EIII'I-lIII'I-DPII. W7 2 O ‘6 ‘6 “1 '1 CHIC AN ST TE UNIVERSITY LIiBRARIES|l ||| |||||||| | | |||| ||| 3 1293 00585 7200 u\ [ LRBRARY w‘sichigfifl State 1111 University | This is to certify that the dissertation entitled Theoretical formulation and experimental investigation of a cylindrical.cavity loaded with lossy dielectric materials pnmmfiaiby Haw-Hwa Lin has been accepted towards fulfillment of the requirements for Doctoral d . Electrical Engineering egree 1n Major professor L‘ | | 11171 2 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before one due. DATE DUE DATE DUE DATE DUE t‘ ’ MAE/33% 394 5 wigs—Yum ‘ ‘ M SU Is An Affirmative AetiorvKual Opportunity Institution Theoretical Formulation and Experimental Investigation of a Cylindrical Cavity Loaded with Lossy Dielectric Materials by Haw-Hwa Lin A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and System Science 1989 ABSTRACT Theoretical Formulation and Experimental Investigation of a Cylindrical Cavity Loaded with Lossy Dielectric Materials by Haw-Hwa Lin This dissertation presents a theoretical and experimental study of a cylindrical microwave cavity loaded with lossy dielectric materials. Theoretically the material- loaded cavity was formulated as a boundary value problem of a coaxially loaded waveguide connected to two homogeneous waveguides each with one end shorted. By means of mode-matching method, a set of homogeneous characteristic equations of the material-loaded cavity were derived. The resonant frequency (resonant length) associ- ated with each cavity mode is then found as the zero of the set of the chameteristic equations. As an essential part of the cavity problem, the propagation characteristics and field patterns of waveguides coaxially loaded with lossy dielectric and isotropic plasma rods were studied numerically for various waveguide modes. These waveguide solu- tions were used to solve the lossy material rod loaded cavity problem. In particular the cavity-short type configuration was solved numerically yielding cavity resonant lengths, resonant frequencies and cavity Q vs the complex dielectric constant and size of the load for TMm, T150ll and HE,” modes. Two different experimental systems were employed in this research work. The first is a low power system, which was designed to characterize the empty and material loaded cavities. The second is a high power system, which was designed for efficient microwave heating of materials. The resonant frequencies, resonant lengths and qual- ity factors measured with the low power system were used to verify theOretical results with materials of nylon and teflon rods, and a quartz tube containing ethylene glycol. The influence of the graphite fiber orientation in composite epoxy slabs on the cavity electromagnetic field intensities and patterns was also studied with the low power sys- tem. High power heating experiments were conducted to investigate the coupling of elecnomagnetic energy into lossy dielectric materials, including isotropic and anisotro- pic materials. Isotropic materials consisted of nylon and wet wood cylinders, and sili- con wafers. Anisotropic material included graphite reinforced epoxy slabs. The heat- ing experiments with nylon and wet wood cylinders demonstrated high coupling efficiency and the ability of on-line diagnosis of the complex dielectric constants of the load materials during heating using single-mode cylindrical cavity applicator. Heating uniformity and the factors that affect heating cycles were examined in these heating experiments using only a few watts of input power. To my parents J yne-ming and Yu-ying Lin and My wife Meij un g iv ACKNOWLEDGEMENTS The author wishes to thank Dr. Jes Asmussen for his encouragement and gui- dance during the course of this investigation. Thanks are also extended to Ben Manring and Ron Fritz for their friendship and help with many of the experi- ment 8. The author wishes to express his gratitude to his family for their emotional and financial support. Finally the author wishes to thank his wife, Meijung, for her understanding and encouragement throughout the duration of this work. This research was supported in part by DARPA (US. Army Grant DAAG46-85-k-0006). TABLE OF CONTENTS LIST OF TABLES ........................................................................................................ viii LIST OF FIGURES ...................................................................................................... ix CHAPTER 1 INTRODUCTION ............................................................................. 1 CHAPTER 2 LITERATURE REVIEW .................................................................. 6 2.1 Intorduction ........................................................................................... 6 2.2 Applications of microwave cavity applicator ....................................... 7 2.2.1 Permittivity measurements ....................................................... 7 2.2.2 Microwave heating .................................................................. 22 2.2.3 Microwave plasma applications ................................................ 25 2.3 Method of analyzing coaxially loaded cavity ....................................... 27 2.3.1 Theoretical formulation ............................................................ 27 2.3.2 Mode designation ...................................................................... 38 CHAPTER 3 THEORETICAL DEFINITION OF THE DIELECTRIC- LOADED CYLINDRICAL CAVITY ............................................... 40 3.1 Introduction ........................ -- ......... 40 3.2 Theoretical formulation of the dielectric-loaded cavity ....................... 40 3.3 Review of theoretical formulation of cylindrical waveguide ............... 44 3.3.1 Cylindrical wave potential ........................................................ 44 3.3.2 The homogeneous cylindrical waveguide ................................. 48 3.3.3 The coaxially loaded cylindrical waveguide ............................ 54 3.4 Derivation of the cavity chaIaCteristic equation ................................... 70 3.4.1 Cavity-open type ....................................................................... 70 3. 4. 2 Cavity-short type ....................................................................... 76 3. 4. 3 Cavity-image type .................................................. 79 3.5 Stored energy, absorbed power and quality factor for the dielectric-loaded cavity ......................................................................... 82 3.5.1 Introduction ............................................................................... 82 3.5.2 Stored energy ............................................................................ 82 3.5.3 Absorbed power ........................................................................ 87 3.5.4 Quality factor ......................... . .................................................. 90 CHAPTER 4 NUMERICAL SOLUTIONS AND TECHNIQUES ......................... 91 4.1 Introduction ........................................................................................... 91 4.2 Numerical technique ............................................................................ 92 vi 4.3 CHAPTER 5 5.1 5.2 5.3 5.4 CHAPTER 6 6. l 6.2 6.3 CHAPTER 7 7.1 7.2 4.2.1 Muller’s method ............... 92 4.2.2 Root-searching algorithm ......................................................... 94 Numerical solutions .............................................................................. 96 4.3.1 Introduction ............................................................................... 96 4.3.2 Radial boundary value problem (dielectric-loaded waveguide problem) ................................................................. 96 4.3.3 Axial boundary value problem (dielectric-loaded cavity problem) .................................................................................... 186 EXPERIMENTAL SYSTEMS ANI) TECHNIQUES ...................... 235 Introduction ........................................................................................... 235 Experimental cavity design considerations and description ................. 235 5.2.1 Experimental cavity design considerations ............................... 235 5.2.2 Experimental cavity description ............................................... 243 Experimental systems and circuits ........................................................ 249 5.3.1 Experimental systems ............................................................... 249 5.3.2 Equivalent circuit of a material-loaded cavity applicator ......... 256 Experimental procedures and techniques ............................................. 261 5.4.1 Definitions ................................................................................ 261 5.4.2 Low power swept-frequency measurements ............................. 263 5.4.3 High power single-frequency measurements ............................ 267 EXPERIMENTAL RESULTS .......................................................... 272 Introduction ........................................................................................... 272 Low power measurements (verification of theory) ............................... 272 6.2.1 Introduction ............................................................................... 273 6.2.2 TMmz mode ............................................................................... 276 6.2.3 HE,“ mode ................................................................................ 278 6.2.4 Discussion ................................................................................. 284 High power heating measurements ....................................................... 289 6.3.1 Heating of nylon 66 cylinders ................................................... 289 6.3.2 Heating of wet wood cylinders ................................................. 295 6.3.3 Heating of silicon wafer ............................................................ 299 6.3.4 Low power and high power heating of graphite reinforced epoxy slabs ................................................................................ 301 6.3.5 Discussion ................................................................................. 313 SUMMARY AND RECOMMENDATIONS ................................... 314 Summary of results ............................................................................... 314 7.1.1 Theoretical formulation and numerical analysis ....................... 314 7.1.2 Exprimental verificaition of theory and microwave heating of materials ............................................................................... 315 Recommendations ................................................................................. 318 vii 7.2.1 Theory ............... 318 7.2.2 Experiments .............................................................................. 318 APPENDIX A FORMULAS FOR EVALUATING THE MATRIX ELEMENTS OF EQUATION 3.102 .............................................. 320 APPENDIX B IMPORTANT INTEGRALS USED IN THE ANALYTICAL FORMULATION OF A CYLINDRICAL CAVITY LOADED WITH LOSSY DIELECTRIC MATERIALS ................................ 324 REFERENCE ................................................................................................................ 326 viii Table 6.1 Table 6.2 LIST OF TABLES Comparison between theoretical and experimental results for 'I'Mmz mode, (a) constant resonant length of 15.5 cm, (b) constant resonant fi'equency of 2.45 GHz ................................................. 277 Comparison between theoretical and experimental results for HE,“ mode, (a) constant resonant length of 6.94 cm, (b) constant resonant frequency of 2.45 GHz ................................................. 283 Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 3.1 Figure 3.2 Figure 3.3 Figrue 3.4 Figure 3.5 Figure 3.6 Figure 4.1a Figure 4.1b Figure 4.2a Figure 4.2b Figure 4.3a LIST OF FIGURES Side-view of diffemet cavity types, (a) cavity-open type, (b) cavity-short type and (c) cavity-image type ......................................... 3 Cylindrical cavity loaded with dielectric rod ........................................ 9 Cylindrical cavity loaded with dielectric disk ...................................... 11 Block diagram of the active system ...................................................... 18 Cross-section of a waveguide applicator .............................................. 24 Side-view of a multilayer dielectric resonator ...................................... 32 Geometry of a dielectric resonator ........................................................ 36 Dielectric-loaded cavity dimensions ..................................................... 41 Coordinate system of the dielectric-loaded cavity ................................ 43 Cylindrical coordinates for the homogeneous waveguide .................... 49 Coordinate system for a coaxially loaded cylindrical waveguide ........ 55 Coordinate system for the cavity-short type ............ I ............................. 7 7 Coordinate system for the cavity-image type ....................................... 8O to-B and (0-0. diagram for TMm mode, where a=l.27 cm, b=7/62 c, and e=3.03-j0.39 ................................................................... 102 (tr—[3 and (0—0: diagram for TEOl mode, where a=l.27 cm, b=7/62 c, and 8=3.03-j1.0 ..................................................................... 102 (tr—[3 diagram for eight waveguide modes, where b/a=6 and 8=3.03-j0.039 ........................................................................................ 104 m—a diagram for eight waveguide modes, where b/a=6 and 8=3.03-j0.039 ....................................................................................... 105 (0—13 diagram for eight waveguide modes, where b/a=6 and 8=3. O3-j0. 39 .............. 107 Figure 4.12f Figure 4.13 Figure 4.14a Figure 4.14b Figure 4.14c Figure 4.15a Figure 4.15b Figure 4.15c Figure 4.16a Figure 4.16b Figure 4.17a Figure 4.17b Figure 4.170 Figure 4.17d Figure 4.17e , b/a=6 and Koa=2.5 ............... 146 Electric field patterns for TE01 mode, where e=3.03-j 10.0 , b/a=6 and Koa=2.5 ............................................................................... 147 Electric field patterns for TEO1 mode for various KOa, where e=3.03-j10.0 and b/a=6 .............................................................. 149 Electric field patterns for TED1 mode, where e=3.03-j0.039 , a=6.35 cm, b=7.62 cm and Koa=3.26 .................................................. 150 Electric field patterns for TE01 mode, where e=3.03-j0.039 , a=2.54 cm, b=7.62 cm and Koa=3.26 .................................................. 151 Electric field patterns for TE01 mode, where 8= 3. 03-j0. 039 , a=0. 635 cm, b=7. 62 cm and r<0a=3. 26 ................................................ 152 (0'5 diagram for HE“l mode for various 8”, where b/a=6 and e’=3.03 .................................................................................................. 154 Low fiequency portion of the (JD-B diagram for HE11 mode shown in Figure 4.15a ........................................................................... 155 co—a diagram for HEu mode for various 8”, where b/a=6 and 8’=3.03 .................................................................................................. 156 to-B diagram for HEn mode for various a/b, where e=3.03-j0.039 ........................................................................................ 158 (o-a diagram for HE“ mode for various a/b, where =3.03-j0.039 ........................................................................................ 159 Electric field patterns for HE" mode, where e=3.03-j0.039 , b/a=6 and Koa=2.0 ............................................................................... 160 Electric field patterns for I-IE11 mode, where e=3.03-j0.39 , b/a=6 and Koa=2.0 ............................................................................... 161 Electric field patterns for HEll mode, where e=3.03-j1.0 , b/a=6 and Koa=2.0 ............................................................................... 162 Electric field patterns for HEll mode, where e=3.03-j2.0 , b/a=6 and Koa=2.0 ............................................................................... 163 Electric field patterns for HEn mode, where (i=3. 03-j2. 5 ,b/a=6 and K0a=2. 0 ............................................................................... 164 xiii Figure 4.17r Figure 4.17g Figure 4.17h Figure 4.18a Figure 4.18b Figure 4.18c Figure 4.19a Figure 4.1% Figure 4.190 Figure 4.20a Figure 4.20b Figure 4.200 Figure 4.20d Figure 4.21a Electric field patterns for HEll mode, where e=3.03-j3.9 , b/a=6 and Koa=2.0 .............................................................................. 165 Electric field patterns for HE“ mode. where e=3.03-j5.0 , b/a=6 and Koa=2.0 ............................................................................... 166 Electric field patterns for HE” mode, where e=3.03-j10.0 , b/a=6 and Koa=2.0 ............................................................................... 167 Electric field patterns for HE, 1mode, where e=3.03-j0.039 , b/a=6 and Koa=l.8 ............................................................................... 170 Electric field patterns for HE“ mode, where e=3.03-j0.039 , b/a=6 and Koa=0.65 ............................................................................. 171 Electric field patterns for HE“ mode, where e=3.03-j0.039 , b/a=6 and Koa=0.26 ............................................................................. 172 Electric field patterns for HEnmode, where e=3.03-j0.039 , a=6.35 cm, b=7.62 cm and Koa=3.26 ............... Electric field patterns for I-IEumode, where e=3.03-j0.039 , a=2.54cm, b=7.62 cm and Koa=l.3 ..................................................... 174 Electric field patterns for HEnmode, where e=3.03-j0.039 , a=0.635 cm, b=7.62 cm and Koa=0.326 _- ar—a and to-B diagrams for TMn mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=4.75 cm, N=10ll and v/03=0.1 .............................................................................. 178 to—a and 01—13 diagrams for TM.)l mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=4.75 cm, N=1o“ and v/co=1.o .............................................................................. 179 arr-0t and (D-B diagrams for TMm mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=4.75 cm, N=1o12 and v/co=0.l .............................................................................. 180 to-ot and co-B diagrams for TMOl mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=4.75 cm, N=1012 and v/m=l.0 .............................................................................. 1181 (o-a and (11—13 diagrams for TMo, mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=8.89 cm,- xiv Figure 4.21b Figure 4.210 Figure 4.21a Figure 4.22a Figure 4.22b Figure 4.220 Figure 4.23a Figure 4.23b Figure 4.24a Figure 4.24b Figure 4.25a Figure 4.25b Figure 4.26 Figure 4.27 Figure 4.28 N=10” and v/o)=0.l .............................................................. -. ............... 182 H and (tr-[3 diagrams for TMOl mode waveguide loaded with isotropic plasma rod, where a=l.25 cm, b=8.89 0m, N=10“ and v/(n=l.0 .............................................................................. 183 m—a and tar-B diagrams for TMO1 mode waveguide loaded with isotropic plasma rod, where a=l.25 0m, b=8.89 0m, 11:1012 and vlm=0.1 .............................................................................. 184 (11—01 and (rt—B diagrams for TMm mode waveguide loaded with isotropic plasma rod, where a=l.25 0m, b=8.89 0m, 11:1012 and v/co=1.0 .............................................................................. 185 Resonant frequency vs. 8’ for TM(n2 mode for various e”, where a=0.6350m, b=7.62 0m and L,=15.5 cm .................................... 187 Low dielectric constant portion of Figure 4.22a ................................... 188 Resonant frequency vs. 8’ for man mode for various 8”, where a=l.27 cm, b=7.62 cm and L,=15.5 0m ..................................... 190 Resonant frequency vs. 8’ for HE," mode for various 8”, where a=0.635 0m, b=7.62 cm and L,=6.94 cm ................................... 191 Resonant frequency vs. 8’ for HE", mode for various 8”, where a=l.27 0m, b=7.62 cm and L,=6.94 0m ..................................... 192 Resonant frequency vs. 8’ for TED" mode for various 8”, where a=0.635 cm, b=7.62 cm and L,=30.8 0m ................................... 193 Resonant frequency vs. 6’ for TED" mode for various 8”, where a=l.27 0m, b=7.62 0m and L,=30.8 0m ..................................... 195 Resonant frequency vs. 8’ for TMm mode for various 8”, where a=0.625 cm, b=8.89 0m and L,=14.407 0m ............................... 196 Low dielectric constant portion of Figure 4.25a ................................... 197 Resonant frequency vs. 8’ for HEm mode for various 8”, where a=0.625 0m, b=8.89 cm and L,=6.69 cm ................................... 198 Resonant frequency vs. 8’ for TEm1 mode for various 8”, where a=0.635 0m, b=8.89 cm and L,=11.255 0m ............................... 199 Resonant frequency vs. a/b for various modes, where b=7.62 cm XV Figure 4.29 Figure 4.30a Figure 4.30b Figure 4.31 Figure 4.32a Figure 4.32b Figure 4.320 Figure 4.32d Figure 4.33a Figure 4.33b Figure 4.330 Figure 4.33d Figure 4.348 Figure 4.34b Figure 4.340 Figure 4.34d and 8=3.03-j0.039 ............... 200 Resonant frequency vs. a/b for various modes, where b=8,89 cm and 8=3.03-j0.039 ................................................................................. 201 Resonant length vs. 8’ for TMO12 mode for various 8”, where a=0.635 cm, b=8.89 cm and fo=2.45 GHz ................................. 