based on their teaperature dependent electrical conductivity. Suitable dopants alght be used to change teaperature dependent conductivity and thus enhance the slnterablllty of certain low loss materials. NOTES AMD REFERENCES 1. Y.L. Tlan, D.L. Johnson and M.E. Brodwln, "Ultraflne Mlcrostructurs of A1,0, Producsd by Microwave Sintering," Proceedings of 1st International Confsrsncs on Ceramic Powder Procssslna. v. l, Orlando, PL, Nov. 1987, p 925-932. 2. M.E. Brodwln and D.L. Johnson, "Microwave Sintering of Ceramics," IEEE MTT-S Symposium. May 25-7, 1988, New York, NY, p 287-8. 3. D.L. Johnson and M.E. Brodwln, Microwave Sintering of Ceramics. EPRI EM-5890, Project 2730-1, Pinal Report, 1988. 4. A .J . Bertaud and J.C. 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Reno, NV, April 4-8, 1988, M3.3. 2S4 62. A.S. Barker, J.A. Dltzenberg and J.P. Reaelka, "Lattice Vlbratlone and Ion Traneport Spectra In 0-alualna, II Microwave spectra," Phve Rev B. v. 14, no. 10, p 425465, IS Nov 76. 63. S.L. McGill, J.W. Walklewlcz and G.A. Sayres, "The Effects of Power Level on the Microwave Heating of Selected Chealcals and Minerals," Material Research Society 1988 Soring Meeting. Reno, NV, April 4-8, 1988, M4.6. 64. P.J. Schneider, Conduction Heat Tranefer. AddlsonWesley, 1955. 65. M.E. Brodwln, D.L. Johnson, Y.L. Tlan and K.Y. Tu, "A Circular Cylindrical Applicator for Enhanced High Teaperature Processing," 22nd Microwave Power Svaposlua Suaaarlea. International Microwave Power Institute, 1987, p 11. 66. N. Robertson, "A Resonant Cavity Systea for Microwave Sintering of Ceraalcs," Masters Thesis, Northwestern University, 1980. 67. J. Schwlnger, Discontinuities In Waveguides; Notes on Lectures bv Julian Schwlnaer. Gordon and Breach, 1968. 68. N. Marcuvltz, Waveguide Handbook. MIT Rad Lab Series, v. 10, 1961. 69. R.A. Waldron, Theory of Guided Electroaagnetlc Waves, D. Van Nostrand, 1969. 70. K.K. Clarke and D.T. Hess, rn»»nnicatlon Circuits: Analysis and Design. Addlson-Wesley, 1978. 71. J.L. Altaan, Microwave Circuits. D. Van Nostrand, 1964. 72. M. Moreno, Microwave Transalsslon Design. McGraw Hill, 1948. 73. S.M. Sze, Physics of Sealconductor Devices. 2nd Edition, Wiley, NY, 1981. 74. Raao, et al, problea 10.06d, p. 545. 75. J.R. Dygas, "Study of Electrical Properties and Structure of Naslcon-type Solid Electrolytes," Ph.D. Dissertation, Northwestern Unvlerslty, 1986. 255 76. K.S. Chaaplln and Q.H. Glover, " 'Gap Kffact' in Measurement of Large Paralttlvlty," IEEE MTT-14. 397398, Aug. 1966. 77. K.S. Chaaplln, J.D. Hoi* and G.H. Glovar, "Elactrodalaaa Dataralnatlon of Saalconductor Conductivity fro* TE^-aod* Raflactivlty," J A p p I Phva. 38,1,96-98, Jan.1967. 78. K.S. Chaaplln and G.H. Glovar, "Influanca of Wavagulda Contact In Maaaurad Coaplax Paralttlvlty of Saalconductor*," J A p p I Phva. 37, 6 , 2355-2360, March 1966. 79. J. Dygas and M.S. Brodwln, "Praquancy-Dependant Conductivity of Naslcon Caraalcs In th* Microwave Raglon," Proc. Conf. Ion Conductor*. Lake Tahoa, CA, Aug. 1985. 80. A. Kaczkowakl and A. Mllewakl, "Hlgh-accuracy Widerange Meaaureaent Method for Dataralnatlon of Coaplax Paralttlvlty In Reentrant Cavity: Part B- Exparlaantal Analysis of Measureaent Errors," IEEE MTT-28.. 228, March 1980. 81. A. Kaczkowakl and A. Mllawskl, "Hlgh-accuracy Wlderange Measureaent Method for Dataralnatlon of Coaplax Paralttlvlty In Reentrant Cavity: Part A- Theoretical Analysis of th* Method," IEEE MTT-28.. 225-228, March 1980. 82. H.E. Bussy and I.E. Gray, "Measureaent and Standardization of Dielectric Saaples," IRE Trans. Instrua.. 162-165, Dec. 1963. 83. W.B. 