203 Enlargement of the low 8’ portion of Figure 4.30a ............................... 204 Resonant length vs. a/b for various modes, where b=7.62 cm, 8=3.03-j0.039 and fo=2.45 GHz ............................................................ 206 Q vs. 8’ for TMon mode for various 8”. where a=0.635 cm, b=7.62 cm and fo=2.45 GHz ................................................................. 208 Q,l vs. 8’ for mom mode for various 8”, where a=0.635 0m, b=7.62 cm and fo=2.45 GHz ................................................................. 209 Q“ vs. 8’ for TMo‘n mode for various 8”, where a=0.635 cm, b=7.62 0m and fo=2.45 GHZ ................................................................. 210 Qd vs. 8’ ,for mom mode for various 8”, where a=0.635 0m, b=7.62 0m and fo=2.45 GHz ................................................................. 211 Q vs. 8’ for HE", mode for various 8”, where a=0.635 0m, b=7.62 0m and fo=2.45 GHz ................................................. 213 Q: vs. 8’ for HE," mode for various 8”, where a=0.635 cm, b=7.62 0m and f0=2.45 GHz ................................................................. 214 Qcl vs. 8’ for HE“, mode for various 8”, where a=0.635 cm, b=7.62 0m and f0=2.45 GHz ................................................................. 215 Q; vs. 8’ for HE," mode for various 8”, where a=0.635 cm, b=7.62 0m and f0=2.45 GHz ................................................................. 216 Q vs. 8’ for TEO" mode for various 8”, where a=0.635 0m, b=7.62 0m and fo=2.45 GHz ................................................................. 217 Qcl vs. 8’ for TEou mode for various 8”, where a=0.635 0m, b=7.62 cm and fo=2.45 GHz ................................................................. 218 Q,“ vs. 8’ for TED“ mode for various 8”, where a=0.635 cm, b=7.62 0m and fo=2.45 GHz ................................................................. 219 ch vs. 8’ for TE011 mode for various 8”, where a=0.635 cm,. xvi Figure 4.35a Figure 4.35b Figure 4.350 Figure 4.35d Figure 4.36a Figure 4.36b Figure 4.360 Figure 4.36d Figure 4.37a Figure 4.37b Figure 4.370 Figure 4.37d Figure 5.1 Figure 5.23 Figure 5.2b Figure 5.3a =7.62 cm and f0=2.45 GHz ............... 220 Q vs. 8” for various modes, where 8’=3.03, a=0.635 0m, b=7.62 0m and fo=2.45 GHz ................................................................. 222 Q¢l vs. 8” for various modes, where 8’=3.03, a=0.635 0m, b=7.62 0m and fo=2.45 GHz ................................................................. 223 Qcl vs. 8” for various modes, where 8’=3.03, a=0.635 cm, b=7.62 cm and fo=2.45 GHz ................................................................. 224 ch vs. 8” for various modes, where 8’=3.03, a=0.635 cm, b=7.62 cm and fo=2.45 GHz ................................................................. 225 Enlargement of the small 8” portion of Figure 4.35a ........................... 226 Enlargement of the small 8” portion of Figure 4.35b ........................... 227 Enlargement of the small 8” portion of Figure 4.350 ........................... 228 Enlargement of the small 8” portion of Figure 4.35d ........................... 229 Q vs. a/b for various modes, where 8=3.03-j0.039, b=7.62 0m and fo=2.45 GHz ................................................................................... 230 Qd vs. a/b for various modes, where 8=3.03-j0.039, b=7.62 0m and f0=2.45 GHz ................................................................................... 231 Qcl vs. a/b for various modes, where 8=3.03-j0.039, b=7.62 cm and f0=2.45 GHz ................................................................................... 232 Qa vs. a/b for various modes, where 8=3.03-j0.039, b=7.62 0m and fo=2.45 GHz ................................................................................... 233 Transverse electric and magnetic field patterns for various waveguide modes .................................................................................. 238 Resonant frequncy vs. resonant length for a seven-inch cavity for various cavity modes ....................................................................... 239 Resonant frequency vs. resonant length for a seven-inch cavity for various cavity modes in the frequency range of 1.6 to 3.2 GHz ....................................................................................... 240 Resonant frequency vs. resonant length for a six-inch cavity for various cavity modes ....................................................................... 241 xvii Resonant frequency vs. resonant length for a six-inch cavity Figure 5.3b for various cavity modes in the frequency range of 1.6 to 3.2 GHz ....................................................................................... 242 Figure 5.4 Photograph of a seven-inch cavity ........................................................ 244 Figure 5.5 Side-view of a seven-inch cavity with fixed bottom plate .................... 246 Figure 5.6 Side-view of a seven-inch cavity with removable bottom plate ........... 247 Figure 5.7 Photograph of a six-inch cavity ............................................................ 250 Figure 5.8 Cut-away view of a precision six-inch cavity ....................................... 251 Figure 5.9 Low power swept-frequency mode system diagram ............................ 252 Figure 5.10 High power single-frequency mode system diagram ........................... 253 Figure 5.11 Equivalent circuit of the material-loaded cavity ................................... 257 Figure 5.12 Q measurement using Q—curve ............................................................. 265 Figure 6.1 Configuration of the base plate ............................................................. 274 Figure 6.2 Microcoax probe and base plate dimensions ........................................ 275 Figure 6.3 Field patterns for 0.635 cm teflon rod, TMmz mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 279 Figure 6.4 Field patterns for 0.635 cm nylon rod, 'I'Mm mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 280 Figure 6.5 Field patterns for 1.27 cm nylon rod, TM012 mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 281 Figure 6.6 Field patterns for 0.2 cm ethylene glycol, TMm mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 282 Figure 6.7 Field patterns for 0.635 cm teflon rod, HE,“ mode, (a) actual magnitude and (b) comparison between theoretical and experimental results ........................................................... .............. 285 xviii Figure 6.8 Figure 6.9 Figure 6.10 Figure 6.11a Figure 6.11b Figure 6.12a Figure 6.12b Figure 6.13a Figure 6.13b Figure 6.130 Figure 6.14 Figure 6.15 Figure 6.16 Figure 6.17a Figure 6.17b Figure 6.18a Figure 6.18b Figure 6.19a Field patterns for 0.635 cm nylon rod, HE," mode, (3) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 286 Field patterns for 1.27 cm nylon rod, HEm mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 287 Field patterns for 0.2 cm ethylene glycol, HE,“ mode, (a) actual magnitude and (b) comparison between theoretical and experimental results .............................................................................. 288 8,’ of nylon 66 versus time .................................................................... 291 8,” of nylon 66 versus time ................................................................... 292 8,’ of nylon 66 versus temperature ........................................................ 293 8,” of nylon 66 versus temperature ....................................................... 294 8,’ of wet wood versus time .................................................................. 296 e,” of wet wood versus time .................................................................. 297 Coupling efficiency of wet wood heating versus time ......................... 298 Hearing of 3-inch silicon wafer ............................................................ 300 Equilibrium temperature versus input power ........................................ 302 Six inch cavity applicator loaded with graphite-epoxy slabe ............... 304 Radial electric field versus 4) at z=2.8 cm for 'I'E""u2 mode, graphite- epoxy slab on bottom of cavity ............................................................. 305 Radial electric field vs. 41 at z=4.65 cm for TE"‘"2 mode, graphite-epoxy slab on top of cavity ..................................................... 307 Equilibrium temperature vs. temperature probe position for TE""112 mode, graphite-epoxy slab on bottom of cavity ........................ 308 Equilibrium temperature vs. temperature probe position for TE‘“211 mode, graphite-epoxy slab on bottom of cavity ........................ 309 Temperature vs. time for the heating of graphite-epoxy slab . xix , input power=5W ................................................................................ 31 1 Figure 6.19b Temperature vs. time for the heating of graphite-epoxy slab for power levels of 5 and 10 W, fiber orientation is perpendicular to the coupling prob .............................................................................. 312 XX Chapter 1 INTRODUCTION The microwave cavity applicator has many current and potential applications in industry and fundamental research, such as permittivity measurements, the heating and curing of epoxy and composite materials, the heating of food, semiconducting and bio- logical materials as well as the use as microwave plasma and ion sources. Accord- ingly in recent years intense attention has been paid to the design and modeling of the microwave cavity applicator. The design of a cavity applicator that is capable of coupling electromagnetic energy efficiently into the load is complex because energy transfer is a function of many variables that change as processing takes place. These variables include changes in dielectric properties, dimensions and temperature of the load material, the variations in the dimensions of the cavity applicator and the electromagnetic field patterns. For instance, as the material is heated during processing, the complex dielectric constant, temperature and the dimensions of the load change, which in turn alters the field pat- terns. Because of these changes, a mismatched, inefficient processing system often results. The energy transfer is also influenced by the operating frequency, the relative position of the load within the applicator and the shape of the load. For efficient energy transfer to load, the factors that influence the energy transfer need to be under- stood and taken into account in the design of the cavity applicator and some mechan- ism must be found to compensate for the varying material parameters to keep the material-loaded cavity matched during processing. Hence it is useful to develop a theoretical model to Study the energy transfer mechanism as a function of the parameters mentioned above. Based on the informa- tion attained from the model, it may then be possible to design an optimized cavity applicator and efficient and controllable microwave heating systems.‘ The research 2 presented here is concerned with the development of efficient and controllable microwave applicators and associated heating systems. In particular it focuses on the theoretical modeling of idealized material loaded cylindrical cavities and the associated experimental verification with selected material loads. This dissertation consists of theoretical and experimental parts. The objective of theoretical part is to develop a model of the cylindrical cavity loaded with lossy materials. From this model, the resonant frequency, resonant length, quality factor and the electromagnetic field patterns are investigated numerically as a function of the complex dielectric constant and size of the load material for a given mode. The objec- tives of the experimental part are to: (1) verify the theoretical model using selected load materials and resonant modes in the low power experiments, (2) design low power experiments to study the effect of anisotropic materials on the field patterns and (3) design high power heating experiments for efficient heating of rod and slab materi- als, including regular dielectric materials, semiconducting wafers and composite materials. The overall layout of this dissertation is as follows. Chapter 2 presents a detailed literature review of the applications of the microwave cavity applicator as well as the methods that have been used to analyze dielectric-loaded cavity. The theoretical for- mulation of the cavity loaded with lossy dielectric materials is derived in Chapter 3. In general, the loaded cavity is treated as a boundary value problem of a coaxially loaded waveguide connected to two homogeneous waveguides each with one end shorted. The cavity walls are assumed to be perfectly conducting in this formulation. According to the cylindrical load material dimensions and positions in the cavity, three different material loaded cavity types, as shown in Figure 1.1, are classified: (1) cavity-open type, (2) cavity-short type, (3) cavity-image type. The classical EM analysis of the waveguide coaxially loaded with lossy dielectric rod was reviewed [87] and then a transcendental waveguide characteristic equation is derived. By means of W ) ( (b) W (c) cavity-image type mode-matching method, a set of the cavity characteristic equations is derived by enforcing axial boundary conditions of continuous tangential electric and magnetic fields over the waveguide-waveguide interfaces and vanishing tangential electric fields on the conducting walls. Also included in Chapter 3 is the derivation of the stored energy, absorbed power and quality factor of the material loaded cavity. The overall quality factor of the loaded cavity consists the quality factor due to the dielectric losses and the quality factor due to the conducting wall losses. Since in the theoretical model the cavity walls were assumed perfectly conducting, it is necessary to use perturbation method [87] for the calculation of the power absorbed by the conducting walls. The description of the numerical algorithm and the numerical solutions of the characteristic equations derived in Chapter 3 are presented in Chapter 4. By solving the waveguide characteristic equations the (r) - B and co - a diagrams are studied for eight waveguide modes, including TM», 7750, and hybrid modes. The or - B and co - or characteristics and field patterns for TMm, TE.” and HE” modes are examined in detail as a function of the complex dielectric constant and radius of the dielectric material. In addition, a special case of the propagation characteristics of the waveguide loaded with isotropic cold plasma rod is investigated as a function of the electron density, normalized plasma collision frequency (relative to excitation frequency) and cavity radius for TM.” mode. Slow-wave and fast-wave regions are identified on the a) -B diagrams. The cavity characteristic equations of the cavity-short type are solved numerically for the resonant frequency and resonant length for TM 012, T150ll and 11sz111 mode, with the complex dielectric constant and rod radius as the running parameters. In addition, the overall quality factor, the quality factor due to the dielectric losses and the quality factor due to conducting wall losses are also investigated as a function of the complex dielectric constant and rod radius. The description of experimental cavity applicators, experimental systems and techniques are introduced in Chapter 5. The design considerations of the cavity appli- cator are discussed and the structures of the seven-inch and six- inch cavities are described briefly. Two experimental systems were employed for the experimental measurements. The first is a low power swept-frequency system used for the charac- terization of the empty and loaded cavity applicators. The second is the high power single-frequency systems for the microwave heating experiments. An equivalent cir- cuit description of the material loaded cavity applicator is derived. Detailed descrip- tions of experimental procedures and techniques for the low power and high power experiments are also included. Chapter 6 presents the results of the low power experiments designed to check the theoretical calculations and the results of the high power heating experiments designed to investigate the microwave heating of solid materials. In the low power verification experiments, load materials included teflon, nylon and ethylene glycol rods. In the high power heating experiments, load materials consisted of nylon and wet wood cylinders, silicon wafers and graphite reinforced epoxy slabs. A summary of the work presented in the thesis is given in Chapter 7. Recom- mendations for future research are also included. Chapter 2 LITERATURE REVIEW 2.1 Introduction The utilization of microwave energy for the generation of heat was discovered accidently during testing of magnetrons at the Microwave and Power Tube Division of Raytheon in 1950. [1] Since then, microwave heating technology in industry has dou- bled every ten years. In the late 60’s and early 70’s microwave energy was used to dry potato chips and paper match books. In the 70’s microwave heating were mainly for food processing, rubber processing and a few for the heating of solid materials [1], [2]. For the past ten years, microwave cavity applicator has been used widely for the heating of composite, semiconducting and biological materials, for sealing cartons and for the use as microwave plasma sources. Due to its growing applications in industry, extensive attention has been paid to the design and modeling of microwave cavity applicator. It was not until 1951 that the first commercial available microwave oven was built as a floor model restaurant unit which led to the microwave home oven in 1960. All of the early industrial and domestic microwave ovens had a multimode resonant structure. The disadvantages of the multimode applicators are inefficiency, no on-line diagnosis and control of the heating processes. Although single mode cavity was util- ized for measuring permittivity as early as 1960 [3], applications of single mode microwave applicators for rapid heating of nylon monofilament [2] and generation of plasma were reported in the 70’s [4]. Since the advent of single mode applicator, the investigation of the coupling between electromagnetic and material and the develop- ment of on-line diagnosis during heating process have been made possible. Theoretical modeling of material loaded microwave cavity is important for the understanding of electromagnetic/material interaction. Applying the theoretical model to experiments, material properties during processing then may be diagnosed and preci- sion control mechanism may be achieved. Many rigorous methods of analyzing dielectric loaded resonators have been developed during the last two decades [7], [8], [56] - [79]. Since most of the resonators have been used for such applications as tem- perature stable oscillators, low noise microwave synthesizers and bandpass filters, dielectrics considered in the modeling are generally low loss and high dielectric con- stant. In microwave heating, however, materials of various sizes, shapes and dielectric pr0perties are encountered. Complex dielectric constant needs to be taken into account in the modeling. Although modeling approaches are similar, the numerical results of the lossy dielectric cases are quite different from those of the low loss (or lossless) dielectric cases. In this chapter, applications of the single mode microwave cavity applicators dur- ing the past 25 years are reviewed and a literature survey of various methods of analyzing dielectric loaded cavities is presented. 2.2 Applications of microwave cavity applicator 2.2.1 Permittivity measurements Interest in permittivity measurements at microwave frequencies has grown dramatically in the last few decades in order to characterize the properties of materials. For instance, in semiconductor industry, the permittivity of thin dielectric films has been received considerable attention due to its crucial role in the properties of sem- iconductor thin films and integrated circuits. The properties of thin biological films are important because their behavior at microwave frequencies is related to microwave biological hazards. Knowledge of the permittivity of industrial materials permits the optimization of the microwave heating process and real time control of the material properties to achieve desirable quality of the products. Various methods for measuring the permittivity of dielectrics at microwave fre- quencies have been seen in literature. These methods can be divided into two categories: (1) the transmission line methods and (2) the resonant cavity methods [84]. Since the permittivity measurement using microwave cavity applicator is of major con- cem in this dissertation, all the papers cited here make use of resonant cavity methods. Careful study of the literature reveals that the complex dielectric constants of materials are determined either by exact formulation of the material loaded cavity or by cavity perturbation technique. Hence, the resonant cavity measurement techniques are further divided into exact formulation and perturbation technique two groups. 2.2.1.1 Exact formulation The first tunable cylindrical cavity used for the measurements of the complex dielectric constant of a fairly low loss dielectric rod was demonstrated by J. K. Sinha and J. Brown in 1960 [3]. The cavity was excited at 9.375 GHz in TMo, or T1301 modes. As shown in Figure 2.1, a rod was placed into the cavity through a hole on the center of the bottom plate of the cavity but did not contact the top plate. The cav- ity resonant length L was plotted as a function of rod length I within the cavity. The propagation constant in the dielectric was obtained from the mean slope .of above curve incorporated with the empty waveguide separation equation. Then, the relative dielectric constant was calculated by solving the characteristic and separation equations of the coaxially loaded waveguide. From the theoretical formulation of Q factor of the material loaded cavity and the Q measurements, the loss tangent was determined. The Q factor was measured by the insertion attenuation of the cavity at resonance. Meas- urements were made on polystyrene and titaniue dioxide but no comparison with pub- lished data was given. Sinha and Brown failed to point out that one approximation was actually made when calculating the complex dielectric constants. That is, the real part of the dielec- tric constant was evaluated from the characteristic equation of coaxially loaded waveguide without including any losses. ,N |......._... | 2 | Z l _r_ A%/ j /—*1 21 I‘— dielectric rod Figure 2.1 Cylindrical cavity loaded with dielectric rod 10 In 1964, D. T. Paris used a mode matching technique to formulate a slab—loaded T50“, cylindrical cavity [5]. With this model, the complex permittivity of the sample materials was derived. Experiments were conducted using a right-circular cylindrical cavity in 7201,, mode. The configuration of the loaded cavity is depicted in Figure 2.2. A disc-shaped dielectric slab of thickness d completely filled the cross section of the cavity. The cavity was kept at resonance by adjusting the plunger position to the resonant length 1. Two measurements, resonant frequencies and propagation constants were required to determine the complex dielectric constant. No experimental data was provided. Eight years later, J. Hanfling and L. Botte [6] also measured the dielectric constant of a disk in a cutoff circular waveguide cavity. But relation between the measured cavity resonant frequencies and the dielectric constant was established by means of the transverse-resonance procedure. The cavity was 0.9 inches in diameter and 1.895 inches long. Only real part of the dielectric constant of beryllia was meas- ured and 0.1% accuracy was reported. The dielectric properties of powder under controlled pressure were first measured by G. Roussy and M. Felden using a T501, mode cavity [7]. The material under test was contained in a tube of quartz, placed along the axis of the cavity. The lengths of the material and cavity were equal. They solved the loaded cavity problem as a sec- tion of a waveguide, loaded with multilayers of materials. Experiments were per- formed in two resonant modes, T80“ and TED], modes, respectively. All the materials tested have dielectric constants less than 8-j0.01. Experimental results were within $0.398 for the real part and within 0.5% for the imaginary one. In a manner similar to Roussy and Felden, S. Li, et a1. [8] also considered a lossy dielectric rod situated in the center of a cylindrical cavity, filling the whole length of the cavity. The cavity was operated in TMmo mode. A dispersion equation was derived by field matching technique, in which field components were expressed in series forms. For the first time in literature the perturbation introduced by the insertion 11 ”N I | plunger d i dielectric disk cylindrical cavity 1 to source —_1 LT to _l detector g t .. r Figure 2.2 Cylindrical cavity loaded with dielectric disk 12 hole was taken into account in the formulation of the problem. Still, the cavity wall losses were calculated by means of cavity wall perturbation procedures. Measurements of resonant frequencies and cavity quality factors were taken for evaluating e’ and e”. Reported experimental results show that the real and imaginary parts of the permittivi- ties of various materials could be measured up to 77.4 and 13.1, respectively. The measurement errors arising from the theory. and measurement system were less than 1 percent in z’ and 5 percent in e”. In 1982, C. B. Roseberg, N. A. Hermiz and R. J. Cook [9] demonstrated a reli- able permittivity measurement technique on low-loss liquid. A single crystal quartz disc was fitted in an adjustable cylindrical mo, cavity as a separator between the liquid and air filled regions. The disc was arranged to be half-guide wavelength thick and was positioned an integral number of half-guide wavelengths from the fixed end of the cavity. Resonance condition occurred by adjusting the movable top plated to an integral number of half- guide wavelengths above the separator, i.e., quartz disk. Mode matching technique was used to derived the relationship between the loaded quality factor and loss tangent. As are the cases in most papers, perturbation technique was involved in evaluating the cavity wall losses in order to reduce the complexity of cal- culation. The real part of the complex permittivity was calculated from the usual guide wavelength formula with the measurements of the guide wavelength in the liquid filled region. Systematic errors arose from the uncertainties associated with the instruments used, fi'om lack of uniformity of measured parameters over the system (e.g. tempera- ture) and from practical departure from the theoretical model. It was shown that loss tangent could measured to an uncertainty of 1: 3x104 and relative permittivity to and uncertainty of i' 0.001. The largest e’ and loss tangent measured were 2.034 and 894x10“, respectively. Unlike most experiments described in literature, Farooq [10] used a rectangular 13 cavity instead of cylindrical one. The dielectric test material was placed on the broad wall of the cavity and completely filled its cross section. Farooq demonstrated that this experiment had the advantage that the change in resonant parameters, i.e. resonant frequencies and cavity quality factors, were enlarged and the measurements of thin low loss material became easier. The derivation of the relationship between quality factor and loss tangent made use of mode matching technique. The range of application was considered best for values of loss factor from 10“ to 10". Materials under test were Polymethyl methacrylate and Polystyrene. Experimental results of the loss tangent were in good agreement with those of the published data for these materials. As can be seen from the experiments described above, the shapes of the samples are either a long rod which is of the same length as that of the cavity, or a disk which completely fills the transverse cross section of the cavity. It is evident that one of the reasons why these sample shapes were chosen is to achieve mathematical simplicity. However, these configurations pose several serious problems. For disk-shaped sample, in many cases it is hard to get large enough samples to fill the cavity cross section, for example, in measurements on single crystals. Small air gaps exist between the sample and cavity walls, even with tight fitting. This gap effect was shown to give erroneous results when materials under test have large 2’ and small a” [l 1]. For long rod sample, the sample was placed along the cavity axis parallel to the electric filed. As before, the fit of the sample is critical. For hard, brittle materials, attaining exact fit required is a problem because the ends of the sample rod tend to chip and crack [12]. For both configurations, the cavity must be disassembled for correct positioning of the sample. This leads to difficulties in reproducing the reference Q factor and the resonant fre- quency corresponding to the empty cavity. 2.2.1.2 Perturbation technique Cavity perturbation techniques have been widely used to measure the complex dielectric constant e of materials at microwave frequencies. These measurements are 14 performed by inserting a small, appropriately shaped sample into a resonant cavity and determining the properties of the sample fi'om the resultant change in the resonant fre- quency and Q factor. The basic assumption of the cavity perturbation is that the change in the overall geometrical configuration of the electromagnetic fields upon introducrion of the sample must be small [l3], [14]. In other words, the percentage change in the resonant fre- quency must be small. This assumption imposes limitations on the size and dielectric constant of the sample. For materials with large 2’ (or large a”), the size of the sample must be very small compared to that of the cavity in order to have minimal perturba- tion to the cavity resonance. On the other hand, if the complex dielectric constant of the material is small, the limitation on the ratio between the sizes of the sample to cav- ity may be relaxed. According to the methods by which the frequencies of the microwave energy sources are controlled, two different experimental systems using perturbation tech- niques can be classified. A passive system is a system in which the frequency of the microwave source is constant in either CW or swept modes throughout experiments. The system is referred to as "passive" because the cavity applicator does not contribute to the generation of the microwave signals. An active system is a system in which the cavity resonance determines the frequencies of the microwave sources. Often, the active system is a close-loop circuit and the cavity resonance is tracked by phase-lock loop (PLL) technique. It has been shown that measurement automation can be easily accomplished in active frequency systems. Passive systems The earliest treatment of cavity perturbation using TMm mode was conducted by E. F. Labuda and R. C. LeCraw [12]. A thin rod was placed in the center of a resonant cylindrical cavity. The length of the sample was of quarter-guide wavelength of the resonant cavity mode. Quasi-static approximations to the fields inside the 15 sample were made in the perturbation formula [15]. Reported data shows the accuracy of e’ to within 3% and e” to within 0.1%. In 1971, W. Rueggeberg first showed the application of cavity perturbation to determine the complex dielectric constant of arbitrary dimensioned dielectric modules at microwave frequencies [16]. In his work, the sample under study need not meet the small volume requirement demanded by cavity perturbation. A mathematical relation that describes sample loss tangent was derived by the modification of the perturbation formula. In this modified formula, the derivative of the cavity resonant frequency with respect to dielectric constant of the material of a particular configuration under investi- gation, i.e. 5%, was included and no approximation to the field inside sample was made. Accordingly, there was no limitations on the volume and e of the sample. A resonant frequency vs dielectric constant curve for a particular sample shape was esta- blished experimentally by measuring the resonant frequencies of a set modules with well known dielectric properties. The evaluation of an unknown sample requires the measurements of the resonant frequencies and the Q factors. The dielectric constant and derivative 53%, evaluated at f r, are determined graphically. Then, using the modified perturbation formula, the loss tangent is readily determined. Eldumiati and Haddad [11] showed that a reentrant cavity can be used to measure the conductivity and dielectric constant of a bulk semiconductor material by perturba- tion technique. In their experiments, a rectangular semiconductor sample was placed on the center of the bottom plate of the cavity, i.e., under the center post of the reen- trant cavity. This sample arrangement has the following advantages: (1) the electric field is maximum in this region, which results in greater sample/electromagnetic interaction, and (2) the field is essentially uniform over the sample, provided the skin depth is larger than the sample dimension along the electric field. They demonstrated an revolutionary method that does not require Q factor measurements. The Q factors in 16 the perturbation formula for calculating a” and 2’ were replaced by the change of the reflected power and the empty cavity coupling factor. Hence, a knowledge of the empty cavity coupling factor together with the measurements of the reflected power and resonant frequency are adequate for the determination of the complex dielectric constant e. As before, the field inside sample was approximated by quasi-static approximation. The empty cavity coupling factor, the change in reflected power and the resonant frequency shift were measured for the determination of the material com- plex permittivity. Maria A. Rzepecka [17] presented a new method of using frequency counter for the determination of a cavity Q-factor instead of frequency markers techniques used by most investigators. She measured the dielectric constant e of teflon, carbon tetra- chloride, amyl alcohol, methyl alcohol, nitrobenzene and water by means of a rec- tangular cavity in Tim and ram, modes at 2.45 GHz. The Q-factor of the material loaded cavity in perturbation formula was expressed in terms of the empty cavity Q- factor, transmission coefficient for the empty cavity and transmission coefficient for the material loaded cavity. The uncertainties associated with the measured parameters and system instability in evaluating e was examined in this paper. A useful in depth literature review of cavity perturbation techniques and errors analysis were performed by Shuh-han Chao [l4]. Chao employed the quasi-static approximation to formulate the total induced electric moments in the perturbation for- mula. He assumed that the electric field applied to the sample was uniform, provided a small sample was placed at the electric field maximum. In the error analysis, an analytical formula for calculating the errors caused by the measurement errors in the resonant frequency and the Q-factor were derived. The errors that have to do with the approximations made in the determination of the electric field inside the sample were not considered. He notes that, due to the conflicting requirements of small sample size for small perturbation error and large sample for small errors in the 6f and SQO, an 17 optimum sample size for minimum errors can be found. Experiments were conducted by measuring the permittivity of small spheres in a iris-coupled T5103 rectangular cav- ity. The sample was held in a cylindrical Styrofoam sample holder, which was placed at the geometrical center of the cavity. Measurements on the Q-factor and resonant frequency were made at 9.5928 GHz in swept frequency mode. Active systems An automatic digital method for measuring the dielectric constant of thin dielec- tric film using 71501. cylindrical cavity at 9.3 61-12 was reported by M. Rzepecka and M. Hamid [ 18] in 1972. The most significant improvement that they achieved over past investigators is the new Q factor measurement technique they proposed. The Q factor was determined by measuring the phase shift of the envelope of an amplitude modulated microwave signal when this signal was transmitted through a resonant cav- ity at resonance. The block diagram of the active system is shown in Figure 2.3. As is shown in this figure, the microwave signal was split into two parts. One part was amplitude modulated and passed through the test cavity, the other part went through a phase shifter and an attenuator. These two signals were mixed in the directional coupler and then fed into the detector. The output signal of the detector had a modu~ lation component whose amplitude and phase depended on the difference between the microwave oscillator fiequency and the cavity resonant frequency. This signal was compared with a reference signal from the modulation oscillator in a phase detector. The output dc signal from the phase detector, which was a function of the frequency difference mentioned above, was used to control the frequency of the microwave oscil- lator. This technique made the automation of the experimental system possible. How- ever, the drawback of the phase measurement technique is that the circuitry is much more complex than that using resonant curve methods. The authors analyzed the errors present in the perturbation formula and in the measurements. The accuracy of evaluating e’ and e” relies on the accuracy of measuring the film thickness, cavity 18 Bough 0533 one no Bonus? xooum 9m charm hoaeaaofie— _ ganja 3.3a... _ m as .3238 henna—ooh .8393 3.33023 havoc , -uoae—avofi nonmaono .33 EA 25.5.3.8 nuance houoouov 1303023 < gouging—d 53260.: ”3334605 _ . 333m fl 0923 _ n 3.6033 human amend _ _ apron 19 resonant frequency shift and Q factor. The main sources of measurement error are the errors related to frequency stabilizing circuit, variations of the cavity resonant dur- in g measurement, and the frequency counter. Cavity dimensions remained constant throughout experiments and cavity resonance was maintained by continuous adjustment of the frequency of the microwave oscillator via phase lock loop (PLL) circuit. Based on this active frequency measurement scheme, C. Akyel, et a1. developed microprocessor control techniques for dynamic measurements of complex dielectric constant using cavity perturbation [19]. In the experimental system, an electronic phase shifter introduces a periodic phase shift inside a closed loop circuit containing the cylindrical cavity applicator. An mechanical phase shifter adjusted by a micrOpro- cessor kept the cavity applicator at resonance. Both the resonant frequency shift and Q factor of the cavity were determined from the system oscillating frequencies, pro- vided the phase shift caused by the resonant cavity was less than 15° . Limited by the small phase shift requirement, high accuracy can be achieved only if the frequency shift is smaller than 20 MHz and the loaded Q falls in the range from 500 to 6000. One year later, C. Akyel and R. Bosisio [20] improved the above system by intro- ducing Q-multiplier measurement mode. In this mode, the quality factor of the cavity applicator is increased whenever it is excessively decreased by sudden increases in the dielectric loss of the material load undergoing dynamic changes. The new system was able to measure the complex dielectric constant of sample material over a wider range of dielectric constants. No experimental results were reported in this paper. Above work was again extended by Akyel, Labelle, Berteaud and Bosisio [21] to the computer aided permittivity measurements (CAPM) of moisten and Pyrolized materials. By means of this method, it is possible to directly relate the moisture con- tent of sample materials exposed to strong electromagnetic fields to the dynamic changes observed in the complex dielectric constant at the test frequency. A rectangu- lar TEN, mode cavity was excited at 2.45 GHz by two in-phase electric couplers to 20 prevent interference from higher order modes. The test sample material was a small piece of white bond paper which was first moistened and then rolled into a cylindrical form. The sample was held in the center of the cavity by a teflon sample holder. An AIM-65 microprocessor was able to calculate the percent moisture from the measured sample weights and the complex dielectric constants from the resonant frequencies and Q factors measurements by perturbation technique. Experimental results show that both 2’ and 2” decrease essentially uniformly until the zero percent moisture level is reached. Beyond this level the 2" increases rapidly while 2’ remains nearly constant. It implies that the loss tangent tans undergoes a radi- cal change when the O-percent moisture level is reached. This rapid change in the value of m8 offers a convenient means of detecting the 0-percent moisture level. The authors claim the precision of the moisture measurements is better than 0.1% and the precision of the dielectric measurements is better than 1% for e’ and better than 5% for II 8 . Using microwave cavity for the measurement of the complex dielectric constant of high loss liquids has been seen in several papers. In H.-J. Blume’s report [22] a cylindrical cavity operated at 1.43 GHz in swept mode was used to measure the com- plex dielectric constant of polluted waters. Very small water sample in quartz tube of known volume was placed in the center of maximum electric field. Measurements on the input power, reflected power, coupling coefficient at resonance, frequency shift and resonant frequency are needed to calculated Q factor. H. A. Buckmaster, et al. [23] used a variable length, nonresonant reflection cavity to measure e of dilute aqueous KCI solutions. Comparison was made between the reflection and transmission type cavities. The complex permittivity was calculated from the attenuation per unit length and the phase shift per unit length. They demon- strated that high loss liquids could be determined more accurately, using a variable- length transmission sample cell than using a variable length reflection sample cell. 21 Two different methods, single mode active cavity perturbation technique (SMAC) and time domain spectroscopy (TDS) were studied by C. Akyel et a1. [24] to heat and measure, at the same time, the complex permittivity of various liquid dielectrics. The sample materials under test included ethanol, various solutions of methanol and decanol and mashed potatos. Comparison between these two methods was given. In SMAC methods, cavity was fixed in dimension [19]. Hence, some coupling varia- tions were found during microwave heating cycles, which slightly affected the meas- urements. These changes were neglected in the calculation of the complex dielectric constants with perturbation formulas. In TDS methods, a step voltage pulse produced by a fast-rising tunnel diode pulse generator was transmitted toward sample material. The voltage pulse propagated in the coaxial line until it reached the air-dielectric interface where a part of the pulse was reflected and the rest was transmitted into sample. The reflected signal was then detected by a wideband sampling oscilloscope. e was determined by the scattering coefficient formula with the measured reflected voltage. In comparison with the exper- imental results obtained by SMAC, TDS may be more accurate. But the temperature permittivity hysteresis effect was observed from the SMAC results. It is not possible using TDS because of the nature of the pulse voltage heating. In 1985, M. Martinelle, P. A. Rolla and E. Tombari [25] used a T50“ mode cylindrical cavity at 9.5 GHz for dynamic dielectric measurements during polymeriza- tion reaction. Possible volume or density changes of samples were taken into account in the perturbation formulas. Only 2’ was measured in the experiments. Active fre- quency close-loop circuit was built to determine the frequency shift required in the for- mula for the evaluation of 2’. Later, in the same year, they proposed a method to determine the e” by power reflection coefficient measurements [26]. A first order per- turbation formula was employed to calculate e’ and e”. Measurements of the standing wave ratio (VSWR), empty cavity Q factor, empty cavity coupling coefficient and 22 filling factor were required for the calculation of e”. Experimental errors were analyzed for e’ and s", respectively. The error of e’ was mainly due to the Q factor changes which contributed more resonant frequency shift but was neglected in the first order approximation of the perturbation formula. The determination of e” by means of VSWR measurements was affected by a relative error, depending on the reflection coefficient value. The complex dielectric constant of chlorobenzene was measured and the error due to Q change was about 10", due to reflection coefficient measurements was about 3x10". 