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(In Russian) APPENDIX A ER99IMH T9B PQHPyiHW TRAIT?I HIT TPgjBAIgMLgRgrifcH Iff A SEHI-INFIKITX 5PAC1 KITH CONSTANT CONDUCTIVITY 1 PLANE HAVE HEATING OF SEMI-INFINITE SPACE ' EXPLICIT METHOD SOLUTION OF HEAT EQUATION I DEFINT I ,J ,N ’ T STORES TEMP DATA, T 1 STORES ELECTRIC FIELD DATA DIM T(2,1000),T1(1000) OPEN "0",#1,"PROFIL.DAT" INPUT "INTERVAL SPACING";D INPUT "LENGTH";L INPUT "TIME STEP";DT N-INT(L/D+.5) ' SET CONSTANTS USING MKS UNITS SO-.002 •ROOM TEMP ELEC CONDUCTIVITY ER-10 'DIELECTRIC CONSTANT OF SPACE PI-3.14159265 F-2.45E+9 'TEST FREQUENCY W-2*PI*F C-2.998E+8 'SPEED OF LIGHT EO-8.854E-12 'FREE SPACE PERMITTIVITY K-20 'THERMAL CONDUCTIVITY 'HEAT CAPACITY CP-1000 RHO—3970 •DENSITY 'EMISSIVITY EL-.6 'STEFAN-BOLTZMANN CONSTANT S-5.67E-8 'CONVECTION CONSTANT HH-10 'AMBIENT TEMPERATURE (KELVIN) T00-300 R-DT/D* 2 *K/CP/RHO PRINT R 'R MEASURES STABILITY < .025 PN-2*10*5 'INCIDENT POWER LEVEL J-l Jl-2 FOR 1-0 TO N 'SET SAMPLE TO AMBIENT TEMP T (J ,I) - TOO: T (J 1 ,I) - TOO NEXT I FOR K2-1 TO 1000000 'COMPUTE INTERIOR FIELDS IF K2 > 1 THEN 15 'FIRST PASS ONLY FOR 1-1 TO N X-SO/W/EO/ER RT-W/C*SQR( ER)• (1+X' 2 )".25*SIN(-. 5*ATN (X) ) T1(I )-377*PN*EXP(2*RT*I*D)*S0 NEXT I 10 M-(l+X‘2)‘.25 'COMPUTE REFLECTION COEFF 257 258 MM«M*SQR(ER) PH— 5*ATN(X) NUM-SQR((MM* 2-1)* 2+4*MM“2*SIN(PH)“2) DEN-MM* 2+1 + 2 *MM*COS(PH) P-l-(HUM/DEN)‘2 'P IS ABSORBED POWER RATIO PRINT p 15 T-T+DT 'INCREMENT TIME PRINT "TIME";T;"SEC" H1-S*EL*T(J,1)‘3+HH 'LINEARIZE HEAT TRANSPER T(J,0)«T(J,2)-(2»H1/K)»T(J,1)•D+2/K*TOO*(HH+S*EL*TOO*3)*D T(J,N)-T(J,N-2)-2*T(J,N-1) 'COMPUTE BOUNDARY VALUES POR 1-1 TO N-l T(J1,I)-R*(T(J,1+1)+T(J,1-1))+(l-2*R)*T(J.I)+ + T1(I)*DT/CP/RHO*P NEXT I IP K2/500 > INT((K2+.5)/500) THEN 20 LPRINT "TIME ";T;"SEC" 'STORE RESULTS OCCASIONALLY POR 1-1 TO 100 LPRINT USING "#####.#♦ ";T(J1,I) - TOO; LPRINT #1, USING "###♦#.## ";T(J1,I) - TOO; NEXT I LPRINT 20 J-Jl J1-3-J NEXT K2 MTiwm p EB99RAH T9 gQBPffll TBAWSIIWT TPff MATVEI PR9KM? IB MHIIHTIBITI tfAVI WTB IBfflBATVEI PI FIEPBfT-KISIRISAIr CONDUCTIVITY PLANE NAVE HEATING OF SEMI-INFINITE SPACE EXPLICIT FINITE DIFFERENCE SOLUTION OF HEAT AND WAVE EQUATIONS T STORES TEMPERATURE DATA, T1 STORES ABSORBED POWER DATA KR AND KI ARE COMPLEX PROP CONSTANTS IN EACH GRID REGION EPR, EPI, ENR AND ENI ARE COMPLEX ♦ AND - TRAVELING WAVES IN EACH GRID REGION. DIM T (2,500),T1(500),XR(500),KI(500),EPR(500),EPI(500) + ENR(500),ENI(500) OPEN "0 " ,# 1, "PROF.DAT" INPUT "INTERVAL SPACING";D INPUT "LENGTH";L INPUT "TIME STEP";DT 'REM SET CONSTANTS IN MKS UNITS N-INT(L/D+.5) •ENDPOINT FOR SG(T) CALC NH-N . 'FOR HIGH ATTENUATION NN-N SO-.002 EXPONENTIAL CONSTANT Cl-.0026 SG-SO ER-10 PI-3.14159265 F-2.45E+9 W»2*PI*F C-2.99E+8 EO— 8 .8S4E-12 K-20 CP-1000 RH0-397C EL-.6 S-5.67E-8 R-DT/D“2 *X/CP/RHO H-10 T0-300 'R MEASURES STABILITY, <.25 PRINT R PN-10'6 J-l Jl-2 259 260 POR 1-0 TO N 'SET SAMPLE TO AMBIENT TEMP T (J ,I) - TO : T(J1,I) - TO NEXT I POR K2-1 TO 1000000 'COMPUTE NORMAL PIELD ATTENUATION ON PIRST PASS IP K2 -1 THEN 5 'IP DIPPERENTIAL CONDUCTIVITY IS HIGH, COMPUTE USING GOSUB IP ABS(1 - EXP(Cl*T(J,1))/EXP(C1*T(J,2))) < .008 THEN 5 ’IP CONSTANT CONDUCTIVITY SKIP APTER PIRST PASS IP K2 > 1 AND Cl-0 THEN 18 GOSUB 40 GOTO IS 8 POR 1-1 TO N SG«S0*EXP(C1*T(J,I)) X-SG/W/EO/ER RT«W/C*SQR(ER)*(1+X*2)“.25»SIN(-.5»ATN(X)) Tl#(I)-377*PN*EXP(2*RT*I*D)*SG NEXT I 10 'COMPUTE ABSORBED POWER RATIO ON PIRST PASS SG-S0«EXP(C1*T(J,1)) M-(l-»-SG‘2) ‘ .28 A-SQR(ER)*M»C0S(-.3*ATN(SG)) B-SQR(ER)*M*SIN(-.8 *ATN(SG)) DD«(A+l)“2+B“2 NR-A‘2+B‘2-1 NI-2»B P"l - (NR“2+NI"2 )/DD‘2 15 T-T+DT 'INCREMENT TIME IF K2/100 > INT((K 2+ .5 )/ 100 ) THEN 16 PRINT "TIME";T;"SEC" 16 H1-S*EL*T(J,1)‘3+H T( J ,0) *T{ J ,2)“ (2*D/K)*(T(J,1)*H1 - TO*(H S*EL*T0‘3)) T (J ,N ) * T (J ,N - 2 )-2 * T (J ,N - l ) POR 1-1 TO N-l T(J1,I)-R*(T(J,I+1)+T(J,I-1))+(l-2*R)*T(J,I)+ Tl#( I)*DT/CP/RHO*P NEXT I IP K2/500 > INT((K2+.5)/SOO) THEN 20 POR 1-1 TO N-l LPRINT USING "#####.## ##.##### ";T(J1,I), S0*EXP{C1*T( J1,1)) . PRINT #1. USING "#♦##♦.## ##.##### ";T (J1,I )i ♦ S0*EXP(C1*T(J1 ,1)), NEXT I LPRINT 20 J-Jl J1-3-J NEXT K2 ' SG(T) CALCULATION 40 TD# - S0*EXP(C1*T(J,N))/W/EO/ER N-NN ' SET FIELDS TO TRAVELING AT DEEP INTERIOR POINT KR#(N ) - -H/C*SQR(ER)•(1+TD#“2)“.25*SIN(-.5*ATN(TD#)) KI#(N) - -W/C*SQR(ER)*U+TD#‘2) ‘ .25*COS(-.5*ATN(TD#)) KR#(0) - 0 KI#(0) - -H/C EPR#(N ) - EXP(KR#(N)*N*D)*COS(KI#(N)*N*D) EPI#(N) - -EXP(KR#(N)*N*D)*SIN(KI#(N)*N*D) ENR#(N ) - 0 ENI#(N) - 0 FOR II - N-l TO 0 STEP -1 SG# ■ SO*EXP(Cl*T(J,11)) TD# - SG#/H/EO/ER IF 11 « 0 THEN 45 KR#(11) - -W/C*SQR(ER)*(1+TD# ‘2)‘.25*SIN(-.5 •ATN(TD#)) KI#(11) - -H/C*SQR(ER)*(1+TD# ‘2)‘.25*COS(-.5 *ATN(TD#)) KR#(11) Ql# Q2# KI#(11) Q3# KR#(I1+1) KI#(I1+1) Q4# Q5# EPR#(I1+ 1 ) EPI#(I1+1) Q 