2.2.2 Microwave heating In microwave heating, microwave energy is utilized to heat materials to achieve certain purpose. Wide variety of applications of this technology has been reported in the literature. Two categories can be classified among the most common applications: (1) dehydrating (water removal); and (2) enhance heating. Dehydrating Dehydrating of various materials is one the most traditional areas in which microwave heating has been applied. The forest industry has been using microwave to process large bundles of woods and plywood veneer, such as the redrying of veneer [27] and the drying of the glue between veneer sheets [28]. Since Sixties, microwave energy has been used to dry potato chips and paper match books [1]. More recently Thiebaut et al. [29] examined the dehydration of 13X zeolites using microwave energy in a TED“ cylindrical cavity. An evaluation of possible use of microwave and rf drying as a pretreatment step in the acid hydrolysis of com stover was conducted by G. Lightsey’s research group [30]. They conclude that microwave drying greatly reduces the process time compared to conventional drying. The drying of plastic materials has drawn extensive attention.' Since the tensile strength of specific plastics depends on the moisture content prior to extrusion, 23 moisture reduction is necessary. Lightsey [31] investigated the microwave drying of four common polymers. It is found that, compared to conventional drying techniques, reductions in drying time of 85 to 90 percent were achieved. In rubber processing and ceramic processing, using microwave energy for dehydration is becoming more and more attractive [l], [28]. Enhanced heating Besides heating the water inside a material, microwaves have been used to heat the dielectric material in order to change the physical characterisfics in known and controllable ways. This is the purpose of enhanced heating. Applications have been found in semiconductor processing, food processing and chemical processing. A typical procedure in which semiconductor devices are fabricated from wafers entails heating the wafers during several process steps. In contemporary practice, wafers are routinely heated with resistance furnaces, infrared lamps, electron beams and lasers. Sirkis and Conrad [32], [33] used traveling waves to heat semiconductor ribbons and wafers directly without susceptor. Figure 2.4 displays the cross-section of a waveguide applicator. The semiconductor wafers were supported at each side by ceramic retainers. Experimental results show that 15 semiconductor ribbons or wafers could be heated uniformly at the same time. In food processing, bacon cooking and meat tempering accounted for 79% of the installed microwave heating capacity as of 1984 [28]. Recently applications of microwave heating for stored-grain insect control and agriculture seed treatment have been reported [34]. An improvement of the nutritional quality of soybeans by continu- ous microwave heating was demonstrated by Vetsuypens and Loock [35]. In chemical processing, plastic processing, rubber processing and ceramic pro- cessing are the most promising areas of microwave heating applications. In addition to the drying of plastics mentioned previously, microwave energy has been proposed to enhance the polymerization process [25], [36]; to heat prior to extrusion; and to 24 rectangular waveguide applicator waveguide wall \ VI” \ semiconductor ceramic holder wafer Figure 2.4 Cross—section of a waveguide - ‘ applicator 25 stimulate specific chemical reactions [28]. Lightsey, George and Russel [31] proved that the use of microwave energy for curing plastics allows development of fast, more efficient and better controllable processing system than the processes that are com- monly used at present. Strand [37] showed that the cure times of epoxy could be reduced 30-fold using microwave heating rather than conventional thermal heating. The utilization of microwave energy in rubber processing has been expanded from the initial application of extrusion curing to every facet of the rubber process. It includes tempering bales, devulcanizing, preheating prep and curing [l], [38]. Since microwave heating gives better heating control and efficiency and is not limited by thermal diffusion process, theoretically a material can be heated to any temperature wanted. This phenomena has been recently applied to the sintering and the joining of ceramics [39] - [42]. More recently researchers have paid attention to the application of microwave energy in composite materials. Park [43] investigated the sinterin g of zirconia- aluminia composites under 2.45 GHz microwave radiation. R.B. James [44] studied the microwave-induced surface melting of carbon composites and graphite by high power microwave pulses. 2.2.3 Microwave plasma applications Using a microwave cavity applicator to generate microwave plasma has received much attention lately. The reason is that microwave plasma has been found many applications in industry and in fundamental researches. These applications include (1) microwave electrothermal and ion engines for spacecraft propulsion, (2) ion sources, (3) plasma etching and oxidation in semiconductor processing, (4) plasma induced chemical vapor deposition for growing diamond thin films. Recent high pressure microwave discharge experiments [45] - [47] conducted at Michigan State University have demonstrated the potential application of microwave arcs as an electrothermal thruster concept. In these experiments, a high pressure 26 microwave discharge was established in a flowing gaseous propellant. The gas was heated to high temperatures and was passed through a nozzle, converting the gas ther- mal energy into thrust. Two different applicators have been used to sustain the discharges: a cylindrical cavity applicator and a compact coaxial applicator [46]. This electrodeless engine concept eliminates the electrode erosion and lifetime problems experienced in DC discharge. It is also expected to achieve much greater thruster level than can be obtained fiom electrostatic ion thruster. Microwave plasma applicators [48] have demonstrated the ability of producing ion beams. A microwave cavity plasma disk ion source was shown to be able to operate over a large pressure, flow rate and absorbed power density ranges. The advantages of using microwave ion sources are no electrode problems such as discharge contamination with electrode materials, electrode lifetime constraints and the ion beam current density limits due to the presence of electrodes. Polymer thin films are widely used in the semiconductor industry as insulators, planarizing layers and for patterning other materials. It has been shown that these polymers could be etched by using microwave plasmas. Hussla, et al. [49] used argon-ion plasma in their investigation of the silicon wafer temperature during the plasma etching process. Hopwood, et al. [50], [51] showed that a 2.45 MHz microwave plasma disk reactor (MPDR) could be used to generate plasmas for silicon etching. In their work, both CFJO; and SFé/A, plasmas were studied. It is found that by operating high-density plasmas at low pressures, ions were accelerated vertically to the wafer surface. Hence, considerable less undercutting was achieved than at high pressures. Microwave plasma oxidation for integrated circuit processing has also been demonstrated by using microwave plasma disk reactor [52], [53]. The oxidation rate was comparable to conventional thermal oxidations at 1000 C. However the tempera- ture of the silicon wafer was less than 200 C. Low temperature is very important in VLSI processing since it reduces impurity motion, wafer warping and defect genera- 27 tion [52]. Recently considerable attention has been paid to the growth of diamond thin films. Because of all the useful properties diamond has, the potential applications of diamond thin film are numerous, such as high temperature and high power electronics and sensors, microwave and millimeter wave power device, printed circuit board, vacuum UV laser, cutting tools and tribological applications. The major difference between chemical vapor deposition (CVD) and plasma induced CVD is the energy source. The energy for the plasma CVD is supplied by plasma instead of heat in CVD. Basically, plasma CVD is performed by passing a mixture of hydrogen and methane into the deposition chamber and applying microwave power to induce glow discharge [54]. Microwave power breaks the methane into carbon and hydrogen. Atomic carbon deposits itself on the substrate in a configuration resembling the tetrahedral formation of diamond. Hydrogen atoms pick up dangling bonds on the sur- face from collapsing into a graphite form. In a pulsed microwave plasma CVD sys- tem, the introduction of hydrogen into deposition chamber is not necessary because the cooling process is rapid enough to prevent the transformation of metastable phases into graphite [55]. 2.3 Methods of analyzing coaxially loaded cavity 2.3.1 Theoretical formulation During the last two decades many rigorous methods have been developed to analyze dielectric resonators, such as the mode matching method, the finite element method, the Rayleigh-Rize method, moment method, and the transverse resonance method. Some of these methods are complicated and take up a great deal of computa- tion time to achieve high accuracy. While some of them may be simplier and faster in computation but with less accuracy. There is no certain rule of choosing the methods. Nevertheless, depending on the resonator configuration, applications, dielectric 28 properties, and computation time and accuracy requeirments, one can select the most appropriate method for his application. The purpose of this section is to introduce those methods that are most frequently seen in literature. No attempt is made to com- pare them and to determine the most significant one. Mode-matching method Mode-matching method has been widely used for the analysis of open dielectric resonators and dielectric resonator with metallic boundaries. Basically, the structure of the resonator is divided into subregions. In each subregion the electromagnetic fileds are expressed as a linear combination of all possible eigenmodes with unknown coefficients. By enforcing the continuity conditions of the transverse electric and mag- netic fields over the interfaces between subregions. a system composed of infinite number of homogeneous equations where each equation is described by infinite sum of terms is obtained. The vanishing determinant of the coefficient matrix of the system equations establishes the characteristic equation of the problem. Together with the separation equation associated with each subregion, the resonant frequency correspond- ing to various modes are solved numerically fiorn the characteristic equation. For computation simplicity, a truncated system is usually considered. The size of the trun- cated determinant is determined by the convergence condition of the resonant fre- quency. Using a mode-matching method to analyze microwave cavity loaded with a long dielectric rod has been reported by many researchers. A long dielectric rod is a rod which has the same length as that of the cavity. Axially independent modes were examined by Rossy and Felden [7], and by Shihe Li et al. [8]. Rossy and Felden stu- died the lossless case, while Li investigated a low loss dielectric rod with e" = 0.04. More complete analysis with the inclusion of hybrid modes was done by Kobayashi et al. [56], and by Zaki and Chen [57]. Only lossless dielectric materials were con- sidered in both papers. In general, the loaded cavity was treated as a section of coaxi- 29 ally loaded waveguide, shorted at both ends. Kobayashi provided the design criteria for a dielectric resonator to accomplish possible highest quality factor for largest fre- quency separation between adjacent modes. In their analysis, the dielectric rod and the cavity walls were assumed lossless. Zalci and Chen not only calculated the resonant frequencies, Q-factors and mode charts, but also studied the effects of the relative dielectric sample dimensions on the Q-factor. Sphicopoulos, Bemier and Gardiol [58] solved the problem of a cylindrical cavity loaded with many concentric dielectrics. The cavity was regarded as a waveguide loaded with multilayer dielectric tubes. Application of the boundary conditions on the conducting walls and continuity conditions between dielectric layers gave rise to a NxN homogeneous linear system equation. The resonant frequencies are the zeros of the determinant of the coefficient matrix. N is determined by the number of dielectric layers present in the cavity resonator. In this paper, losses in the dielectric and in the cavity walls were taken into account through the use of complex frequency and com- plex permittivity. The real part of each zero is the actual resonant frequency, while the imaginary part yields the unloaded Q-factor. In addition to the resonant frequency, mode conversions from homogeneous cavity to cavity loaded with two layers of dielectrics was examined and the field distributions for no" and HE,” modes were calculated. The lossy dielectric material considered has a complex dielectric constant of e = 13.1 - j 12xw'2. Theoretical results were verified by experiments. The earlies study of a microwave cavity centrally loaded with a finite dielectric cylinder using mode matching method was done by D. M. Bolle [59]. Bolle compared the results with those obtained from Slater’s perturbation theory. No loss due to the dielectric material was considered. Later, Kobayashi, et al. [60], and Zaki and Chen [61], [62] investigated the same problem. They divided the resonator into two homo- geneous waveguides and one dielectric-loaded waveguide three subregions. In other words, the resonator was treated as a three-layer problem. Boundary conditions and 30 continuity conditions were matched on the conducting walls and the interfaces between layers. Due to the structural symmetry, magnetic wall and electric wall boundary con- ditions were employed over the symmetry plane by some researchers [60], [61], [63]. It is worth noting that the resulting system equations is half of those without using the electric and magnetic walls boundary conditions. Kobayashi also studied two extended cases:(1) long rod and (2) disk loaded cavi- ties. Mode charts and resonant frequencies were calculated for various cavity modes. In zaki’s paper [61], the transverse electric and magnetic field patterns at different axial positions for different modes were plotted. A rectangular cavity loaded with low loss dielectric materials was analyzed by Leong, Kooi, and Yeo [64]. It is found that for low dielectric constant, e'=2.58, three eigenmodes in each region was enough for good convergence, while for high dielectric constant (=95, convergence was achieved using six eigenmodes. The effect of dielec- tric loss on the resonant frequency and Q-factor was examined. Theoretical results show that the dielectric loss does not affect the resonant frequency for low loss dielec- trics (tan8 < 0.001). Resonant frequency increases of less than 0.08 MHz was observed for a medium value of loss tangent of 0.03. The authors studied the temperature dependence of the resonant frequency and the output power variation of a cavity loaded with teflon over a temperature range fi'om 23° C to 72° C . Over this temperature range, the frequency decreased by 16 MHz. Although the existence of complex modes [65], [66] in waveguide loaded with lossless dielectric material have been known since 60’s, most researchers assume that all the modes in each of the subregions are either propagating or evanescent modes when mode matching method is applied. Not until 1988, Chen and Zaki [67] first included complex modes in the field expansions in the analysis of the lossless dielec- tric resonators using mode matching method. They successfully predicted all the resonant modes that were observed experimentally, which was impossible without 31 including the complex modes. Studies of the multilayer dielectric resonator with metallic boundaries have been seen in literature [68] -[70]. As is shown in Figure 2.5, in the multilayer structure, a dielectric cylinder is placed in a metallic cylindrical cavity, which includes a dielectric support to hold the dielectric cylinder, one or two layers of dielectric substrates, and/or a dielectric post to tune out fabrication tolerance. Hence, the cavity is often divided into subregions of homogeneous waveguides and dielectric-loaded waveguides. Each layer corresponds to one of the waveguide subregions. Two basic assumptions were made as follows: (a) dielectric materials involved are isotropic and lossless, and (b) the metallic boundaries are perfectly conducting. As mentioned earlier, the electromagnetic fields in each subregion are expanded in terms of all possible eigenmodes existing in it. Boundary conditions are then made at the interface between dielectric layers (subregions). Hong and Jansen [69] considered a 5-layer resonator structure. They found that 4 or 5 eigenmodes present in each layer were adequate for good convergence of resonant frequencies. Bonetti and Atia [68] investigates a 4-layer TE,“ mode resonant cavity. Following the formulation of Zaki et al. [62], Vigneron and Guillon [70] studied a 3- layer asymmetric dielectric resonator. They solve for the resonant frequencies for both hybrid and axially symmetric modes. Generally, 6 expansion eigenmodes were required to achieve reasonable convergence. Rayleigh-Ritz method Recently Rayleigh-Ritz method [71] have been used to determine the resonant frequencies and Q—factor of dielectric-loaded microwave cavity with remarkable suc- cess. By using this method, the accuracy of solutions depend highly on the selection of basis functions. The basis functions are the solutions of Maxwell equations subject to appropriate boundary conditions, depending on what cavity structure .is under study. Empty cavity solutions have been frequently employed as the basis [72] in the analysis % /____ layer a layer 2" ' % y cavity nu — 'ew of a multilayer dielectric 33 of lossless dielectric resonators. For the first time in 1983, the solutions of a partially filled cavity were used as the basis function by J. Krupka [73]. In this paper, Krupka demonstrated that the basis could be optimized through a proper choice of N basis functions from its infinite set to achieve the most accurate solutions. The optimization is easy because the resonant frequencies calculated by Rayleigh-Ritz method are lower bounded by the accurate solution [71]. According to this rule, N functions having the lowest frequency values among the infinite set are chosen to form the most significant basis. No loss was included in his analysis. In addition to the choice of the basis, the dielectric constant of the basis cavity (partially filled cavity) also affects the convergence of the solutions. The optimal dielectric constant for certain cavity dimension is the one which has the lowest resonant frequency when only one basis, the predominant one, is used in the Rayleigh-Ritz method. The resonant frequency corresponding to this dielectric con- stant is taken as the starting point of the Rayleigh-Ritz method. With the optimized N basis functions and the optimaized dielectric constant of the basis cavity, the best con- vergence could be achieved by experimenting with N. Krupka calculated the resonant frequencies of a no” mode cavity as a function of N. It is found that for N > 10, significant convergence can be achieved. By optimization of the basis, a high calculation accuracy of the resonant fi-equen- cies of the dielectric-loaded cavity is obtained. However, as the author noted, some difficulties arise because of the optimization. First, N transcendental equations need to be solved in order to determine the basis. Second, the new basis functions are more complicated than empty cavity basis functions. Therefore Krupka recommended that an optimized procedure be employed in the following cases: (1) when a very high accuracy of calculations is required, and (2) when the computation time must be short or/and the available computer memory is limited. In the interest of calculating the resonant frequencies and Q-factor of MIC dielec- 34 tric resonators, Krupka took the electromagnetic fields of an auxiliary post dielectric resonator as the basis of Rayleigh-Ritz method [74]. The complex dielectric constants of the dielectric resonator and the substrate are 36.2 - j3.t52x10’3 and 3.03 - j 1.65x10‘2, respectively. Because only nonradiating TED... was looked for, the distance between conducting plates was smaller than half of the free space wavelength. Theoretical results were compared with the calculations obtained by mode-matching and Weinstein variational methods. It is intriguing to note that both mode-matching and Weinstein variational methods provides lower bounds for the accurate solutions, while Rayleigh- Ritz method provides upper bounds for them. The convergence of the Rayleigh-Ritz method is worse than mode-matching and Weinstein variational methods. However, on the other hand, Rayleigh-Ritz method leads to a simple eigenvalue problem which can be solved faster than the problem of searching the roots of the vanishing deter- minant which must be solved when those two methods are used. A detailed comparison of the Rayleigh-Ritz method and the mode-matching method for computations of 123:0... -mode resonant frequencies and Q-factors of dielectric-loaded microwave cavity was presented again by J. Krupka in 1988 [75]. In his analysis, all the dielectric materials were assume lossless. Computations were made for the dielectric resonator structure shown in Figure 2.6. Krupka demonstrated that, to achieve good convergence, 100 basis functions were required for Rayleigh-Rize method, while nine eigenmodes were used in the modal expansion in each subregion for mode-matching method. The computation time were approximately identical for both methods for various material/cavity dimensions and dielectric constants ( 10 seconds on a CDC 6600 computer per frequency value). It is seen that the discrepancy between resonant frequency values obtained by these two methods is not greater than 0.6 percent for the first six modes if the dielectric constant of the dielec- tric material is 10. However, the differences increase when the dielectric constant is 35 and the ratio of the length of the dielectric material to that of the cavity is less than 35 0.2, i.e., p < 0.2 as shown in Figure 2.6. This is due to the poor convergence of the Rayleigh-Rize method under these resonance conditions [73]. Krupka also examined the influences of the thickness of the substrate and the relative diameter (ratio of the diameters between dielectric material and cavity) on the resonant frequencies. Moment method Moment method has been used in the analysis of dielectric resonators in the last few years. In an attempt to find the 715m mode resonant frequencies of a rectangular cavity containing a dielectric sample, Kedzior and Krupka made use of Galerkin method [72]. This is a well known moment method in which the set of testing func- tions is identical to the set of basis functions that expands the electric field (or mag- netic field). Similar to mode-matching method a truncated system of linear equations was established in the formulation of the problem. The resonant fi-equencies were the zeros of the determinant of the coefficient matrix of the system equations. Glisson and Kajfez [76] utilized moment method to formulate a microwave cavity loaded with lossless dielectric material with arbitrary cross section. Their approach was based on the solution of a surface integral. Numerical computations was made for a dielectric cylinder resonator without metallic boundaries. The values of the resonant frequencies and the values of the corresponding Q-factors were determined by the complex frequency technique [77]. The convergence condition of the solution was stu- died by increasing the number of points (N) on the generating are. As was expected, the resonant frequencies and Q-factors showed good convergence as N is increased. Magnetic and elecnic surface currents, n'ansverse electric and magnetic fields, and mode chart for various resonant modes were presented. Finite element method In order to investigate the cylindrical dielectric resonator in MIC applications, Kooi, Leong and Prakash [7 8] carried out the analysis using the finite element method. The resonator is a three layer problem, including dielectric post, substrate and empty 36 cavity wall / L dielectric material T h l _L_ 1‘ 2a —-I +- .. ~| p = h/L Figure 2.6 Geometry of a dielectric resonator 37 regions. The dielectric materials considered were lossless. Each region was divided into three-noded triangular elements. First-order polynomials were selected to approxi- mate the variation of the electric field within each element. More elements were taken in the dielectric post region as most of the electromagnetic energy is confined in this region. Galerkin weighted residual approach was used to derive a system equations of each element. Putting together the system equations of each element yielded the overall system equations. The zeros of the determinant of the coefficient matrix are the resonant frequencies. The convergence condition was tested by increasing the number of elements. The authors suggested that higher accuracy in the computation of the resonant frequencies be. achieved by using higher-order elements, instead of three- noded triangular elements. The resonant frequencies and Q-factors of 12015 mode were evaluated and compared to published data. Transverse resonance method The first report of using transverse resonance method to analyze a dielectric- loaded cylindrical cavity was done by Hanfling and Botte [6] in 1971. Recently, El- Sayed and Farghaly used this method to perform more detailed analysis of the influence of the load position on the resonant frequency of a dielectric-loaded rec- tangular cavity. [79] The cavity was loaded with a lossy dielectric slab, completely filling the cross section of the cavity. By means of the equivalent transmission line circuit of the loaded cavity, the characteristic equation was derived by matching the impedance in a properly selected reference plane. The resonant frequencies were com- puted for Him, TM 112 and TM,11 modes for various load locations within the cavity. They also examined the effects of the dielectric loss on the resonant frequencies. It is found that the resonant frequency vs dielectric constant curves exhibit oscillatory behavior. In general, the amplitude of oscillation is higher for a raised slab than when the slab is positioned at the bottom of the cavity. The dielectric loss factor has a rounding effect on the rapidly rising portions of resonant frequency vs dielectric con- 38 stant response. 2.3.2 Mode designation Resonant modes of dielectric-loaded cavities are more complicated to designate than in homogeneous cavities. For axially symmetric modes, the classification of modes are simple. Since these modes are either pure TE or TM relative to cavity axis [80], they are denoted by 130,, or TMOP, as in the case of homogeneous cavity, where p and q are integers. For nonaxially symmetric fields, the resonant modes are neither TE nor TM to any coordinate. These modes are termed hybrid modes. Several methods have been proposed for classifying hybrid modes, but no method has been universally accepted. Most schemes classify hybrid modes into one of two categories, HE," and EH." . However, the criterion used to determined whether a mode is H8 or EH is still a subject of controversy. One method is based on the relative contributions of E, and H, , i.e. the longitudi- nal field components, to the transverse field components at some reference point [60], [81]. If E, is dominant over H,. the mode is designated as 511,". If H, is dominant over 5,, the mode is classified as HE". Snitzer [82] proposed a method for designating hybrid modes by inspecting the ratio of the coefficients of E, and H,. If the sign of the ratio is the same as for the fundamental mode, which is conventionally referred to as HEm, the hybrid modes are designated as HE“. On the other hand, if the sign of the ratio is opposite to that of H51“, the modes are designated as 5H,". Chou [83] and Sphicopoulos [58] classified the hybrid modes by examining the resonant modes in the limiting case of vanishing dielectric load. That is, the dielectric pemrittivity of the dielectric load is equal to that of the surrounding media or the volume of the dielectric load is forced to be zero. In this case, the dielectric-loaded cavity is in fact a homo- geneous cavity. The hybrid modes are referred to as HE," if , in the limiting case, they are TE," mode. While EH," modes are those modes which cenvert to TM," modes in the limiting case. Zaki and Chen [61] introduced a totally different scheme 39 for mode designation. The resonant modes are classified as HEH", HEB“, TEHO.” TEE“, TMHO, or TMEM. This designation scheme is designed for structural symmetry resonators with electric wall and magnetic wall boundary conditions applied in the symmetry plane. In this dissertation, the homogeneous cavity solutions are used as the initial approximations in the root searching algorithm. Each resonant mode of the dielectric- loaded cavity then corresponds to a mode in the homogeneous cavity. Accordingly, the mode designation scheme proposed by Chou and Sphicopoulos is employed here. In other words, hybrid modes are designated as HE," if they correspond to 119,, in homogeneous cavity and as EH," if they correspond TM" homogeneous cavity modes. This method is simple and clear. It is simple in the sense that no calculation of the field components is required to distinguish the modes. It is clear because the dielectric- loaded cavity mode is related to the one in homogeneous cavity. As a result, the mode inversions can be investigated as the dielectric constant or the relative dimension of the dielectric load is changed. Chapter 3 THEORETICAL DEFINITION OF THE LOSSY DIELECTRIC- LOADED CYLINDRICAL CAVITY 3.1 Introduction This chapter presents the theoretical formulation of a cavity loaded with lossy dielectric materials. The material-loaded cavity is a a cylindrical cavity concentrically loaded with a cylindrical lossy material rod which partially fills the cavity. In the for- mulation, the material-loaded cavity is treated as a section of inhomogeneous cylindri- cal waveguide partially loaded with concentric cylindrical lossy dielectric rod, con- nected to two homogeneous cylindrical waveguides each with one end shorted. The mode matching method described in Chapter 2 is employed to derive the cavity characteristic equations. A review of the classical EM analysis of the homogeneous and coaxially loaded waveguides is presented in section 3.3. The cavity characteristic equations for three different cavity types, i.e., cavity-open, cavity-short and cavity- image types shown in Figure 1.1, are derived in section 3.4. The last section of this chapter is a detailed derivation of the stored energy, absorbed power and quality factor of the dielectric-loaded cavity. 3.2 Theoretical formulation of the dielectric loaded cavity Figure 3.1 shows the geometry of the dielectric loaded cylindrical cavity. A dielectric cylinder of radius a and length 12 is placed coaxially in a metallic cylindrical cavity of radius b and length L. A convenient way of analyzing the problem is to divide the structure of the loaded cavity shown in Figure 3.1 into four regions (I, II, IH and IV). Regions III and IV represent a metallic homogeneous waveguide of radius b which is shorted at one end and is filled with dielectrics of permittivities s, and £4, respectively. Regions I and H represent a section of a waveguide of radius b coaxially loaded with a concentric dielectric of radius a. Region I is the dielectric core of 40 Om) H H H D H (a) ah II ; I 2 Y 7/ // Iv (o,-o.er,) 1 c, 4 __l_ l. (00-h4) Figure 3.1 Dielectric- loaded cavity dimensions 42 permittivity 9,. Region II is the space between the metallic walls of the waveguide and the dielectric rod and filled with dielectric of permittivity £2. As shown in Figure 3.1, I, and I, are the lengths of regions III and IV, respectively, and 12 is the length of the coaxially loaded waveguide, i.e. regions I and II. The problem then reduces to a boundary value problem of a coaxially loaded waveguide connected to two homogene- ous waveguides each with one end shorted. The coordinate system to be used in the analysis of the dielectric loaded cavity problem is depicted in Figure 3.2. In this coordinate system the z-axis coincides with the axis of the loaded cylindrical cavity. The dielectric load is placed symmetrically about i in the cavity but n0t necessarily in the center of the cavity. It may be in any position on the cavity axis. Then, in general, the problem considered here has an asymmetrical structure and the mathematics is more complex if the origin is arranged in the center of the cavity. In order to have simplest mathematical expressions, the origin of the coordinate system is placed at the center of the dielectric load material. The dielectric in each of the four regions is considered to be homogeneous, iso- tropic and lossy. The complex dielectric constant for each region is defined as . n 0'; . , ,, . e, =8.- ~18.- -i(;)=eo(er, we, )=€o€r, 1 =1. 2. 3. 4 (3.1) where N e" = e", - je'i e,’ = dielectric constant a," = dielectric loss factor 2,; = relative dielectric constant a," = effective relative dielectric loss factor e, = relative complex dielectric constant i = l, 2, 3 and 4 are indices corresponding to regions I, II, III and IV, respectively. 43 C ‘— E 1r— m Q ::__‘_C_:BL__,:‘J Q I _ n T Q L .— -------- L~~..‘ :’ _<___> ”,1 1v 6. / _lL. \ > Figure 3.2 Coordinate system of the dielectric- loaded cavity 44 Both damping and conduction losses of the dielectrics are included in the effective relative loss factor, an": Often, the cavity is assumed to be filled with homogeneous dielectric, i.e., £2 = £3 = 3,. As mentioned earlier, the dielectric load material is of radius a , length I; and dielectric permittivity £1. The radius of the dielectric load may be as large as that of the cavity, i.e., 0 < a s b, and the length of the dielectric load may be as long as that of the cavity, 0,125L. When either a=b and lz=L or e, = e; = 83 = 64, the cavity is completely filled with the dielectric load material of com- plex dielectric constant 21. As can be seen from Figures 3.1 and 3.2, different load material dimensions and positions in the cavity result in a different boundary value problem. Variations of the problem can be identified by changing these parameters. In this dissertation, three different material loaded cavity types are classified according to the placement and size of the dielectric volume. In Figure 1.1a, the structure in which the dielectric cylinder load does not contact the conductor is call the W. In Figure 1.1b, both ends of the long dielectric rod are in contact with the cavity conducting end plates. This structure is called W. The structure shown in Figure 1.1c is called W, in which one end of the dielectric disc contacts the cavity end plate. Each of the three different structures is subject to different axial boundary con- ditions. However, it is obvious that the cavity-short and the cavity-image types are two extended special cases of the cavity-open type. The solutions to the cavity-open type can be used to derive those to the other two structures. A detailed theoretical derivation of the characteristic equations for each of these structures is given in sec- tions 3.3 and 3.4. 3.3 Review of theoretical formulation of cylindrical waveguide 3.3.1 Cylindrical wave potential As mentioned in previous section, the dielectric loaded cylindrical cavity problem 45 shown in Figure 3.1 is viewed as a structure which consists of an open-ended waveguide and two homogeneous cylindrical waveguides shorted at one end. Hence, a knowledge of the solutions of the normal modes in these waveguides serves as the basis from which the cavity problem is formulated. In this section the cylindrical wave potentials for the dielectric loaded cylindrical waveguide are derived by solving for the solutions to the scalar Helmholtz equation in cylindrical coordinates. With the cylindrical wave potentials, normal modes in the coaxially loaded and homogeneous waveguides can be obtained by applying associated boundary conditions. Some of the basic and well known equations are given below. The Maxwell’s equations and associated constitutive relationships for a source free region are given by, v x E = -jm§’ (3.2a) V ~13 = 0 (3.2b) V x H = jmb’ (3.2e) v - F = 0 (32d) 15’ = 22‘ (3.2e) F = u}? (3.20 where e is the complex dielectric constant as defined in equation 3.1. By introducing the magnetic vector potential If and electric vector potential 13', the field solutions result in Erma—14x“; (3.3) we g=vm+4vwxr (3.4) 10311 if and I? are the solutions to the following wave equations and are called wave poten- tials. V2,? + x2}? 0 (3.53) 46 V2F+xzf=0 (3.5b) where Note that in cylindrical coordinate system, 2 component of the wave potentials satisfies the scalar wave equation, or Helmholtz equation, V2w+K2w=O (3.6) An arbitrary field in a homogeneous source free region can be expressed as the sum of a TM field and a TB field. A TM field is transverse magnetic to z‘ with no H, com- ponent and a TB field is transverse electric to i with on E, component. A field TM to i can be obtained by setting 15' = 0 and X = ity (3.7) then i=~jtouX+-;-l—-V(V-X) 17=VXX (3.8) 1036 This can be expanded in cylindrical coordinates as, Ep=_l__§v_ Hp=lfl )me 3032 P 34’ __1___3_‘\L ".31 5,-) p a“: H,- 3p (3.9) __1_ _3’_ _ 58"],0.‘ (822 +‘K2)‘V ”8.0 Similarly, a field TE to i can be obtained by setting 5’ = 0 and I? = i \V (3.10) __ =_- 4 . then B. VxF’ 17 ”WNW vwi') (3.11) Expanded in cylindrical coordinates, this is Ep=—-li\y- Hp_—__._l__a_zy_ with 10m 3032 5,=%‘§ H,=-,-‘—-iz‘l’— (3.12) E,=O H=—1—-(a—2+K’)\v ' jmtt 322 Once the scalar wave functions, w’s, are solved from equation 3.6, the electric and magnetic fields can be derived according to equations 3.9 and 3.12. The scalar Helmholtz equation, equation 3.6, written in cylindrical coordinates is, F$®fl>+ p—l-2§12’-+%:‘2K+1c2w=0 (3.13) Using the method of separation of variables, the solutions of \V are expressed in the form w=R(p) ¢(¢)Z(z) (3,14) Substitution of equation 3.14 into equation 3.13 and division by w yields 1 d 1 42¢ 13:2 '3_ 0 (3.15) —)+ p —R To .32» d¢2 2 dz2 After some algebraic manipulations, the wave equation can be separated. Define x, as KP: + x,’ = K2 (3.16) and write the separated equations as He p)+ [(Kp p)2-n’]R= 0 —2:+n2:=o (3.17) The ¢ and Z equations are harmonic equations, giving rise to harmonic functions, in general, denoted by h(n b) and h(x,z). The R equation is Bessel’s equation of order 11, solutions of which are denoted by B,(x,,p). By choosing appropriate h(n 0). M192) and B, (xpp), the solutions to the Helmholtz equations are formed as v = 3.09.13) h(n¢) h(1<.z) (3.18) The choice of the Bessel functions is based on the behavior of the waves in the waveguide. Commonly used Bessel functions are J,(xpp), N,(xpp). H,.“’(xpp) and 48 H, (”(xpp). These functions are linearly independent to one another. 1,, and N, are the Bessel functions of the first and second kinds, respectively. They exhibit oscillatory behavior for real xp and represent cylindrical standing waves. H,.” and H,. (2) are the Hankel functions of the first and second kinds, respectively. They represent cylindrical traveling waves, H,.“) representing inward-traveling waves and H, (2) representing outward-naveling waves. Note that only J,(x,p) functions are finite at p = 0, i.e., J,(0) at co. Hence, if a field is to be finite at p = 0, J,(x,p) must be chosen for the Bessel function. With properly selected harmonic functions and Bessel function, problems having cylindrical surface boundaries, such as the homogeneous and coaxially loaded cylindri- cal waveguides, can be solved by means of equations 3.9, 3.12 and 3.18. 3.3.2 The homogeneous cylindrical waveguide 'As is shown in Figure 3.1, regions III and IV are two empty waveguides, shorted at one end, completely filled with homogeneous and isotropic dielectric. In this sec- tion, the modal fields in the homogeneous waveguide with one end shorted is derived. Figure 3.3 depicts the cylindrical coordinate system for the analysis of the homo— geneous waveguide, which models regions III and IV of Figure 3.2. Since fields are finite at p = 0 and travel along the i direction, the wave functions are of the forrn 1:! = a 1.099) {$3.3 }er (3.19) where a is an arbitrary constant. Either sinn¢ or 003110 may be chosen, except for the n = 0 case. Substituting equation 3.19 into equations 3.9 and 3.12 and applying the boundary conditions that E, = 0 for TM modes and E:p = 0 for TE modes at the conduct- ing walls p = b, the modal fields and eigenvalues for x, are determined 49 m) m Figure 3.3 Cylindrical coordinates for the homogeneous waveguide so (1) TM, modes 5:: =a(—w%)(vxp)1.'(x.p)cosn¢ E: =a (fin? 11.09.13) sinncp 15;” = a ( -l- ) x92 1,,(xpp) cosntt (3.20a) 1018 11'; =a (—%)J.sinn¢ (3.20b) H: =2: (3% )("—p"- )J.sinn¢3+<—x,)1.'(x.p)cosn¢t‘> (3.23) (2) TE, modes 5:, = p H; +611; =(-m% )(vxp')1.'(x.'p) sinmt a +(-mj;)(% )J. (3.24) In general, the transverse electric and magnetic normal modes are denoted as E,- and Ii.- , respectively. They may be either TE or TM modes, where the subscript i represents one of the possible combinations of the mode designations, np. To obtain the fields in regions III and IV, in which waves travel along both + z‘ and - 2 directions, a homogeneous waveguide with one end shorted is considered. The transverse modal fields traveling in the + z‘ direction are given by 1?,” = a,- é‘, cw" (3.25a) 12* = a,- Ii, cw" (3.25b) The transverse modal fields traveling in — z‘ direction are given by E’,‘ = b,- 6, e'" (3.26a) H,- = i5,- ( - ii.