6# Q7# ENR#(I1+1) Q 8# ENI#(11+1) AR# Ql#-Q3# AI# Q2#-Q4# BR# Ql#+Q3# BI# Q2#+Q4# CR# -Ql#-Q3# Cl# -Q2#-Q4# DR# -Ql#+Q3# DI# -Q2#+Q4# HR# EXP(AR#*D)*COS(AI#*D) HI# EXP(AR#*D)*SIN(AI#*D) XR# EXP(BR#*D)*COS(BI#*D) XI# EXP(BR#*D)*SIN(BI#*D) YR# EXP(CR#*D)*COS(CI#*D) YI# EXP(CR#*D)•SIN(CI#*D) ZR# EXP(DR#*D)*COS(DI#*D) ZI# EXP(DR#*D)•SIN(DI#*D) MR# Q5#«(BR#*HR# - BI#*HI#) - Q 6#*(BI#*HR# + BR#*HI# MI# Q5#*(BI#*HR# + BR#*HI#) + Q 6#*(BR#*HR# - BI#*WI# NR# Q7#*(AR#*XR# - AI#*XI#) - Q 8#*(AI#*XR# + AR#*XI# NI# Q7#*(AI#*XR# + AR#*XI#) + Q 8#«(AR#*XR# - AI#*XI# QR# Q5#*(DR#*YR# - DI#*YI#) - Q 8#*(DI#*YR# + DR#*YI# Q5#*(DI#*YR# + DR#*YI#) + Q 6#*(DR#*YR# - DI#*YI# Ql# RR# Q7#*(CR#*ZR# - CI#*ZI#) - Q 8#*(CI#*ZR# + CR#*ZI# RI# Q7#*(CI#*ZR# + CR#*ZI#) + Q 8#*(CR#*ZR# - CI#*ZI# 2*(Q1#“2 + Q2#‘2) DD# 262 EPR#(II) - (Q1#*(MR#+NR#) + Q2#*(MI#+NI#))/DD# EPI#( II) - (Ql#* (MI#+NI#) - Q2#* (MR#+NR#) )/DD# ENR#(II) - (-Ql#*(QR#+RR#) ♦ Q2#*(QI#+RI#))/DD# ENI#(I1) - (-Q1#*(QI#+RI#) - Q2#*{QR#*RR#))/DD# Rl# - EPR#(II) R2# - EPI#(II) R3# - ENR#(II) R4# - ENI#(II) IP II - 0 THEN 50 PI# - EXP(-Q1#*I1*D) P2# - EXP(Q1#*I1*D) P3# - C0S(Q2#*I1*D) P4# « SIN(Q2#*I1*D) SR# - R1#*P1#*P3# + R2#*P1#*P4# ♦ R3#*P2#*P3# + R4#*P2#*P4# SI# » R2#*P1#*P3# - R1#*P1#*P4# + R4#*P2#*P3# ♦ ♦ R3#*P2#*P4# Tl#(II) - 377*PN*(SR#“2 + SI#‘2)*SG# NEXT II 50 P - 1 - (R3‘2 ♦ R4“2)/(Rl“2 + R2‘2) 'ABSORBED POWER CON - SR‘2 ♦ SI‘2 POR II - 1 TO N-l Tl#(II) - T1#(I1)/C0N*EXP(-2*KR#(1)*D) 'CHECK POR HIGH ATTENUATION IP Tl#(Il)/Tl#(l)<lE-3 THEN NN-I1: GOTO 60 NEXT II N-NH RETURN 60 POR 12 - II TO N-l Tl#(12) - 0 'RENORMALIZE POWER MATRIX NEXT 12 N-NH GOTO 55 APPKWDIX C WQW-DIMKKSIOWAL ANALYSIS He scale the heat and wave equations dlaenslonal fora. to a non- This peralts analysis In terms of non- dlaenslonal parameters. The solution we present Is general and can be applied to a variety of geometries. We begin with the heat equation: K7,*T - pcpT, - Ao(T) |E(X) |* (C-la) and tcT„ + h(T - T0) ♦ •CJT 4 - T04) - 0 (C-lb) where the left hand side of (C-la) Is a one-dimensional Laplaclan. The boundary condition (C-lb) Is expressed In terms of a normal derivative. This representation notation Is used so that the scaling can be applied to rectangular, cylindrical and spherical geometries. He scale the length In terms of some convenient physical value. This might be the half-thickness of a planar alab, the radius of a cylinder, or, In the case of the semi-infinite solid, the wavelength in the material: - f dx - JdX (C-2a) The time Is scaled accordingly, 263 264 ( C-2b) and the temperature, using (32) Is T - T0 u - — s— 1 *o (C-2c) Substituting (C-2) Into (C-l) we have 7,‘u u„ ♦ (u) (C-3a) T0’( (u+1)4 - 1) - 0 (C-3b) He define the following constants _ _ hL c. " ir » cf ■ s(.L , o„Lz k T0 , X ■ 1E01 (C-4a,b,c) The heat equation becomes 7 m*u - ut - -Xf(u) (C-5a) un ♦ c,u + Cjttu+l)4 - 1) - 0 (C-5b) For the wave equation, we scale the length to the wavelength In the material at t ■ 0. geometry, We have In a planar 265 “** * ®*<1 - 0 r “ 0 (C-6a) *x ♦ 3YoE - 2 JY0 (C-6b) and, x ■ 6X , dx -6dX (C-6c) The non-dlmenslonal wave equation becoaes E«, ♦ (1 - Jc3f(u))K- 0 ♦ JB ■ 2J wh.re c, - 5 ^ - (C-7a) (C-7b) APPENDIX P DAMPED HARMONIC OSCILLATOR We show the difficulty that arises of the perturbedharaonlc oscillator. In alinearexpansion We thenderive an approximate solution uolng the method of multiple scales.[52] The pertinent equation Is: u’ + (u' ♦ u ■ 0 (D-la) or u’ ♦ u ■ (D-lb) -€u". where € << 1 . We look for an approximate solution using a Taylor series expansion: e es u - Z'Ut)*" (D-2) n*0 Taking the first two terms, u 'v u0 + (u, (D-3) Substituting (D-3) Into (D-lb), we have u0 - e'* (D-4a) and u^ ♦ ui ■ -is11 (D-4b) 266 267 The solution of (D-4b) Is u, - - £te'* (D-5) Combining (D-5) with (D-4a), yields u ~ e" - tte'' (D-6 ) For large values of t, u, > u0, asking the approximation Invalid. However, when t Is small, the solution looks like .n u 'v e “ e 2 (D-7) So, we look for a solution In We express u In terms of u ■ v(t,u,€) , where ti this fora. two paramters, ■ ft (D-8 ) Substituting (D-8 ) Into (D-la) we have v tt + 2£vtn ♦ €2vw + €vt ♦ e2v„ ♦ v - 0 Rewriting (D-9a), eliminating have (D-9a) second orderterms In c, we 26 8 v,f ♦ v - -€ (2V,, ♦ v.) (D-9b) We now look for a solution using s ssrlss expansion In t and T): v ~ v0(t,T») ♦ €v,(t.D) ♦ ... (D-10) The solution of (D-9b) for the first two terns Is as follows: Vo.it ♦ v0 « 0 , v0 - Ao(Tl)e(,t) v .,.» ♦ v. “ -M2A.