- ) av" (3.26b) where a,- and 5, are arbitrary constants'and subscript t represents the transverse field component. The transverse modal fields in the waveguide are the sum of the modal fields traveling in both directions. They are E: = 22* + E.- = a, ( 21.1"" + 5,?“ ) (3.27a) 1?, = 171* + 1?,- = t. (3.6” — 5,8” ) (3.271» As is shown in Figure 3.1, the boundary conditions in regions III and IV are that the transverse electric fields vanish on the end conducting walls. Since the approaches are similar, only region III is considered here. The transverse electric field is zero at the perfectly conducting z = h, plane, which forces equation 3.27a to be equal to zero for each mode, i.e. ' a, (a, ("“3 + Haw") = o for all e , and o < p < b (3.28) An examination of equations 3.21 and 3.22 shows that a, is not identically zero for all :3 and 0 < p < b. Thus, above equation is satisfied only if a; 84“, + 5; Cw”, = O (3.29) ai _ ev‘h’ Then .5.- - - FT; (3.30) Since both a,- and 5,- are arbitrary constants, it makes no difference how they are chosen as long as they possess the relationship expressed in equation 3.30. One of the possible choice is v.53 d,- =Cg (3.3la) " ‘Vrhs b, = - c, _ (3.31b) Substitution of a,- and 5,- into equation 3.27a yields the transverse electric modal field 53 in region III. [752,” = 2 c,- é‘, sinhv,(h3 - 2) Similarly, the transverse magnetic field in region III is obtained by substituting equa- tion 3.31 into equation 3.27b. The resulting magnetic field is 172"” = 2c, 5,- coshv;(h3 - z) where superscript 111 represents region 111. Since 2c, is an arbitrary constant corresponding to mode 1', it can be replaced by some cther arbitrary constant, say 21,-, without affecting the results. Rewriting the above equations below E’ ‘r ’2' = a, a,- sinhv,(h3 — z) (3.32a) Rim = d,- 5; COShV; (’13 — Z) (3.32b) Except subject to the boundary condition that the transverse electric field vanishes at z = - 11,, the derivation of the transverse modal fields in region 1V is identical to that for region III. Substitutions of — h. for ’13 and - 5, for a, in equation 3.32 result in the transverse electric and magnetic modal fields in region IV. These fields are at” = 5; £3 Sinhv" (’14 '1' 2) (3.333.) 17,,” = - 5,- 12,- coshv;(h4 + z) (3.33b) The total transverse fields are the sum of all possible TE and TM modal fields. According to equations 3.32 and 3.33, the total transverse fields in regions III and IV are given in the following, 1?,” = z a, a, sinhv,(h3 - 2) (3.3421) Fl)” = z d,- 6, coshv,(h3 - z) (3.34b) 5",” = Z 5,- é‘, sinhv,(h4 + z) (3.34C) [7,” = - 2 5,- 11,- coshv,(h4 + z) (3.34d) 54 3.3.3 The coaxially loaded cylindrical waveguide The coordinate system and cross-section of a coaxially loaded cylindrical waveguide is shown in Figure 3.4. The metallic waveguide of radius b has perfectly conducting walls. Region I is a lossy dielectric rod of radius a. It has a complex rela- tive dielectric constant an and is placed symmetrically within the waveguide. Region II is the concentric area of complex relative dielectric constant 6,2 between the dielec- tric rod and the conducting walls. The electromagnetic fields in region I and II are the solutions to the source free Maxwell’s equations subject to the boundary condition that the tangential electric and magnetic fields are continuous over the dielectric interface and the boundary condition that the tangential electric and magnetic fields are identically zero on the perfectly con- ducting surface at r = b. In section 3.3.1, the derivation of the general solutions of the source free Maxwell’s equations was introduced. By choosing the appropriate forms of wave function V’s, the EM fields in each region can be determined readily from equations 3.9 and 3.12 for TM and TE modes, respectively. According to equation 3.18, the wave functions in each of the two regions are given by (1) Region I, 080 ‘09,,9) 15;" = 0 £31 =A _1_ 2 B ml (jme1 Km) 0 09,9) 11;" = 0 Hz” =A <- xp, )Bor‘kxpy) 11:“ = 0 Region (II) (1) n at 0 ( nonrotationally symmetric TM modes ) 5?: =C (Toi— ) ( thy)8.""(xp,p)oosn¢ 2 End =C L H B“. ' . (1“sz ) ’(szpwnw .. l u E.2=C(E)(Kpf)8. 2(v<,,,p)cosm1> H32 =C (- % >8."2(x,,p)sim¢ Hz."2 =C (-rp,)B.""(Kp,p)cosn¢ Hrz=o (2) n = 0 ( rotationally symmetric TM modes ) 53" = C ( 36L ) < w) Bo"2'B."'Bo"'(x.,p) 15:1 = o 113‘ = B (If; ) < w) Bo“' H;1 = o H.“ = 3 (7305-1- ) ( x9} ) Ramp) Region (II) (1) n a: 0 ( nonrotationally symmetric TE modes ) E32 = D (— % )B,‘2(Kp2p) cosno (3.42) (3.43) 59 5:2 = D (sz ) B.'2'(x,2p) sinn¢ 5:2 = o (3.44) ”32 = D (a: ) ( szv ) B.‘2'(Kp,p) mm H? = D (To: ) < 1‘} )B.'2s.~* = c asleep,» (3.46) l (2) ”11 = ”:2 B (irwmmw (flwfltxpa) (3.47) K9: 1 “2 2 (3) 15,l = 5,2 n M C l A (17);? )B. ‘(vcp,a)+B x,,B. l(w) 60 = C ( 17:2; )B,"2(sza) + D sz B,‘2(sza) (3.48) (4) HQ! = ”.2 A (— x91)8.'"" = c (— x., > 3:10.92.) + D (J— ) < 51 )B."(x,,a) (3.49) W2 (1 where 5,1, 5,}, H”, H9: = field components of hybrid modes in region I 5,2, 5,2, 11,2, 11,2 = field components of hybrid modes in region II Equations 3.46 through 3.49 are four linearly independent homogeneous equations with four unknown constants A, B, C, and D. A nontrivial solution exists only if the deter- minant of the coefficients of A, B, C, and D vanishes. For mathematical simplicity, define mm = B.“‘(K.,p) . F. = Na) . F.’ = Fr'(a) = als,-'11:“) F202) = B.“(Kp,p) . F. = tha) . F; = F2’(a) = mp, B."'(x,,a) F30» = B.“2(Kp,p) . F. = Fa(a) . F; = Fs’(a) = mp, 8:109,» F4(p) = Bn‘Rxpzp) 9 F4 = F4(a) 9 F4’ = F41“) = ‘91 Bndz'(sza) Substitution of these expressions into equations 3.46 through 3.49 gives A (62912)]:1-(3 (éerf)F3=0 3(H1Kp,z)F2'D(lHKp22)F4=0 (3-50) A <.—"7—)F.+B F2'-C(.—"7—)F3—DF.'=0 Jwera Jméza _ ’ J.- H _ —-L 21 = A( humming >F2+CF3 D(o)p.2)(a )F. 0 Hence, the characteristic equation for the dielectric loaded cylindrical waveguide in determinant form is 61 : (22mm 0 — (3 66) N u ’(szb )Jn (K929) - Jr. ’(szb )N3 (929) ”W” “ J ' (“"1“) N. my. (9.0) — J.’ (2)TMmodes(n=0) 2 Kp E." =A(a)(-L)(-K;—)Ro(xp,p) 2 E°m=0 (3.84a) (3.84b) (3.84e) (3.84d) (3.84e) (3.840 69 l E "=A —— 2 R 3.85 :2 (josh )(Kp‘ ) 009329) ( ) e. '9’ H,;'=A(-—)(—1-)Ro'(l=[a,-a,‘ds (3.90) where 3 represents the guide cross section, 6,‘ is the complex conjugate of 6,. Then, the orthogonality properties of the normal waveguide mode can be made as follows, <éj,é‘°>={coj ::5 (3.913) <5).I?.->={_?j 13:5. (3.91b) where a, , e“.- = transverse electric modal fields of TE (TM) mode 1?,- , ii,- = transverse magnetic modal fields of TE (TM) mode c,- , s; = normalization factors It is worth noting that if a, and 5, represent the transverse modal fields of a TB mode while a.- and 5.- representing the transverse modal fields of a TM mode, then (éj ,5; >20 for all i and] (3.92a) <5,- .ii.- >=o forall i andj (3.92b) Taking the dot product of the elecuic field equation 3.8% with a," , and the magnetic field equation 3.8% with £3" and integrating over the waveguide cross section, using the orthogonality relationships of the normal modes, two homogeneous linear equations with unknown constants 0,, A,’s and B,- ’s are obtained. I ‘7.”— T:- a a, c3), sinhv3j(h3- f): 201,- e 2 +3,- e 2 ) (3.93a) l '2 1 I: '71— r- .. . a,- s3} coshv3j(h3— f): 201,- e 2 —B,- e 2 ) ‘ (3.93b) 73 In a similar manner, equations 3.89c and 3.8% reduce to two homogeneous linear equations with unknown constants 5,, A,- ’s and Bg’s. l l 2 2 Y .— —Y.— . ‘ '2+B‘°€ .2) .. , I b" C4} SlnhV4J(h4 - 72) = z ( A; C I C l I .. 12 75"," "Ti-2 . -bjS4ICOShV4j(h4-"§')=Z(Ai€ 2-B,e 2) t where Equations 3.93a through 3.94b can be simplified by defining I 5} = 03, SinhV3,(h3 - '32) _. 12 S} = S3} W310” -" ‘2') . 12 E, = C4, WWII,“ " ‘2') 12 3} = 5'4, COS/4,014 - ?) Substituting into equation 3.93a through 3.94b yields ’2 '2 — '2‘? 2‘7 ~ . d,c,=£(A,e +856 ) ’2 ’2 .. 7"— ‘7.'_ a . —b,3,=Z(A,-e 2-B,-e 2) (3.94a) (3.94b) (3.95a) (3.95b) (3.96a) (3.96b) (3.960) (3.96d) (3.973) (3.97b) (3.97c) (3.97d) The a, ’s and 5,’s can be easily eliminated from above equations by dividing equations 74 3.97a and 3.970 with equations 3.97b and 3.97d, respectively. ’2 .‘2 A-e '2+B,e 2 a F‘;( . ) i}? E‘T: 1’2 7'2 E- l ‘t— — 1' £(Aa 2_Bte‘2) ‘2 -11 A-e 2+3, '2 a] _$( ‘ ).ifl -_" z 11 T 1 Ti ‘ — l‘ 2(A.-e 2-B.e ‘2) where ffi=<‘,,é‘3’> 7,,= 2,.=<‘,~,e‘4j> 9fi=<fii.£4,> (3.98a) (3.98b) (3.98c) (3.98d) Multiplying out equations 3.98a and 3.98b gives a system of equations in A,- ’s and B,- ’s. 0 2A; )7}; +3; iii 3 ’2 Y,,o=e (s,x,.--c,y,.~) 7.1.2. Xfl=€ 2 (3} iii +31”) -713 Yii=‘ 2 (3)217'31' 5‘1?) (3.99a) (3.99b) (3.100a) (3.100b) (3.100c) (3.100d) The matrix form of the system of infinite equations in equation 3.99 is expressed as [22;] [21w (3.101) 75 where the element values of the submatrices if, )7, J? and f' ( i.e., Y,“ 7,, 1?,- and 7,,- ) are determined with the aide of equations 3.100a through 3.100d. The subscript i represents the mode number of the dielectric-loaded waveguide modes and the sub- script j represents the mode number of the homogeneous waveguide modes. A nontrivial solution exists only if the determinant of the coefficient matrix is zero, that is det [1: ;]=0 (3.102) The resonant frequencies of the dielectric-loaded cavity of the cavity-open type are the roots of equation 3.102. In theory, the sizes of the submatrices if, i", J? and Y are infinite because there are infinite numbers of modes existing in both the homogeneous and the dielectric-loaded waveguides, respectively. But in practice, for numerical com- putation, the infinite matrices 1?, i", X and Y are truncated at a certain number N of the homogeneous waveguide and the dielectric-loaded waveguide modes. Then, the fre- quency is varied and the determinant in equation 3.102 is computed for each fre— quency. The frequencies giving zero value of the determinant are approximations to the resonant frequencies. The size of the determinant N is determined by the convergence condition of the roots of the determinant, and the convergence toward the resonant frequency is achieved when the size N is increased. The size of the system is increased by taking more and more modes in the dielectric-loaded cavity. The modes in each of the three subregions are chosen as follows : for nonaxially symmetric modes ( n at 0 ), 2N hybrid modes are taken in regions I and II, the coaxially loaded waveguide region, and N-TE modes and NW modes in regions III and IV; for axially symmetric modes ( n =0 ), N-TE ( N-TM ) modes are taken in regions I and II, and N-TE ( N-TM ) modes in regions III and IV. 76 Analytic expressions for the inner product terms of equations 3.95a, 3.95b, 3.98c and 3.98d, and the element values of the submatrices if, )7, X and f’ of equations 3.100a through 3.100d are given in Appendix A. 3.4.2 Cavity-short type The cavity-shorted type can be treated as a special case of the cavity-open type by forcing the lengths of the two homogeneous waveguides to be zero. As a result, all the analysis in previous section applies here except that 13 and I, are taken to be zero while doing the numerical computation. Even though the homogeneous waveguides do not exist in the cavity-short type, the normal modes of the homogeneous waveguide are still included in the computations. The involvement of the modal fields of the homogeneous waveguides results in more numerical computation time due to the large size of system equations. The analysis of the cavity-short type can be made more straightforward and much simpler if the homogeneous waveguide regions are excluded. As shown in Figure 1.1b, the cavity configuration of this problem can be viewed as a dielectric-loaded waveguide shorted at both ends. The axial boundary conditions require that the transverse electric and magnetic fields vanish on the conducting end plates. The coor- dinate system is defined in Figure 3.5. The transverse electric and magnetic fields of the dielectric-loaded waveguide are given by equations 3.87a and 3.87b. The boun- dary conditions of zero transverse electric field on the end conducting walls lead to z; E,- (A,- + 3, )= o for all ¢ and p (3.103a) z; E,- (A,- e‘” + 3,. :7“ )= o for all ¢ and p (3.103b) According to equations 3.81 through 3.86, it is found that E’s are not functions of z and do not identically equal zero over 2 = L plane. Examination of equations 3.103a and 3.103b reveals that they are identically zero if only if . Z >///////////// a. ///////////// .T . r/ . 78 A; + B; = 0 (3'1043) and A; e-Y‘L + B; CY‘L = 0 (3.104b) From above expressions and knowing that A,- and B,- are not zero for all i, the follow- ing condition is reached, that is sinh‘y,L = 0 (3.105) The subscript i is dropped from now on because above relation holds for any modes. Since 12 = — x3, equation 3.105 can be rewritten as sinx.L = 0 (3.1063) so 10:11:15 q=0,1.2,.... (3.106b) Substituting - x} for 7’ in equations 3.37a and 3.37b. the separation equations become it," + x} = x,” (3.1078) x922 + x} = x22 (3.107b) where x12 = «fine, , x22 = 03211222 (3.1070) From the characteristic equation in 3.71 and the separation equations in 3.107, along with the propagation constant 1:, defined in 3.106, the resonant frequencies of the cavity-short type are readily determined. The present approach without including the two homogeneous waveguide regions avoids the computation of an infinite system equations. Apparently, the numerical computation procedures are simpler and the computation time is less. The solutions of the cavity-short type problem are very useful for the understand- ing of the more general case, the cavity-open type problem. They are utilized to verify 79 the solutions of a special case of the cavity-open type in which the lengths of the homogeneous waveguides are zero. 3.4.3 Cavity-image type The cavity-image type problem can be solved in many ways. It can be treated as a special case of cavity-open type problem by forcing the length of one of the homo- geneous waveguide regions to be zero, say, I, = 0. Electric wall modal analysis also can be used to analyze this problem [60]. As is shown in Figure 3.6a, the transverse electric fields are zero on the conduction wall in the z = 0 plane, this problem becomes a cavity-open type problem with cavity length of 2L and s symmetric with respect to z-plane. In this section, the cavity-image type problem is considered as a coaxially loaded waveguide connected to a homogeneous waveguide, both waveguides are shorted at one end. Following the cavity-short type analysis in previous section, the coordinate system of the cavity-image type is shown in Figure 3.6b. One of the homogeneous waveguide regions, region IV shown in Figure 3.1, is excluded form the cavity-open type prob- lem. This problem now is subject to the boundary, conditions that the transverse elec- . . . I . trrc and magnetrc fields are conunuous over 2 = % plane and the transverse electrrc filed is zero on the conduction end plates in z = 0 plane. The transverse field components in region III are expressed in equations 3.34a and 3.34b. The transverse field components in regions I and H are expressed in equa- tions 3.87a and 3.87b. To satisfy the boundary conditions, three homogeneous linear equations result : :2 r 2 2 x ' 5- i— I ZE, (A, e 7 2 +3.. :7 2 )= 2a, 63, sinhv3i(h3- -23) (3.108a) t I -12. 1.2. I 2 H,- (A, . 2 "' B‘- e . 2 )= z a] £3). COShV3j(h3 - ‘3') (3.108b) i 80 to—b—u-l T :‘13 .31.. 7 7 _ 2L I2 {AM/A .K "' Y L ha ___.__._ “Li ___ I €211 /)¢7// 0.512 * t’Y (0.0) (b) Figure 3 . 6 coordinate system for the cavity— image type 81 z E; (A. + B; )= 0 (3.108C) From equation 3.1080, it is obvious that A,- = — B,- for all i (3.109) Using the orthogonality properties of the modal fields of the homogeneous waveguide and equation 3.109, two linearly independent homogeneous equation are obtained. Theyare - ' 1.2. a,c,-zA,(e ‘2 — 7‘2)<1§,-,a,,> (3.110a) -12 7!"; A A 0,3}=ZA,-(e ‘2 +€ 2 )(H;,h3l> (3.110b) Dividing 3.110a by 3.110b yields I - 2A, sum, 31):,- = 12 (3.111) 24; C0850: '5071.‘ an [on where E}, F,, 3?,- and 5",.- are defined in equations 3.96a, 3.96b and 3.990. After some algebraics, equation 3.111 becomes 2‘: if" - =0 (3.112a) where _ I I X,- = Sinha-é) i}, s“, + comm—23) 7,,- F, (3.112b) Equation 3.112a has a nontrivial solution only if the determinant of the coefficient matrix is zero, i.e., det [rfi ]=o . (3.113) where the element values 11,,- are computed according to equation 3.112b. The 82 resonant frequencies of the cavity-image type problem are the roots of equation 3.113. As was discussed in section 3.4.1, for the purpose of numerical computation, a trun- cated system is considered in equation 3.113. The size of the matrix is determined by the convergence conditions of the resonant frequencies. More cavity modes taken into account will result in larger matrix size but better convergence. Although solving the roots of the determinant of a truncated matrix to obtain the resonant frequencies is needed as is the case for the cavity-open type, there is a major advantage over the analysis described in section 3.4.1. Comparison of equation 3.102 and 3.113 reveals that the matrix size in the present analysis in which only one homo- geneous waveguide region is considered has reduced to one half of that presented in section 3.4.1. As far as the computation time is concerned, smaller determinant size means faster numerical computation. 3.5 Stored energy, absorbed power and quality factor 3.5.1 Introduction The energy stored in the cavity and The power losses due to the conducting walls and the dielectric losses are investigated in the following two sections. These results will be used to calculate the quality factor Q of the dielectric-loaded cavity. The power loss due to the coupling structure is neglected to avoid the complexity of input probe analysis. The cavity structure shown in Figure 3.1 will be considered throughout this section. 3.5.2 Stored energy The energy stored U in the cavity is composed of the energies stored in the two homogeneous waveguide regions, inside the lossy dielectric cylinder, and in the region between the lossy dielectric cylinder and the conducting waveguide, walls, i.e., region (II-2). Thus 83 U=U1+UZ+U3+U4 (3.114) where U1 = energy stored inside the lossy dielectric material, i.e., region I U2 = energy stored in region 11 U, = energy stored in region 111 U 4 = energy stored in region IV The dielectric materials filling regions 11, III and IV are assumed lossless. In other words, 22, e, and e, are actually real-valued numbers. The time average electric and magnetic energies are given by, U, ”213 Irn (”de Irma) (3.115a) U..=?161m(_mjoruoll7|2dv) (3.115b) where e=e’-je” Since the total energy stored in the cavity passes between electric and magnetic fields, it may be calculated by finding the energy storage in electric fields at the instant when these are a maximum. Thus the total energy is u = 2n, = ‘33 Irn jfljme ”2*.de =32”; urn dv (3.1150) The modal fields in each region were derived in sections 3.3.2 and 3.3.3, and are rewritten as: - ’2 ’2 RegronsIandllz-7( mm. as + uc,|2c, D.*) -(q.-1" +q.+1‘)(|cr|’A.‘ 3. + lealzCR D.) +(Et-r-zwr)(€r(311'I A» Cu' +013 C311." C.) +(8n-r-8ur H6163"I 3. D.‘ 4431'I 6'33." D.) -(q.-1-q.+r)(61Ca* A. D.‘ +Cr“ 63A.‘ D.) - (4.4" -q.+r* )(c1c3‘ B. C.‘ +ct" eaB.‘ C.) + H + at: t - 2 [(IA|2+IBI2)S, -(AB +AB)Sz ]>< [(32- W2 x( 01,12; + 13,. fig, -A,, as q, -A,." a, 4*, ) ]} (3.117b) I 2 1 l2 03 =< ‘3 " ><-.,—:?rz->{[v3 (p".+r+p".-r >(sr-s;)]+ 2m2 2 [(K3PR.)(53++53_)]} ———TM ' ~ (3.117C) 86 “63’ I 2 + -— c e =T(Kp)(53 "$3 )(pu+1+pn-l) TE U - ‘7" K" ”"2 s -s ]+ 4 —(:0—_—2)(T_-_84’|2) (Pm n+l+p mn-l)( 4 4.) [I2p do b g. =I IN.(Kp,p)|’pdp b b q. = IJ.(Kp,p) N:(Kp,p) 9 do 4.. * = II: (19,9) N. (rap) 9 do . o 12 , . (2 WIMP-Y )7 srnh(y-y )—2- S += S -= 2 o 2 0 1+7 7-7 sinh( V3+V3. )13 _ sinh (V3 “V3. )13 5.3+: 0 S3 = e V3+V3 V3-V3 S,_sinh(v4+v4' )1: S__sinh(v4-v4')lr 4 - o 4 - o V4+V4 V4-V4 1.09.9) N. «p,» A" = N.(x,,b> J.{(b1.2(x,b11<: +s. )+(p “1+1: an} —m :"2 R: I =( M02 -———2n) 01.209”) (54 +54 )+ 1‘11“‘(544-"Sa-)]‘*‘(b'\’4'2“p'2)(12‘u+1firm-1)}——Tli' where ._ _ 12,12 'Kp__,_‘_‘_| 2 2 01121 (:2- 18112 'Tzlz IIIR(K92b)I+(; )(| 12112 )(0) -_2Lo) “(91" £2 ——IP,, 12-2R —— * x “(92" (xpzb) c{((02u02a|£1|2)( a7) “a" xtl)<———‘—;1R.(x,b)P* b 11:1,2l21qfi"2 = 1 I01 2I2 Pc2=( (0‘ “02 )W an(Kp2b)12 l 90 3.5.4 Quality factor Q for dielectric loaded cavity A quality factor Q for the dielectric-loaded cavity may be defined as U Q-%, where too is the resonant angular frequency, U is the total time-average energy stored, and P is the average power loss. As was mentioned in previous section, the time- average power dissipation consists of two parts, the power loss due to the dielectric material and that due to the metal walls, namely, 1’ = I" + P, Consequently, it is convenient to express the Q in terms Q, and Q4 corresponding to the losses P, and P4 to have a better understanding of the effects of the losses in the various parts of the loaded cavity. i=._1_+_1_ Q Q4 Q: where - _U. Qd-O’ol,‘ - 1L Qc‘wOPC mo = resonant angular frequency It is of great interest to understand how the loss of the dielectric affects the Q. Since the resonant frequency and Q are the two measurable experimental data, knowing the effects of the dielectric constants on them in theory will help interpret experimental results and improve on-line diagnosis microwave material processing techniques. From the measured Q’s and resonant frequencies, the dielectric properties of the material may be predicted and the beginning and termination of the material heating cycles may be determined. Chapter 4 NUMERICAL SOLUTIONS AND TECHNIQUES 4.1 Introduction The characteristic equations for the dielectric-loaded waveguide and the dielectric-loaded cavity were derived in chapter 3. In this chapter these transcendental equations are solved numerically using Muller’s method [71, 86]. Muller’s method has been used on computers to find any prescribed number of zeros, real or complex, of an arbitrary function with remarkable success. In section 4.2.1 a brief description of Muller’s method is given. In addition, the root- searching algorithm and the determina- tion of the initial approximation is described in section 4.2.2. The dielectric-loaded waveguide is known as a W because the material discontinuities occur at the dielectric-dielectric interface in the radial direction. In section 4.3, the changes in the propagation characteristics as a function of efi, e," and rod radius are investigated. The 0H: diagrams for eight waveguide modes, i.e. TMm, TM”, 150,, TE”. HE", H521, EH" and EH” are presented. In order to have a better understanding of the changes of the propagation characteris- tics as one of the waveguide parameters (2;, cf, and rod radius) varies, the m - x, diagrams and electromagnetic field patterns for Tim”, mo, and HE" modes are studied in more detail. The dielectric rod under study includes lossy dielectric materials and isotrOpic lossy cold plasma. The dielectric-loaded cavity is referred to as W due to the fact that the electromagnetic field continuity conditions are enforced over the material discontinuities in the axial direction. In section 4.4, the resonant frequency, resonant length and quality factor of the dielectric-loaded cavity for TMm, T5012 and HE,“ modes are determined as a function of 5;, e: and rod radius. In addition, a com- parison between the exact formulation and the perturbation technique for TM 0,, mode is 91 92 presented. 