* ♦ A0)e(,t) , v, - - j (2A0 (D-lla) + A0)te" He define the constant tern of v, to be zero. (D-llb) This yields .u Ao - a0e * (D-12a) . ii and v0 - SgS 'e" which Is equivalent to (D-7). (D-12b) APPENDIX 1 THERMAL EQOILIBRIOM OF SEHI-IltTIMTE SOLID The following derivation deaonstrates the conaervatlon of power at eteady atate — •eal-lnflnlte solid. theraal equilibrium, for the The heat and wave equations as defined In (31a-d) are used. We fora the conjugate wave equation as follows: E.’. ♦ V '(1 ♦ - 0 , x > 0 " - 2 JV0 , x - 0 . (E-la) (E-lb) We aultlply (31c) by E* and (E-la) by E and take their difference, which yields (E-2) The right hand side of (E-2) is proportional to the right hand side of (31a). We Bake the substitution and Integrate the equations over the length of the solid: 270 / (*'I„ - BE/Jdx o f 0 *< T) | K ( x ) | ’dx (E-3) o T„dx - - O0nopo f f (T) I E(x) I o *dx (E-4) The left hand aide of (E-4) le Integrated and, from (31b) and (33b) we have T- L “ T-l0 “ " h(T “ To> " f f (T) | E(x) | *dx - ~ V) [h(T - T0) ♦ s€JT« - T04)] <E-3) (E-6 ) O Subetltutlng (E-6 ) Into (E-2) we f now have (e'E„- EE^Jdx - J—p^[h(T - T0) ♦ 9 i j r - T/)] (E-7) 0 He Integrate the left hand side of (E-7) and evaluate the solution at the boundaries, using (31d), (33a) and (E-lb): 271 (e'E. - e \ )|~ - 2J#m(E*K.) (E-8 ) | K(0) | * - 2jy0E* , E( 0 ) - 1 ♦ p S'K. - Jy0[2(l ♦ p') - (1 ♦ p)(l ♦ p’)] - (E-9a) jy0 (1- IpIM (E-9b) Substituting (E-9b) Into (E-7), we have P0d - IP I* ) - h(T - T0) ♦ s€JT 4 - T04). This Is a statement of thermal equilibrium. (E-10) The Incident power, less the reflected power Is equal to the heat lost at the surface. I APPENDIX F PROGRAM FOR COHPUTIIfQ TRANSIENT TEMPERATURE PROFILES IK A PLAPAff SfcAP 1 PLANE WAVE HEATING OF PLANAR SLAB DEFDBL A-Z DIM T( 2,500),T1(500) OPEN "0",#1,"SLBPRO.DATINPUT -INTERVAL SPACING-;D INPUT "LENGTH";L INPUT "TIME STEP";DT N-INT(L/D+ .3) SO-.002 'DEFINE CONSTANTS, SEE APPENDIX A Cl-.0026 SG-SO ER-10 PI-3.14159265 F-2.45E+9 W-2*PI*F C-2.99E+8 EO-8.854E-12 K-20 CP-1000 RHO-3970 EL-.6 S-5.67E-8 R-DT/D”2*K/CP/RH0 H-10 T0-300 PN-10‘6 J-l Jl-2 FOR 1-0 TO N 'SET SAMPLE TO AMBIENT TEMP T(J,I) - TO : T(J1,I) - TO NEXT I FOR K2-1 TO 1000000 5 FOR 1-1 TO N 'ASSUME CONSTANT E-FIELDS SG-S0*EXP(C1*T(J,I)) X-SG/W/EO/ER Tl#(I)-377*PN*SG NEXT I M-(1+SG*2)'.23 'CALCULATE % ABSORBED POWER A-SQR(ER)*M*COS(-.5•ATN(SG)) B—SQR(ER)*M*SIN(-.5*ATN(SG)) DD-(A+1)*2+B*2 NR-A" 2"fB“2-1 272 15 20 273 NI-2*B P«1-(MR‘2+NI ‘2 )/DD “2 T-T+DT •INCREMENT TIME IP K2/100 > INT((K2+.5)/100) THEN 15 PRINT "TIME";T;"SEC" H1"S*EL*T(J,1)*3+H 'TWO BOUNDARIES, HI, H2. H2-S*EL*T(J,N-1)*3+H T (J ,0) *»T(J ,2)-(2*D/K)•(T (J ,1)*H1-T0*(H ♦ S*BL*T0'3)) T(J,N)»T(J,N-2)-(2*D/K)*(T(J,N-l)*H2-T0*(H ♦ ♦ S*EL*T0“3 )) POR 1-1 TO N-l T(J1,I)-R*(T(J,I+1)+T(J,I-1))+ ♦ (1-2*R)*T(J,I)+T1#(I)*DT/CP/RHO*P NEXT I IP K2/500 > INT{(K2+.5)/500) THEN 20 LPRINT "TIME ";T;"SEC" POR 1-1 TO N-l LPRINT USING "#####.## ##.##### ";T(J1,I), ♦ SO*EXP(Cl*T(J1,I)), PRINT #1, T(J 1 ,I ) NEXT I LPRINT J-Jl J1-3-J NEXT K2 tfp m i i i a PROGRAM FOR COMPUTING STEADY STAT1 TEMPERATURE PROFILIS IN A PLAWAR 9 L U JtfM 9 A lE H E R IV A b-lSU Q ST IlfQ — tfElEQD In Chapter III, Section C.2.b.l, a "shooting" aethod is described to determine the relationship between steady state temperature and applied power. The program listed below uses the Runge-Kutta method to solve a system of linear differential equations to determine the Incident power level required to obtain a prescribed steady state temperature profile. The program generates a curve relating Incident power to steady state temperature, and Interior temperature profiles at each steady state temperature. The program Is written In Turbo-Basic and MKS units are used. rem rea SHOOTING METHOD PROGRAM USING RUNGE KUTTA rem dim k(4,4), y (500,4) rem DEFINE CONSTANTS 'error margin for convergence zerr* .00001 'convection constant hl- 10 'thickness of the slab 11-4.00000IE-03 'thermal conductivity k- 2 0 'Stefan-Boltzmann Constant s-5.67E-08 'emlsslvity el- . 6 'ambient temperature tO—300 'non-dimensional constants cl-hl*ll/k c2-e*el*ll/k*t0‘3 'elec. cond. constant kk - .78 'currantly exp(kx) 'grid spacing for calculation h - .02 'number of linear diff eqs. n2-4 'number of points nl-99 'file for power v. temp data open "o",#1 ,"shootp.dat" 274 275 l-cl*.l 'first guess for norm pwr, 1 . rem rs» M i n loop, cslculsts power (1) for eech new temp(a) res for a - .1 to 10 step .1 'loop with norm, temp, a. y(0 ,l)-a <s«t initial values y(0,2 )b c 1*s +c 2*((a+1)~4 - l) y(0,3)-0 y(0,4)-0 Sub: GOSUB Runge d b y(nl,2)+cl*y(nl,l)+c2*((y(nl,l)+l)‘4 “ 1 ) 17 ABS(d)<zsrr THEN 640 'test for convergence dp-y(nl ,4)+cl*y(nl,3)+4»c2*(Y(nl,1)+l)*3*y(nl,3) lBl-d/dp 'use Newton's Method to comp GOTO Sub 'next point print #l,L,a 'store power, temp data b$ b str$(int(a*10+.05)) a$ b "shot"+ald$(b$,2 ,len(b9 )-1 )+".dat" open "o",#2 ,a9 for 1b 0 TO nl 'store temp profile print # 2 ,y(1 ,1 ),i next 1 close #2 next a STOP Runge: 'first point xbo for Jb Q yi y2 y3 y4 i-i GOSUB yi y2 ys y* i-2 GOSUB yi y2 ys y* 1-3 GOSUB yi y2 to nl y(J.i) y( J. 2 ) y( J* 3) y(J.*) oopl yl ♦ y2 ♦ y3 + y4 ♦ .5*k(1,1) .5*k(l,2) .5*k( 1,3) .5*k(1 ,4) oopl yl ♦ y2 ♦ y3 ♦ y4 ♦ .5*k( 2,1) .5*k(2 ,2 ) .5*k(2,3) .5*k(2 ,4) oopl yl ♦ k{3,l) y 2 ♦ k(3,2 ) 276 y3 » y3 '♦ k(3,3) y4 - y4 ♦ k(3,4) 1-4 OOSUB loopl for 1 - 1 to 4 Y(J*1.1) - y(d.i) ♦ (k(l,i)+2*k(2,i)+2*k(3,i) +k (4,1) )/6 next 1 x-x-fh 'next point next j return loopl: k(l,l) k(l, 2 ) k(1,3) k(l,4) return - h*y 2 'define functions here -h*l*exp(kk*yl) h*y4 -h* (l*kk*exp(kk*yl)*y3 ♦ exp(kk*yl)) ATfBTPIX B H0D13 IW A PIELECTRICALLY LOADED, CIRCULAR CAVITY Presented below Is s derivation of the fields and nodes within a dielectrically loaded cylindrical cavity. The cylindrical dielectric Is hoaogeneous, lossless, and centsred In the cavity. The cavity geometry Is shown In Pig. 65. He will describe the fields as a summation of trial wave functions for TE and TM waves. [31] For TM waves, (H-la) (H-lb) (H-2a,b) (H-2c,d) H - 0 and for TE waves, 277 (H-2e,f) 278 ZZ to Hour. »T. a.o«trv Ot • U P '1 lB T,‘" c«wity- 279 E - 0 H. ■ 3 uii Laz* + *'J * (H-«e,f) where S(x) Is e Bessel function. The seperetlon relations are: kj, ♦k* - w*€,U, (H-5a) kj, ♦k* - ^*€,11, (H-5b) In the present cavity, k, ■ q*ir* so this becomes, «.»* k;, ♦-^r ’ ;* ‘r (»-«•) where q la an Integer representing the axial order resonance. I H .. I H*. I We now define the boundary conditions: - E•2 I,... A I .... T? B * H- 1 .... * *.* ^ D BJc ♦ <3D^s 281 The following amplifications prove useful: (H-8 a,b) (H-8 c,d) F, - K* <*,•> The solution can be represented In aatrlx fora where the eigenvalues of the solution can be found by setting the determinant to zero. k» 2!■ ¥7P* k n I k n u)U7ip« 282 the determinant Is: [K €,»,»;] 2 [ K, u.r.F, - k,, u,FzF, ] k*n2 r k5. - k?. 1 * - A ? p ' p< p> p< - 0 For n ■ 0 this separates Into TE and TM modes: TE: [ k,, u,F,F, - „ U,F,F,] - 0 TM: [ k,, e.FlF, - k,, £,F,F, ] - 0 (H-lOa) (H-lOb) The fields In the TM,,,*, cavity are easily discovered. We look for a solution of the form 3B„(kp) - a0 J„(kp) ♦ Y„ (kp) (H-ll) Fields must be finite at the origin, so we require that Yn ■ 0 In region I. Furthermore, since the tangential electric field Is zero at the cavity walls, we can write Ft and F,. F, - Jn(k,,a) , F, - - * YJKi*) (H-12a,b) 283 Setting n ■ 0, and using (H-l), (H-2) and (H-lOb), the fields given In (131) are obtained. To determine the fields In the "TE,,," cavity, we must use (H-9). The solution Is a hybrid aode, for n > 0. Again we assume fields of the form (H-ll), and require the same boundary restraints. F » " F z The constants 7,, 7,, 7, and 7, are: ■ (H-13a,b) Y„ (k.,b) " - jt ixlb) (H-13C) Using (H-3), (H-4), and (H-7) along with (H-13), we can describe the fields within the sample: ■ A Jn{k,,p) cos ne cos (H-l4a) (H-14b) 284 w -XdV * u>€, dpdz . 1 P d# ■ t * ¥ i^ 7 ..J . t k . , * ) _ E** " E _ ^ i [ - T SJ .(* '.,P ) ] « • »• • ! " d V ‘. a**' ~5*5i♦ -fr ~a¥ J n (k,,P) ♦ B k^jjl^.p) ] aln n* sin SJ5 (H-lSb) - -4- I -^L ♦ k* I 0;*' “T ■ Laz* J " i^7 Jn(k»iP) coa n* coa ^ ***• " £ l£~ ♦ dr; 1 ^ - H*> ■ ■ ^ y&z H.