4.2 Numerical technique 4.2.1 Muller’s method Most numerical methods for searching roots of an arbitrary function can only find an isolated zero of the function once an approximation to that zero is known. These methods are not very satisfactory when all the roots are required or when good initial approximau'ons are not available. Some methods may need the evaluation of the derivative of the function, which may turn out to be very difficult because of the com- plexity of the function itself. A method, developed by Muller and hence called Muller’s method, can numeri- cally find the complex roots of an arbitrary function on computers. This method is iterative, converges almost quadratically in the vicinity of a root and does not require the computation of the derivative of the function. Both complex and real roots can be obtained even when these roots are not simple. Moreover, the method is global in the sense that an initial approximation needs not to be supplied. In addition, if an approxi- mation to the zeros is provided, less computation time is required using Muller’s method. Muller’s method is an extension of the secant method. In the secant method, a second order polynominal P(x) is established by passing through (x,- , f (x,)] and {rum , f(x,-,1)}, where x, and 153-1 are the approximations to a root of f(x) = 0. The next approximation xm is then determined as the zero of P (x). In Muller’s method, the next approximation x,” is found as the zero of the parabola which goes through the three POintS ix.- .f(x.-)}. {xi-1 .f (xi-1)}. and [xi—2 .f(X.--2)}- The fOUOWing Pm‘abola P00 =f(xi) + (I - 100i +flxt . xi-l . xi-2](X - 102 (4-1) is an unique parabola which agrees with the function f(x) at the three points It. xi-l. xi-z. Wlth 93 C; =f [It . xi-l] +f [xi . Xi-t , 35-210; - Xi-r) (4-2) _ f(xi) -f(x.--1) f Ix.- .xH] - x.- -x.--1 (4.3) __ fix; .xi-tl "flxt-t . xi-7J f[xi 9 xi-l 9 xi-ZJ - xi _ xi-Z (4'4) Thus any zero it of the parabola P(x) satisfies ‘ 2f 0“) (4.5) x- .= x‘ c.- i {cf-quark.- .x.-..x..211m The sign before the radical is selected so that the denominator is the largest in magni- tude. If the right-hand side of equation (4.5) is labeled as hm, then the next approxi- mation to a zero ofo) is taken to be xi“ = xi + hid (46) The process is then repeated using xH, x,, x“, as the three basic approximations. The iteration process is stopped until either of the following criteria is satisfied for prescribed s1, 22: ln-aal (a) T < 8‘ (47) (b) lf (xa)| < 62 (4.8) or until the prescribed number of iterations is exceeded. To find more than one zero, Muller’s method employs a procedure known as deflation. For instance, if one zero 5,, has already been found, the next zero will be calculated by working with the new function f 100 = £% (49) 94 If r zeros 1:1, ..., 1;, have already been found, the next root is obtained by working with the deflated function = for) M" (x - €00: - :2) - - - (x - a.) (4°10) 4.2.2 Root-searching algorithm The complex characteristic equations in equations 3.71 and 3.102 for the dielectric-loaded waveguide are solved numerically using Muller’s method. There are very many roots inherent in these complex transcendental equations and each of the roots corresponds to a unique waveguide mode. As a result, how to locate the correct root for a particular mode is nontrivial. An iterative root—searching algorithm, which uses the homogeneous waveguide as the initial approximations in Muller’s method, is developed here. This algorithm not only identifies each of the infinite modal solutions but also gives clear insight into how the mode changes as a; and e: are varied from 1 and 0, respectively. A detailed description of the algorithm is presented in the follow- ing. The waveguide configuration is shown in Figure 3.5. By letting the dielectric constant of the dielectric rod equal to that of the surrounding media, i.e., e, = 82, the dielectric-loaded waveguide becomes a homogeneous waveguide. Then, using the well known homogeneous waveguide solutions as the initial approximation in the Muller’s method and incrementing £1 slightly, the solution corresponding to the new inhomo- geneous waveguide is obtained. Replacing the initial approximation by the solutions just attained and incrementing e, again, the solution associated with the new dielectric constant of the dielectric rod is evaluated. Solutions obtained at each iteration are util- ized as the approximations for the next iteration. The iterative process continues until the preset dielectric constant of the dielectric rod is reached. Throughout this chapter, region II is assumed to be filled with air, i.e., t2 = so. 95 The dielectric permittivity of the dielectric rod is complex. In other words, the dielec- tric material in region 11 can be either lossless or lossy. For lossless dielectric rod, only the real part of the dielectric constant is incremented in the computation process while the imaginary part is set to be zero. For lossy dielectric material, both the ima- ginary and real parts of the relative dielectric constant are incremented from free space permittivity toward the preset value. However, they do not change simultaneously. The computation process is a two-step process. First, the real part of the dielectric con- stant is increased to a preset value, with the imaginary part equal to zero. Then, the dielectric loss is incremented and the real part remains unchanged for the rest of the computation. There is no special order to which part of the dielectric constant is incremented first. The numerical results are not affected if the imaginary part of the dielectric constant is incremented ahead of the real part and vice versa. Note that only one part of the dielectric constant is changed and only one root is searched for at each iteration in the Muller’s method. This is because there are many roots corresponding to various modes embedded in the transcendental equations. Some of the roots may exist in a very tight neighborhood. A large disturbance in the vari- able parameters, such as the complex dielectric constant, may result in solution corresponding to undesired mode. This is known as modal jump in this dissertation. In addition, if more than one root is desired at one time, it becomes extremely difficult to identify the modes of the solutions. Consequently, in order to avoid modal jump during computation and to be able to identify the root, only one zero at a is investi- gated. In order to remain on this zero a very small step size is added to the variable parameter during each iteration process. Adequate step size of the variable parameter is determined by trial and error. The numerical technique for solving the cavity problem is identical to what just described above except that the homogeneous cavity solutions are used as the initial approximations. A three-layer dielectric- loaded cavity, as shown in Figure 3.1, is 96 considered here. For simplicity, it is assumed the dielectric materials in regions II, III and IV are the same and equal to air, i.e. £2 = e, = e, = so. Then the resonant fre- quency, resonant length and quality factor vs. 6,1, 2:, and dielectric load radius are obtained by following the empty cavity solutions. Note that for simplicity, in the fol- lowing sections 2, represents the complex dielectric constant in region I. 4.3 Numerical solutions 4.3.1 Introduction The numerical solutions given in this section are divided into two categories: (1) radial boundary value problem solutions and (2) axial boundary value problem solu- tions. As mentioned in section 4.1, the radial boundary value problem corresponds to the coaxially loaded waveguide, and the axial boundary value problem corresponds to the dielectric-loaded cavity. In section 4.3.2 the or — 1:, solutions of various waveguide modes of the coaxially loaded waveguide are evaluated. These solutions are computed for different running parameters such as the dielectric permittivity and dielectric material radius. These dielectric-loaded waveguide solutions are essential for studying the dielectric-loaded cavity problem. In section 4.3.3, the resonant fiequency and the resonant length of the dielectric- loaded cavity are calculated as a function of 2;, a: and load radius for various cavity modes. In addition, the influence of the complex dielectric constant and dielectric sample volume on the quality factor is also examined. 4.3.2 Radial boundary value problem (dielectric loaded waveguide problem) In this section, the co - 1:, solutions for the coaxially loaded waveguide are evaluated for eight waveguide modes, which include TMOP, TED, and hybrid modes. The influence of the perfectly conducting walls on the propagation characteristics and 97 electromagnetic field patterns are examined by varying b/a ratio, where b is the radius of the waveguide and a is the radius of the dielectric material. In addition, the changes in the propagation characteristics and EM field patterns as a function of e,’ and e," are studied. The separation and characteristic equations for the coaxially loaded waveguide are given in equations 3.37 and 3.71, respectively. The subscript 1 and 2 represents the regions I and II shown in Figure 3.4. Here, nonmagnetic materials are considered, i.e., it,! = 11,2 = 1. Region II is assumed to be filled with air, i.e., 2,2 = 1. Equations 3.37 and 3.71 are rewritten as it2 = er,‘ + z2 (4.1 la) Y2 = W2 + 22 (4.111)) W’V.W. + U} = o (4.11c) where J.’(X) P.'(Y) V, — X - Y (4.12a) J.’(X) R.’(Y) W, — 6,! —X_ - Y (4.12b) _ .1. - i U, — n 2 J.(X)( X2 Y, (4.12c) X = Itpla Y = K920 Z = ya W = xoa where P. and R. are expressed in equations 3.66 and 3.67, respectively. Substituting 98 equations 4.12a through 4.12c into equation 4.11c gives the following transcendental equation. J '00 P.’(Y) LIX) Ra'O’) 2 n - — —_ u ( x Y )(e,, x Y ) 2 2 2 Yz-Xz 2 +11 Z J.(X)(—X-5YT') =0 (4.133) Since Y and Z can be expressed in terms of X by means of equations 4.1la and 4.1 lb, equation 4.13 is in fact a transcendental equation with one unknown X . X is the root to be determined. For axially symmetric modes, n=0, TE and TM modes decouple, as already been demonstrated in chapter 3. These modes of the dielectric loaded waveguide are still denoted by TMO, or 1730,, modes, in accordance with their physical nature. The characteristic equations for these modes are given as 6,1150!) Y - Ro’o’) X = 0 for TMO, modes (4.13b) Jo’ot) Y - Po’o') x = 0 for 180, modes (4.13c) For nonaxially symmetric modes, i.e. n at 0, due to the existence of both axial electric and magnetic field components, the waveguide modes are no longer TE or TM with respect to any coordinates. These modes are called hybrid modes. They are denoted by HE", or EH” depending on their nature in the limiting case when the relative dielec- tric constant of the dielectric material is one. HE” modes corresponds to TB, homo- geneous waveguide modes, while EH, modes corresponds to TM” homogeneous waveguide. As mentioned in section 4.2.2, equation 4.13 has many roots and each root is associated with one waveguide mode. These roots are searched iteratively starting from empty waveguide solutions . The separation equations for empty cylindrical waveguide are given below. 99 (x20 )2 = K302 + 7202 for TM” modes (4,143) (fig-0')2 = K0202 + 7202 for T8,, modes (4,141)) It is obvious that the initial approximation to X , the normalized radial wavenumber, should be r a for TM... modes 0 for T5,, modes it b i b L where a = radius of dielectric material b = radius of circular waveguide The subscript 0 denotes the initial approximation with the relative permittivity being one, i.e., 2,, =1. X0 is in fact the cutoff wave number of the empty cylindrical waveguide [87], normalized by the rod radius. Complete co - x, solutions are obtained in two steps. First, the computer program searches for roots iteratively at a fixed frequency, starting from X o by incrementing dielectric constant until the desired value is reached. Then, the complete or -- x, solu- tions are evaluated iteratively by varying incrementally the operating frequency, while the dielectric permittivity is kept constant during the computation process. 4.3.2.1 or — tt diagrams and electric field patterns 4.3.2.1.1 Introduction A waveguide loaded with lossy dielectric material has different propagation characteristics and field patterns from those of a homogeneous waveguide. It is of 100 interest to study the changes in the dielectric loaded waveguide modes as a function of the size and permittivity of the dielectric load, as well as the operating frequency. The waveguide under investigation in this section is a cylindrical waveguide of radius b, loaded with a lossy dielectric rod of radius a and dielectric constant e, = a; —je;', as is shown in Figure 3.4. The region between dielectric rod and waveguide walls is assumed to be an air-filled region, i.e. 94 = 80. Nylon is chosen as an example dielec- tric material in this section. Since nylon is a low loss dielectric material with e, = 3.03-j0.039 and commonly used in the laboratory, the characteristics and EM field pat- terns of a nylon rod loaded waveguide can easily be verified by experiments. All the dielectric-loaded waveguide examples demonstrated in the following sections use either nylon rod or variations of nylon which have a; or 2: equal to that or nylon. In order to show the general trend of the changes in the propagation characteris— tics as e," is varied, the a) - x, diagrams of eight waveguide modes are examined. These eight modes cover both axial symmetric and asymmetric modes, including two TM, two TE and four hybrid modes. Following these examples is a detailed study of the propagation characteristics and field patterns of TMm, mo, and HE“ modes as one of the waveguide parameters, such as a/b, ts;1 and 2:1, is changed. These three modes are the frequently used waveguide modes for microwave material processing and for gen- erating microwave plasma in the laboratory. In addition, a waveguide loaded with iso- tr‘Opic cold plasma rod is also examined under a variety of experimental conditions that are of interest to electric propulsion discharges and diamond thin film deposition discharges. Since the dielectric rod is lossy, x, is complex, i.e. it, = B - jar, where B is the pro- pagation constant (B = 21r/ 1,) and a is the attenuation constant in the 5‘ direction. By solving equation 4.13 for non vs. tea with en, 6,2 and b/a specified, xoa vs. Ba and rod vs. an diagrams can be generated for each waveguide mode. Interesting elecu'omag- netic properties can be obtained from such diagrams. Because of the lossy guiding 101 structure, the waveguide modes are either propagating modes but subject to attenuation in i-direction, or nonpropagating modes, which decay exponentially in i - direction with no wave propagation. As will be seen in the following examples, or and [3 do not vanish simultaneously, that is, it, never goes to zero at any frequency. Therefore, there is no longer a well-defined cutoff frequency. In order to identify the propagating, highly damped propagating and nonpropagating modes, in this dissertation a new cutoff frequency is defined. Cutoff is defined as the frequency at which aAo=l, where M is the free space wavelength. According to this definition, a cutoff line, on which aAo=1, can be drawn on the to — ct diagram. A cutoff point is found as the intersection of the cutoff line and co - ct curve. To demonstrate the concept of the new cutoff fiequency and cutoff line, the a) - x, diagrams for a waveguide coaxially loaded with lossy dielectric rod are shown in Figures 4.1a and 4.1b for TM,” and 115:0, modes, respectively, where a=l.27 cm and b=7.62 cm. As shown in Figure 4.1a, the normalized cutoff wavenumber tea is 0.36. Since B is always larger than zero, only propagating and highly damped propagating regions are identified. In the propagating region, i.e. roa > 0.36 in this case, B > 0 and ado < 1. While in the highly damped propagating region, i.e. xoa < 0.36, B > 0 and ado > 1. In other words, in the highly damped propagating region a wave will suffer an attenuation 2 l/e after traveling one fiee space wavelength. Figure 4.1b depicts a case where propagating, highly damped propagating and nonpropagating modes exist. As shown, B=0 for Koo < 0.35 and the cutoff line intersects co - or curve at rcOa=0.625, i.e. x,a=0.625. Hence, propagating region is in the frequency region of 1:00 > 0.625. Highly damped propagating region is in the frequency region of 0.625 > Icon > 0.35. As Koa < 0.35, the TED, mode is in the nonpropagating region. Note that, as demon- strated in this example, a mode below cutoff could be either highly damped propagat- ing mode or nonpropagating mode, depending on if B is zero. [Coo [COO 102 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1— 0.0' 0.0 Figure 4.1a l T—' ' T l 0.2 0.3 0.4 0.5 0.6 0.8 w—B and u—a diagram for Thor b=7.62 cm and c=3.0-j0.39 q l 0.1 0.7 mode, where a=l.27 cm, 0.8 I ' I ' I '/ _ 1// 0X7: propagating region // 0.5— “r"“r' Q:\T\<\X<\”K\ 0.5:. \ )X‘ /\\ \\ ‘ _ ‘\ highly damped region 04- 0.3—_ ' . 0.2;;4» are 0.1 _. ,//’ 77? :- —' 0.0 r I I I r I 1 l x I l l l “—I 0.0 0.1 0.2 0.3 0.4 0.5 0.6. 0.7 Figure 4.1b w—B and w-a diagram for TEor mode, where a=l.27 cm: b=7.62 cm and c=3.0—j1.0 103 4.3.2.1.2 to —- it, diagrams for eight waveguide modes Figure 4.2a shows the roar - Ba diagram for a waveguide cenually loaded with a nylon rod with, e, = 3.03 - j0.039 and b/a = 6. Eight loaded waveguide modes, i.e. TMm, TMm, T501, T802, HE“, H521, EH1, and EH21, are presented in this figure. The co - B diagram is separated into three regions by the free space light line and the light line in dielectric material, in this case nylon. Region 1 is the fast wave portion of the curves, in which the phase velocity v, of electromagnetic modes is faster than the speed of light in free space. Region 2 is the slow wave region where v, is smaller than the speed of light in free space. The electromagnetic modes illustrated in Figure 4.2a exhibit both slow wave and fast wave characteristics. HE“, TMm, If.” and HE” cross the free space light line and become slow waves before xoa = 2.1. While TM”, 1730,, EH21, and EH" remain in fast wave region for Koa < 3.6. NOt until xoa reaches 4.9, do all these four modes cross into the slow wave region. Above xoa = 4.9, all the slow wave modes shown in Figure 4.2a approach the nylon light line asymptotically as fi'equency is increased. It is important to note that even though T50, and EH 1, are degenerate modes in homogene- ous waveguide, they possess different dispersion characteristics and different cutoff frequency as will be seen later in Figure 4.2b in the dielectric loaded waveguide. As shown, TED, mode becomes a slow wave at Icon = 1.6, while EH1, does not cross the free space light line until Koa = 4.9. Figure 4.2b depicts the or - a diagram for the case of Figure 4.2a. It is seen that the relative order of the cutoff frequencies among these eight modes are essentially the same as in the limiting case when these modes are empty waveguide modes, i.e. when 8,1 = 6,2 = 1.0. However, the cutoff frequencies are shifted down from the empty waveguide mode. For instance, the normalized cutoff wavenumber xca is 0.4 for empty waveguide TMm mode but is 0.368 for loaded waveguide TMO, mode. When compared with Figure 4.2a, it is found that for propagating modes, slow waves are 104 cklzr.sdf — ck82r.sdf 7/14/89 6 ‘1 I I I I I /' / _ e / C! as e a» \ 5_ 1. Fast Wave Region 12 Figure 4.2a w-B diagram for eight waveguide modes, b/a=6. e=3.03-j0.039. 105 . . .mno.o«rne.nlu oua\n «coca monsoo>a8 usodo nan Eauquo at: nu.v museum OXYI m... . 