* “ He now aolve for conatanta. (H-15c) (H-i5d) (H-15e) (H-15f) Fro* (H-7a) and (H-7b), we have 285 Substituting Into (H-16a,b) Into (H-7d) we obtain: B * * ? iS^S [ ] F.p. / [ Equation (H-15) can be simplified — E.. " A, [ kM J„-, - n(Bi/ r,r, - k.. r,r: ] <h-i7) (H-15a-c) become 2) J„ ] cos n* »ln *Sr (H-lBa) r n(B. -fl) 1 OTTZ E*i ■ Ai [ Ki*x Jn-.-----v --- Jn ] *in nb sin 3^- (H-18b) E .» ■ A . “ $ r k .*i J r> c o s ( H- 1 8 C ) n* cos wh.r. A, . * 3* Jinrl n B‘ m n r « r— — 1 J F lE « [ k " r;», The lowest order mode, HE F.p;) is obtained by setting n 286 ■ 1. The electric fields Inside the saeple ere listed In (136). MTHfPIX-1 PROGRAM TO STUDY OAF EFFECTS 03IWO TOT METHOD Of TRAWSVRRS1 RESONANCE REM REM TRANSVERSE RESONANCE METHOD FOR GAP EFFECT 3 REM 4 REM SET CONSTANTS 10 LET PI-3.1415926538# 20 LET C0-3E+10 25 LET BO-8.854E-14 DIMENSIONS OF COAX, SAMPLE 30 LET A-.152 40 LET C-.35 45 LET B-.345 46 INPUT "DIELECTRIC CONSTANT AND CONDUCTIVITY OF INNER + REGION "'El SI 47 LET E2-l! S2-0 'AIR GAP IN OUTER REGION 50 LET F-2 'FREQUENCY OF OPERATION IN GHZ 51 PRINT F 60 LET W«2*PI»F*lE+09 70 LET T1-S1/W/E0/E1: T2-S2/W/EO/E2 72 REM 74 REM COMPUTE CAPACITANCE MODEL SOLUTION FOR COMPARISON 75 REM 76 LET L1-L0G(C/B): L2-L0G(B/A): L3-L0G(C/A) 77 LET EM-(E2*E1‘2*L1»L3*(1 + Tl*2) ♦ E2‘2*B1*L2*L3*(1 + ♦ T2‘2) )/( (L1*E1 ♦ L2*E2)*2 ♦ (L1*B1*T1 ♦ L2*E2*T2)‘2) 78 LET TM-(E1*L1*T2*(1 + T1‘2) ♦ E2*L2*T1*(1 ♦ + T2 “2))/(El*Ll*(l ♦ T1‘2) ♦ E2»L2*(1 ♦ T2‘2)) 79 LET SM«TM*EM*EO*W 00 LET A4-W*SM/E0/C0‘2 85 LET B4-W‘2/C0‘2*EM 86 LET ZR-SQR(.5*B4+.5*SQR(B4*2+A4“2 )) 87 LET ZI-A4/2/ZR 90 PRINT "EM ";EM;" SM ";SM;" ZR ";ZR;" ZI ";ZI 95 INPUT G1,G2 96 LET ZR-G1: LET ZI-G2 100 LET A1#-W»S1/E0/C0‘2-2*ZR*ZI 110 LET A2#-W*S2/EO/CO‘2-2*ZR»ZI 120 LET B1#-W‘2»E1/C0‘2+ZI‘2-ZR*2 130 LET B2#-W“2*E2/C0* 2+Z1‘2-ZR“2 140 LET VI#— .5*B1#+.5*SQR(A1#*A1#+B1#*B1#) 145 LET V2#-Al#*Al#/4/Vl# 150 LET Wl#— .5*B2#+.5*SQR(A2#»A2#+B2#*B2#) 155 LET W2#-A2#*A2#/4/Wl# 160 LET J3-SQR(V2#) 1 2 288 165 170 175 180 185 199 200 205 210 220 230 231 240 250 251 260 270 280 281 290 300 301 310 320 330 331 340 350 351 360 365 370 380 385 390 392 400 410 411 420 430 435 440 441 450 460 461 470 480 485 LET J4-SQR(V1#) LET K3— SQR(W2#) LET K4«SQR(W1#) PRINT J3"2“J4"2-B1#,K3“2-K4“2-B2# PRINT 2*J3*J4-A1#,2*K3*K4-A2# REM COMPUTE COMPLEX BESSEL FUNCTIONS LET N*0 IF J3-0 THEN LET TH«PI/2*SGN(J4): GOTO 220 LET TH-ATN(J4/J3) LET R-SQR(J3‘2+J4‘2)*A GOSUB 1000 REM JO(KCIA) LET Ql-SR#: Q2-SI# GOSUB 1500 REM YO(KCIA) LET Q3-SR#: Q4-SI# LET R-R*B/A GOSUB 1000 REM JO(KCIB) LET Q5-SR#: Q 6 -SI# GOSUB 1500 REM YO(KCIB) LET Q7-SR#: Q 8 -SI# LET N-l GOSUB 1000 REM Jl(KClB) LET Ll-SR#: L2-SI# GOSUB 1500 REM Yl(KClB) LET L3-SR#: L4-SI# IF K3-0 THEN LET TH-PI/2*SGN(K4): GOTO 380 LET TH-ATN(K4/K3) LET RaSQR(K3“2+K4‘2)*B IF T2>1 THEN 410 GOSUB 1000 REM J1(KC2B) LET L5-SR#: L 6 -SI# GOSUB 1500 REM Y1(KC2B) LET L7-SR#: L8 -SI# LET N-0 IF T2>1 THEN 460 GOSUB 1000 REM JO(KC2B) LET Rl-SR#: R2-SI# GOSUB 1500 REM YO(KC2B) LET R3-SR#: R4-SI# LET R»R*C/B IF T2<1 THEN 490 289 486 487 490 491 500 510 511 520 530 531 540 541 550 560 561 570 571 580 590 600 601 610 611 620 630 631 640 641 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 805 810 820 825 LET R5-1: R6-0: R7-0: R8-0 GOTO S30 GOSUB 1000 REM JO(KC2C) LET R5-SR#: R 6 -SI# GOSUB 1500 REM YO(KC2C) LET R7-SR#: R 8 -SI# REM COMPUTE CYLINDRICAL COTANGENTS REM J1(KC1B)*Y0(KC1A) LET DR-L1*Q3-L2*Q4: DI-LI*Q4+L2*Q3 REM Y1(KC1B)*J0(KC1A) LET ER-L3*Q1-L4*Q2: EI-L3*Q2+L4*Q1 LET MR-DR-ER: MI-DI-EI REM JO(KC2B)YO(KC2C) LET DR-R1*R7-R2*R8: DI-R1*R8+R2*R7 REM YO(KC2B)*JO(KC2C) LET ER-R3*R5-R4*R6: EI-R3*R6+R4*R5 LET NR-DR-ER: NI-DI-EI LET N1-NR*MR-NI*MI : N2-NR*MI+NI *MR REM JO(KCIB)*Y0(KC1A) LET DR-Q5*Q3-Q6*Q4: DI«Q5*Q4+Q6*Q3 REM YO(KCIB)*J0(KC1A) LET ER-Q7*Q1-Q8*Q2: EI«Q7*Q2+Q8*Q1 LET MR-DR-ER: MI-DI-EI REM J1(KC2B)*Y0(KC2C) LET DR-L5*R7-L6*R8: DI-L5*R8+L6*R7 REM Y1(KC2B)*J0(KC2C) LET ER-L7*R5-L8*R6: EI-L7»R6+L8»R5 LET NR-DR-ER: NI-DI-EI LET D1»NR*MR-NI*MI: D2-NR*MI+NI*MR LET D-Dl“2+D2*2 LET HR— (N1*D1+N2*D2)/D: HI-(N2*D1-N1*D2)/D REM COMPUTE COMPLEX PROP CONSTANT LET D-E2"2+(S2/W/E0)“2 LET NR- (El *E2-fSl *S2 / (W*EO)“2 )/D : NI-(El *S2-E2*S1)/D/W/EO LET N1-NR*HR-NI*HI: N2-NR*HI+NI»HR LET AR-N1*2-N2“2: AI-2*N1*N2 LET D-{1-AR)*2+AI“2 LET ER-E2*AR+S2*AI/W/E0: EI-E2»AI-S2*AR/W/E0 LET MR-E1-ER: MI-S1/W/EO-EI LET PR-(MR*{1-AR)-MI*AI)/D:PI-{MR*AI+MI*(1-AR))/D LET A3-W“2*PI/C0*2 LET B3-W“2*PR/C0“2 LET XI— .5*B3-»-.5*SQR(A3*24-B3‘2) LET X2-A3‘2/4/Xl LET KR-SQR(X2) LET KI-SQR(Xl) 830 PRINT "REAL GUESS ";ZR;M REAL CALC";KR 840 PRINT "IMAG GUESS ";ZI;" IMAG CALC";KI 850 LET EN«CO‘2/W‘2*(KR‘2-KI‘2) 860 LET SN«CO“2/W*EO*2*KR*KI 870 PRINT "EN ";EN;" SN " ;SN 875 PRINT "K3 ";K3;" K4 ";K4;" J3 ";J3;" J4 ";J4 880 PRINT "B1 ";B1#;" B2 ";B2#;" B3 ";B3 890 PRINT "T1 ";T1;" T2 ”;T2 920 GOTO 90 990 END 1000 REMCOMPUTE COMPLEX BESSEL PUNCTION OF 1ST KIND 1010 REMN-ORDER OF J 1020 REM R-MAG OF ARG, TH-PHASE OF ARG 1025 REM RETURNS SR# AND SI# 1030 LET SR#-0 1035 LET SI#-0 1040 FOR M-0 TO 1000 1041 GOSUB 1050 1042 GOTO 1165 1050 REM COMPUTE FACTORIALS 1060 LET Fl#-0 1080 IF M-0 THEN 1120 1090 FOR U-M TO 1 STEP -1 1100 LET F1#-F1#+L0G(U ) 1110 NEXT U 1120 LET F2#-F1# 1130 IF N-0 THEN 1161 1135 LET F2#-0 1140 FOR U-M+N TO 1 STEP -1 1150 LET F2#-P2#+L0G(U ) 1160 NEXT U 1161 RETURN 1165 LET S#—SR# 1166 LET T#-SI# 1169 LET Pl#-(-l)‘M*EXP(M*L0G(R/2)-Fl#) 1170 LET P2#-P1#*EXP(M*L0G(R/2)-F2#): P#- P2#*(R/2)‘N 1171 LET SR#-SR#+P#*C0S((2*M+N)*TH) 1172 LET SI#-SI#+P#*SIN((2*M+N)*TH) 1175 IF ABS(S#-SR#)<.0001 AND ABS{T#-SI#)<.0001 AND M>0 + THEN 1200 1190 NEXT M 1200 RETURN 1500 REM COMPUTE COMPLEX BESSEL FUNCTION OF 2ND KIND 1510 REM N-ORDER OF Y 1520 REM R-MAG OF ARG, TH-PHASE OF ARG 1525 REM RETURN SR# AND SI# 1530 GOSUB 1000 1535 LET JR#-SR# 291 1537 LET JI#-SI# 1540 LET GAMMA -.577215665# 1545 LET P-LOQ(R/2)+GAMMA 1547 LET YR#—2/PI*(P*JR#-TH*JI#) 1548 LET YI#-2/PI* (P*JIF+TH*JR#) 1550 LET TR#-0 1551 LET TI#-0 1555 IP N-0 THEN 1800 1560 FOR M-0 TO N-l 1570 LET Fl#-1 1580 LET F2#-l 1585 IF M-0 THEN 1620 1590 FOR U-M TO 1 STEP -1 1600 LET F1#-F1#*U 1610 NEXT U 1620 IF N-M-1-0 THEN 1660 1630 FOR U-N-M-l TO 1 STEP -1 1640 LET F2#-F2#*U 1650 NEXT U 1660 LET P#—F2#/F1#*(R/2)"(2*M-N) 1661 LET TR#-TR#+P#*COS ((2 *M-N)*TH) 1662 LET TI#-TI#+P#*SIN((2*M-N)*TH) 1665 NEXT M 1666 LET SR#-0 1667 LET SI#-0 1670 FOR M-0 TO 1000 1660 LET Kl#-0 1690 LET K2#-0 1700 FOR K-l TO M+N 1701 LET K3#-l/K 1705 IF K>M THEN 1720 1710 LET K1#-K1#+K3# 1720 LET K2#-K2#+K3# 1730 NEXT X 1740 GOSUB 1050 1741 LET F1#-EXP(F1#) 1742 LET F2#-EXP(F2#) 1745 LET S#-SR# 1746 LET T#—SI# 1749 LET Pl#-(-l)*M*(Kl#+K2#)*(R/2)‘M/Fl# 1750 LET P2#-P1#*(R/2)'M/F2#: P#-P2#*(R/2)“N 1751 LET SR#»SR#+P#*COS((2*M+N)*TH) 1752 LET SI#-SI#+P#*SIN((2*M+N)*TH) 1760 IF ABS(SR#-S#)<.0001 AND ABS(SI#-T#)<.0001 THEN 1780 1770 NEXT M 1780 LET SR#-YR#-TR#/PI-SR#/PI 1785 LET SI#-YI#-TI#/PI-SI#/PI 1790 RETURN 292 1800 1801 1809 1810 1815 1820 1830 1840 1850 1860 1870 1880 1881 1890 1891 1895 1896 1900 1920 1940 1945 1990 LET SR#«0: SI8-0 FOR M«1 TO 1000 LET Kl#-0 FOR X-l TO M LET K3#-l/K LET X1#-E1#+K3# NEXT K LET Fl#-1 FOR U-M TO 1 STEP -1 LET F1#-F1#*U NEXT U LET S#-SR# LET T#-SI# LET Pl#-(-l)‘(M+l)»Kl#«(R/2)*M/Fl# LET P#-Pl#*(R/2)*M/F1# LET SR#-SR#4P#*C0S(2»M*TH) LET SI#-SI#+P#«SIN(2*M*TH) IF ABS(S#-SR#)<.0001 AND ABS(T#-SI#)<.0001 THEN 1940 NEXT M LET SR#-2/PI*SR#+YR# LET SI#-2/PI*SI#+YI# RETURN AffBTPIE J 8AMPL1 RKS0WAWC1 Ilf COAXIAL LIHI The quality factor, or Q of a resonating saaple can be determined from the measured reflection coefficient and la a meana of determining the conductivityof samples. be shown ualng(148). We note This can thatfor lowconductivities: ~ jr ( i - , J ' 1 ) At the resonant frequency, w r, from (157), ~ j* (> - <J-*> The reflection coefficient Is a minimum at the resonant frequency. When the reflected power Is doubled, at frequencies w r + Aw and w r + A w : yd ~ j, (i , * it?) ' ) [l - ( J - 3 ) Substituting into (148), 293 294 p(wf t Aw) (£' * ( W , 4 + *&[*■ ■ J2 w r°£0(r) | p(wr ± Aw) | * - I > - »>• + O f r ) 1] [ m [ ; I M ' ] 18«. ♦ 8 JTl«, ♦ 1 ) :rfer '- o 'r We know that 2 | p(wr) | * ■ | p(wr t Aw) | 2 (J-6) and thus W W £«£ - -4 ^ Which la by definition [74] the unloaded Q. (J-7) AEPIWPIX K 1LICTRQPLATIWQ PROCESS To fora an electroplated layer of copper on the non conducting samples used In aaterlal characterization we follow a four step process outlined In [8 8 ]. Plrst the surface Is "roughened" using an etchant solution; next a "sensitizing" solution Is used to prepare the sample surface to receive a metallic film:; a thin film of silver Is then deposited; and finally copper Is electroplated onto the sample using the thin silver film as a conducting surface. After the electroplating process Is complete, the sample Is sanded to remove metallization from the flat surfaces and to fit the sample holder. The chemical solution used to etch the sample Is 100 ml Sulfuric acid 15 g Potassium dlchromate 50 ml Hater It Is recommended that the sample be Immersed In this solution for 2 minutes. solutions are available. For delicate samples other etching It Is Important to keep the sample free of grease and other contaminants. To prepare the sample to receive the metallizing film, a sensitizing solution Is needed In order to Improve surface adhesion, producing a more even deposition. 295 296 1 g Stannous chloride 4 al Hydrochloric acid 100 al Water After lnerslng the saaple for 1 to 2 alnutes In this aqueous solution, with agitation, the saaple should be rinsed In distilled water. A silver flla can now be deposited on tbe saaple. The procedure Is to prepare two solutions a "slivering" solution and a "reducing solution." When these two solutions are alxed sliver will preclpate onto the saaple foralng a thin aetalllc layer. This Is the "alrror” process. Silvering Solution 6 g Silver nitrate 100 al Water 6 al Aaaonlua hydroxide (28k) Reducing Solution 6.5 al Poraaldehyde (40%) 100 al Water When the reaction is coaplete, an oha aeter can be used to verify the presence of a conductive flla. The saaple should be dryed overnight. Copper aay now be electroplated on the saaple. It Is recoaaended that a "flashing" bath, containing weak acid, be used to fora a thin copper layer to protect the silver flla froa daaage. The saaple can then be laaersed in the “plating" bath to build up to the required thickness. Two 297 electrodes arc Immersed Into tha solution, one connected to the saaple, and one a sheet of pure copper. The electrodes are connected to a battery capable of producing about 1 aap. Plashing bath (pH - 3) 40 Plating bath 200 15 - 100 g/1 Copper sulfate 2.5 g/1 Free sulfuric acid (pH ■ 1) -300 g/1 Copper sulfate - 40 g/1 Pree sulfuric acid We add a small quantity of gun arable to the solution to Inhibit the formation of dendrites and to aid the formation of a smooth copper layer. Techniques for depositing nickel and other metals onto non-conducting surfaces are also given In [81]. h i a Name: David Qray Hattara Date of Birth: Place of Birth: November 25, 1960 Bvanaton, IL Educational Background Master of Science In Electrical Engineering Northwestern University Evanston, IL 60208 Project: "An Automatic Scalar Network Analyzer for Microwave Materials Characterization" Graduated: June 1985 Bachelor of Science in Electrical Engineering Northwestern University Evanston, IL 60208 Graduated: June 1983 Publications 1. D.G. Watters and M.E. Brodwln, "Automatic Material Characterization at Microwave Frequencies," IEEE Instrumentation and Measurement Technology Confsrence, April 27 -29, 1987, Boston, MA, p 247. 2. D.G. Watters, M.E. Brodwln and G.A. Krlegamann, "Dynamic Tsmeperature Profiles for a Uniformly Illuminated Planar Surface," Material Research Society 1988 Spring Meeting. Reno, NV, April 4-8, 1988, M2.2. 3. D.G. Watters and M.E. Brodwln, "Automatic Material Characterization at Microwave Frequencies," IEEE Trans on Instrumentation and Measurement, IM-37, June 1988, p 280-285. 4. D.G. Watters, M.E. Brodwln and G.A. Krlegsmann, "Dynamic Temperature Profiles for a Uniformly Illuminated Planar Surface," 23rd International Microwave Power Symposium. Ottawa, Canada, August 29-31, 1988, p 45-47. 5. G.A. Krlegsmann, M.E. Brodwln and D.G. Watters, "Microwave Heating of a Ceramic Half-Space," Tech Rept 8729. Dept of Engr Science and Appl Math, Northwestern University, Evanston, IL, Aug 1988. 298 3 5556 019 165 778

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