0.7. b c _ r N... at of _ _ _ IT. . _ \r p r. _ M“ or??? eooemxo .. coerce 33:3 106 subject to larger attenuation than fast waves at a fixed Koa, while each mode is highly damped below cutoff due to ado > 1.0. Also seen is that above cutoff or increases when the mode crosses over into the slow wave region. Because the operating frequency considered in this dissertation is primarily at 2.45 GHz and the largest radius of the dielecuic materials is less than 5 cm, icon for all cases is well below 3.0. As a result, all the (D - x, diagrams under investigation later are limited to K00 less than 3.0. The dispersion characteristics of a waveguide loaded with lossy dielecuic rod varies as the dielectric loss of the dielectric rod changes. Figure 4.3a displays the (D - B diagrams for a rod having 9, = 3.30 -j0.39 with b/a = 6. The dielectric loss is ten times larger than the case of Figure 4.2. It is observed that HE", TM“, 1301, and 811;, still become slow waves as frequency increases. In comparison with Figure 4.2a, the crossings where these modes enter slow wave region change little for HE", IMO, and T50, modes but become lowered for EH21. How the crossings are influenced by the change in a: will be more clear later in this section when the propagation characteris- tics of TM”, T50, and HE“ are examined for a wider range of cf. TM”, TE”, H521 and EH11 stay in fast wave region for iron < 3.0. A close look at the low frequency portion, as shown in Figure 4.3b, discloses that the propagation constants for TMOI, Tina; and EH" modes do not vanish for any frequency. This type of phenomenon has been pre- viously found in plasma loaded cylindrical waveguide [88]. In comparison with Figure 4.2b, the a) - or curves displayed in Figure 4.3c show that the cutoff fiequencies for all modes are shifted up slightly as e," is increased. Except for H521 mode, the attenuation constants above cutoff for each mode is increased. At high fi'equency, i.e. W > 2.0, the increases in a for HE“, TMOI, T50, and EH2, are much larger than those for EH”, TM”, and T1502. For example, at Koa = 2.5, the attenuation era for TMO, grows from 0.025 to 0.22, while the (m for. TMO2 increases from almost zero to approximate 0.006 as a," is changed from 0.039 to 0.39. By 107 fig420.5pf elzr.sdf — e82r.sdf , 3'O'I'T‘l'l'l“ITIII / _ I / _ / / 2.5— / — / _ 1. Fast Wave Region / .. / i O / 2.0— / _. / / ‘ , 2. ~. owWave/Region ‘ / // ‘4 / Q50 _ Gr NR“ 8 -- TEfi 7— TM“ (9 / é -—- EH” -‘ / d?-- EH" -6) // 4 — TEor _ / free space light line- a _ H521 / 2.-—- TM“1 oo ’ _ ”Eu 0 T T T I U l I I I l r I I I I l l l I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ba Figure 4.3a w-B diagrah for eight waveguide modes, b/a=6. e-3.03-jo.39. . 108 . .c Ino.nlu .mla\a .uoUOE ovaaoo>¢s usudo acumwauwaav at: onu «o unequuoa hocosvouu 30a An.v ou=o«h on 8.0 3.0 8.0 3.0 . 8.0 8.0 .Ju — p — - P in — F I- 0.0 MI Ola , _ . as: I _ .«MI BIG 0 . rt EMF III _ PO - :1“. To §IU I _ .1 32h ole _d NAV 1 32. I ~ .. . ..L. . t u. ._.. .. _ _.___ Yo I TE. d .. . _ . fit I. mAu _ r . r r “ od ._ l 50 _ _ _ _ . _ _ _ Cemdflo I “.3420 camdmvoc 0°3/ 109 .an.cntno.nlu .ole\n .noooe ocazoo>ax anode nan flouoaao or: on.v cannah _ . _ . b . earmo .. sofa Badman: 110 comparing Figures 4.2a and 4.3a, it is found that in both cases, in the frequency range of Koa > 2.0, HE“, TMm, T501 are slow waves, and EH 1,, IMO; and T502 are fast waves. Above Koo = 2.125, an increase in a," gives rise to the transformations of EH;1 fast wave into EH2, slow wave and H521 slow wave into H152, fast wave. According to the above observations, above cutoff the attenuation will decrease when the mode is transformed into a fast wave mode from a slow wave mode as e: is increased. On the other hand, the attenuation increases if a fast wave mode becomes a slow wave as a result of an increase in 8:. Because the propagation constants for TM m, IMO, and EH” are always greater than zero, the three modes are highly damped propagating modes below the cutoff frequency. Figure 4.4a through Figure 4.4c display the ar—B and co-ct diagrams of a waveguide loaded with a rod where b/a = 6 and which has losses of e," = 10. As can be seen in Figure 4.4a, in the high frequency range, where Koa > 2.0, all dispersion curves are asymptotic to the he space light line. Accordingly, no slow wave exists at higher frequencies. However, as depicted in Figure 4.4b, which is the enlargement of the low frequency range of Figure 4.4a, the m-B curves for TMm, TM” and EH“ modes are in the slow wave region at very low frequencies and do not go to zero. Figure 4.4c, which is the or - or diagram, shows that, at high frequencies the attenua- tions for mosr modes are reduced when compared with the case of Figure 4.3c. How- ever, the attenuation for TM 0, mode in the frequency range between cutoff and K00 = 2.5 is increased. More significant increase in a is seen for 8H,, mode when a: is increased fi'om 0.39 to 10. In comparison with Figure 4.3c, the cutoff frequency for HE” is lowered, while for other modes the cutoff frequencies are shifted up. As will be demonstrated later, the changes in the attenuation for each mode is dependent on the changes in the EM field distribution within the waveguide as one of the waveguide parameters is varied. In the following examples, it will be seen that the variations in the field patterns resulted from an increase in e: are different among TM or, TEm lll 3.0 1 I T I I rj ' I I 1 I 1 T l V l 1 A ‘ / / / 2.5-k1 1. Fast Wave Region . / .. 2.5flothvelkgflnr/ / - / /! / _ / ./ / . / ”hd’ /¢ .. V '0 {5° 8— TE. 7— TM” 5— EH," — 5— EH" 4— TE“ 3— HE” ‘ 2“ LE“ 4 I— ll 0-0 TI'I'I'I'I'T'I'I‘I' 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ba Figure 4.4a w-B diagram for eight waveguide modes. b/aaé. £83.03-j10.0. 112 / I — - light line 31311111 5 0.2 0.3 0.4 0.5 Figure 4.4b Low frequency portions of the w-B diagram for eight waveguide modes, b/a-G. t-3.03-310.0. 113' .o.oufitmo.nlu .muoxn .nouot ovusoo>az anode now aeuodqv at: oq.v casuam ool ©._I 0....1 N41 m.» ..w E 2 . z o T; / I I _ / m P n / _ / . / : x, . . _ _ 0 ON N. //.9. ”UP IN ”A... , 6o 3. IL . .0.A // $00 ”HT—U lulu¢ . A ...,. ..m 11m . / mm mw Q , .52. 11w. . ,, .Mux lm // 33:. IN 0.” _ F _ _ /L _ p _ . L “—...— 1" Cowman. I 8:3 3339.. 114 and HE” modes. From Figure 4.2 through Figure 4.4, the dispersion characteristics for various modes have been demonstrated from low loss to high loss cases. It has been shown that different modes undergo different changes in the or - x, characteristics as the dielecuic loss of the dielectric material changes. It is interesting to find if there are certain ways of predicting the trend of these changes for each mode. To do so, the a) - x diagrams and electromagnetic field distributions within the waveguide for TMOI, TE 01, and HE" modes are examined in detail below. 4.3.2.1.3 (o — it. diagrams and field EM field patterns for ma, mode (A) or — ac, diagrams Figures 4.5a, 4.5b and 4.5c are the a) - B and m - a diagrams for TMO, mode with a = 1.27 cm, b = 7.62 cm and e; = 3.03. In each diagram, there are a family of curves where each curve corresponds to a specific dielectric loss 2:. A wide range of dielec- tric losses, fi'om e: = 0.039 to 10.0, is investigated. As shown in Figure 4.5a, the (0 - B curves can be divided into two groups: (1) a" s 1.5, and (2) e: 2 3.9. For the group where a: s 1.5, the dispersion curves will cross free space light line and approach the nylon light line asymptotically at high frequencies. Crossings occur at about xoa = 2.0. For those curves that cross free space light line, the crossing xoa is higher for larger 2,". If a," s 1.5, TM 0, mode may be either slow wave or fast wave, depending on the Operat- ing frequency. Below w = 2.0, the co - B curves lie in the fast wave region, while when rod is larger than 2.5, the m - B curves lie in the slow wave region. As shown, the dispersion curves for group (2), where e: 2 3.9, are in the fast wave region and become asymptotic to the free space light line in the high frequency range. In Figure 4.5b, which is the enlargement of the low frequency portion of Figure 4.5a, it is found that as a: > 0.039, B is always larger than zero. In the next example, it is seen that although B > 0 in the low frequency range, these modes are highly attenuated modes 115 oncofl'.0 .muo\n .su nsowuo> uOu ovoe .oza you aquacuc at: um.v annuah on we on. mm, om” ohm mnodflb ..l a r mnduzo I: N 0.0"..u I m .l o.Tu:w .11 e 0.71....» II. n m.mu..u II o I 0.0"..Q ..lll m. 0.9"..u .l. m i 8: £9. .1 .n eoeom 0:: :03: II o _..|rlri—'.L1..LLF_L»_._.\c_.___..._c. 68...”? :6 .. 5359.8 - - :23”. 063 “no“. .9 __r_._ Emdevmzsc lmN Oh 116 ON no. nae olc\a . an asoauu> new even .oze uOu agenda at: on» ma coauuon aocoavouu :04 Am. v museum on .3 m; or n.0, ed to No we _ p _ moo.ou..o oneneo 0.0":Q o. 2...... n. Pres moneo oou..o oéui oe: 22. P IIIIIIIIIJD. _ _ _ _ _ _ _ _ .mm mm? Bonnie .. aooefiozo a anemone TN 117 .no.nl.u ‘— .muu\n .au macaua> now 0002 .oza now Eeuuouo at: om.c «Nanak 06.1 o... m... QT NT a. _ c _ _ _ . _ L mflodflza 01.0 AKWOflzu 9|. \\ 0.0":u 0'0 O. Fuzu 1 m. Puzu I m.n":Q I O.W":Q I O.O—.":Q I 0:: tOuDO ...II _ p p _ L F 823:8 t coafiozo 33393: 118 due to large attenuation constant. From the m - a diagram shown in Figure 4.5c, some interesting phenomena asso- ciated with increasing e," are observed. In the range of rod > 2.0, the attenuation con- stant increases with increasing 2,". The largest attenuation occurs at a," = 1.5. Further increase in e," results in decreasing attenuation, such as the cases for 2: equal to 3.9, 5.0, and 10.0. Comparing Figures 4.5a and 4.5c reveals that group (1) is in slow wave region, and group (2) is in fast wave region, provided 1:00 > 2.0. It indicates that attenuation for slow waves increase with increasing 2,", but for fast waves attenuation reduces with increasing 2:. Interestingly, it is found a mode may have more than one cutoff fi'equency as is shown in Figure 4.5c. For example, as e: = 1.5, the a) - a curve intersects with the cutoff line at two points, i.e. xoa = 0.875 and 1.875. For too > 1.875 or < 0.875, the TM.” mode is highly damped propagating mode since ado > 1. Although in the frequency range of 1.875 > too > 0.875 (no is less than one, the attenuation is still large. Figures 4.6a and 4.6b show the to - B and m— a diagrams, respectively, for vari- ous a/b, with e. = 3,03 -j 0.039. As is indicated in Figure 4.6a, no matter what the ratio of a/b is, TMm mode enters slow wave region at so. = 1.7. In other words, the crossing is identical for all a/b ratios. As shown in Figure 4.6b, when Icon < 1.25, TM.” are subject to larger attenuation for larger rod radius at a fixed roe . The cutofl‘ wavenumbers are shifted up with increasing rod radius. However, as compared with the empty cavity cutoff wavenumber, all the cutoff wavenumbers are lowered. Note that, by comparing Figure 4.6a and 4.6b, the transition from below cutoff to above cutoff is seen to become more gradual as rod radius increases. (B) EM Field patterns Two steps are involved in the computation of the field patterns. The first step is to evaluate the propagation constant 7 for given frequency to and complex dielectric constant e, using the computer program developed for the computation of the (1)—K, 119 fig450.spf olzr.sdf —- onr.sdf 3.0 I I I 1 I I 1 1 1 1 1 / / .1 / I- ¢°// x #7 _ o 215 $,/ . ‘ / 1 . Fast Wave Reglon Qe° / .1 09 / / 1/ '1 \(c / .l’. / 1.x 2.0— /,-/‘ — _ line ‘ A H o/b = 1/121 v—v o/b = 1/10 0—0 o/b = 1A6 H o/b = 1/5-4 H o/b = 1/4 H o/b = 2/6 a—a o/b = 35/6 " o—o o/b = 4/6 I T 0-:0 o/b = 516 1.5 210 215 3.0 60 Figure 4.6a £83.03-j0.039. w-B diagram for TMo: mode for various a/b. 120 003] m.- 0.? 0.? new .n\e uncuue> u0u 0005 ext _ p .mnc.0fitno.nlu .925 new Echoeao at: n«.« casuah 051 ,9... QT .3: 4.7 O- x. .. fi> u a} I m. oi. u a} I ,0 m) u {a I H / «w {P u {a I n / ac ...> u {a To . / D\N «I n\0 I , {nun} mum / o\¢ u n\o ele _ _ r _ . _ . a . _ o.\m n_n\o.olo , _ - _ ,_ g .. - a- - . n - o - . . . u n o ‘ o . — u o o -.- evwfimo n .6350 8.849.. 121 diagram. The field patterns of the coaxially loaded waveguide are then obtained by substituting 1 into equations 3.81 through 3.86. The field patterns of E,, E,. and E, presented below are the radial variations of the square of the magnitude of each com- ponents at a certain waveguide cross section. Field patterns are normalized with respect to the largest value among the three components. For instance, electric fields are normalized by the maximum of IE,I3,,,, IE, 13..., and IE,I},,,. TMm mode has only two electric field components, i.e. E, and E,. Since E, is the dominant component, IE, I2 and IE,I2 are normalized with respect to IE, I},,,. The dashed line shown in the following figures represents the dielecuic rod surface. Figures 4.7a through 4.7g display the electric fields with ma = 2.7, b/a = 6, and e,’ = 3.03, for various 2”. Two different y-axis scales are shown in these figures, the one on the right hand side is for the p component and the left hand side is for the 2 com- ponent. The main reason why roa = 2.7 was chosen is to illustrate the transition between mo, slow wave and fast wave as 2," changes. The operating frequency is 8.9 GHz for waveguide dimensions of a = 1.27 cm and b = 7.62 cm. Figure 4.7a is the field patterns of a waveguide centrally loaded with a nylon rod where e, a 3.03 - j0.039. It is seen that a large portion of the electric energy is confined in the dielecuic rod. BOth IE, I2 and IE,I2 vanish on the conducting walls. Also found is the large difference between the magnitude of E, and E,. As shown, the relative IE, Im=l and the relative IE,I,,,, = 6.5x10'5. As the dielectric loss a: is gradually increased up to 1.5, the relative field patterns, shown in Figures 4.7b through 4.7d, undergo hardly any changes. In other words, for a: s 1.5 most of the EM energy is distributed in the dielectric rod region. As demonstrated in Figure 4.7e, when a: = 3.9, the relative field patterns are altered and most of the EM energy is now distributed in the air-filled region. Note that the field in the rod is no longer a standing wave pattern but decays from the boundary. Further increasing the dielectric loss, as is shown in Figures 4.7f and 4.7g, two major changes in the field patterns are observed: (1) rapid decay of 122 wauodwoo 0’ .F.«ra.x .onuxn .mmo.oatmo.muu .0002 :29... new 3:0...an Ododu OAHHOOHN ah.v ouaodh Ace; 0500.65 ..o .5200 05 895 o. m n m w .v n o 0.0 I. 734 1.0.0 m .. molwoNl . IN 0 molmo.¢l _ . u . Ito _ WOIMO.©1 _ z I _ 1 _ 1.0.0 moIQOl " .. . _ .. _ Imd votmogul _ _ r h _ _ I . volmmgl _ o P .. 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N _ _ _ . _ p _ . . _ _ b . _ . .6082 :E .. 00302 :2 08.00000 N P iuauodwoo z 129 IE,I in the rod from the boundary, e.g., more and more IE, I2 is "expelled" from the dielectric rod and (2) IE,I,fu, has grown dramatically. According to Figures 4.7e through 4.7g, most of the EM energy is distributed in the air-filled region as a: 2 3.9. However most of the energy remains around the rod and E, and E, are still tightly bound to the region around the rod. The explanation for the rapid decay of IE, I in the rod as e," is increased is the skin effect. As 2: gets larger, the dielectric rod becomes more like a conductor. As a result, the field in the rod is quickly damped and confined to a region closed to the boundary. Skin depth is the distance where the field in the rod is attenuated to l/e of its magnitude on the dielectric boundary. The skin depth is determined by the equivalent conductivity, i.e. 0:02". By knowing that at xoa = 2.7, TM.“ is a slow wave if 12:51.5, a fast wave if e," 2 3.9, two observations are obtained from Figures 4.7a tlu'ough 4.7g. First of all, for slow waves, most of the EM energy is confined in the dielectric rod and for fast waves most of the EM energy is distributed in the air-filled region. Secondly, a change in e," will result in a redistribution of the EM field patterns within the waveguide. The EM energy distributed in rod may increase or decrease with increasing 2,". Based on the observations, the changes in a as e: is increased at high frequency, as is shown in Fig- ures 4.2b, 4.3c, 4.4c and 4.5c, can be explained from the view point of electromagnetic energy redistribution within the waveguide. For a slow wave, if an increase in e," still results in a slow wave and the EM field distributions do not undergo much alteration, and attenuation is increased since most of the EM energy is confined in rod. If an increase in a," transforms a slow wave into a fast wave, attenuation is reduced because of the removal of EM energy from rod into air-filled region. On the other hand, attenuation becomes larger if an increase in 2," yields a transition from fast wave into slow wave. For fast waves, increasing a: leads to a decrease in attenuation because part of the EM energy is shielded from the rod as a result of an increase in 2,". In short, attenuation is decreased if an reduction of EM energy in rod occurs as e," is 130 changed. Attenuation is increased if a change in 2," focuses more EM energy into rod. Previous examples have showed that changing dielectric loss could move pro- pagating 7M0, mode from slow waves to fast wave because of the redistribution of electromagnetic energy within the dielectric loaded waveguide. The next two exam- ples, shown in Figure 4.8a and 4.8b, demonstrate the influence of the operating fre- quency on the field patterns. The waveguide and dielectric rod dimensions are identi- cal to the cases given in Figure 4.7, i.e. a = 1.27 cm and b = 7.62 cm. Nylon rod was selected as the low loss dielectric material. In Figure 4.8a, W = 0.65, which is resulted from 2.45 GHz operating fiequency. It is shown that almost all of E, is distri- buted in the air-filled region. Although IE, I,,,,, occurs in the dielectric rod, the major- ity of the energy of E, is confined in the air-filled region. This TM,1 mode is a fast wave. By examining Figure 4.5a, the or — B diagram for TMO, mode with b/a = 6, it is indeed a fast wave at xoa = 0.65 if e, = 3.03 - j0.039. Figure 4.8b depicts the electric field patterns of TM,” mode at xoa = 0.3, which corresponds to 1.22 GHz operating fre- quency. Obviously, the energy distributed in the dielecuic rod is much less than that in air-filled region. According to Figure 4.5a, this is a nonpropagating mode. The high frequency example of rod = 2.7 has already been given in Figure 4.7a. From these examples, it is found that changing operating frequency not only changes the field pat- terns but also alters the operating points on the co- 1:, curves, which determines the electromagnetic wave to be a fast wave, a slow wave or even a nonpropagating mode. According to waveguide perturbation theory [89], electromagnetic fields of the loaded waveguide will change as the rod radius is increased. It is of interest to inves- tigate if there is any way of predicting the changes of the field patterns as b/a varies. To do so, the fields of a waveguide loaded with nylon rod are shown in Figures 4.9a through 4.9c for various rod radii. Note that even though the Operating frequency is identical at 2.45 GHz in these examples, W is different corresponding to different rod radius. For small rod radius, TM,l mode is most likely in the fast wave region. .00.0ua.x .muaxn .mmo.0filmo.fluu .Omvofi Noah. 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' € I I I I i I I ' I I I, _ I I II II II I’ _ I: I’ — It 9 _ Id — rd ,. _ v] o _ ll — r1 a _ ll ’1 _ a _ II I; — I1 r1 _ ’I ll _ II [I — II I4 — ll .1 Emiosé 60128.: 3083: wauodwoo z 136 As rod radius is increased, EM wave may become a slow wave due to the increase in W. The waveguide radius considered in Figures 4.9a, 4.9b and 4.9c is 7.62 cm. In Figure 4.9a, the rod radius is 6.35 cm which yields xoa = 3.26. As can be determined from Figure 4.6a, such a propagating TM.” mode is a slow wave. It is con- sistent with the field patterns shown in Figure 4.9a that most of the electric energy 2 'mx IE exists in the dielectric rod. In this example, —-'E' I2 p m is approximately 300. As the rod radius is reduced to 2.54 cm, which is the case depicted in Figure 4.9b, xoa is equal to 1.303. According to Figure 4.6a, the EM wave is still in slow wave region but about to transform to a fast wave. By comparing Figure 4.9b with Figure 4.9a, it is obvious that less electromagnetic energy is confined in the dielectric rod region. Neverthless, the electric energy distributed in the dielectric rod still dominates. Hence, it is still a 2 slow wave. Interestingly, -:§-‘—“'£ is increased to 1250. In Figure 4.9c, the rod radius '2 p mu. is decreased to 0.635 cm and the corresponding too is 0.325. It is seen that the elec- tromagnetic energy in the air-filled region is well beyond that in the dielectric rod region. Obviously, this is a fast wave, which agrees with the conclusion that could be 2 drawn from Figure 4.6a. I_:,I+., is risen up to 5x10“ in this example. I .I... 4.3.2.1.4 (u - r, diagrams and field patterns for 11201 mode (A) a) - 1c. diagrams Figure 4.10a and 4.10b show the m-a and m-B diagrams for T50, mode as a function of s," with a; = 3.03 and b/a =6. As is shown in Figure 4.10a, the dispersion curves can be divided into two groups:(l) a," 50.5 and (2) e," 2 1.0. For group (1) the dispersion curves cross the free space light line and enter the slow wave region at too = 1.6. Eventually these curves become asymptotic to the nylon light line in high fre- quency range. It is found the intersections Koa, at which fast waves cross over the free [Coo fig4100 3'0 I ' l 137 .spfnw401znsdf—-nw4162nsdf 7/14/89 I ' fl ' I 7 l 2. Sloww e Region /' /" 2.5— _1 / / / / _ I /.,.° 2.0— ’3‘ _ a ¢v 1.5— _ 1.0— _ E: - 6 /\' K / (305-L: / / 5 —— e"=10.o — / , s — 9:19 /I ,/ 4 — e"=1.0 / 3 — e"=0.5 ' /’, 2 — s"=0.39 1 — "=0.0.39 0'0 l I I I I I l f l 0 2 3 4 5 60 diagram for TEoi mode for various e", Figrue 4.108 u—B c’= 3. O3. b/a=6. 138 r’i .nOInu .u .. uo\o ...u m30«Lo> Lo» 0005 «om». Lo» EOL00«U 613 £36 6.5.5: 06 om.l cm... Our. omt _om.l _ _ _ _ p _ _ _ . mnoduzu IIH mm.ou..u llm n.0u..o IIm O.—.":Q IV ®.n"..w Ilnm 0.0P"..Q IIIIO m / / . _ _ — _ _ p b p/ P _ P _r P _ _ mm\¢fi\m ..8._Nm:;c I 5933;... 23.0030: 139 space light line, is higher for larger cf. The dispersion curves of group (2) stay in fast wave region and approaches the free space light line asymptotically at high frequency. The (n - a diagram displayed in Figure 4.10b shows that for group (1), increasing 2," yields an increase in attenuation, provided icon is larger than 0.5. On the other hand, the attenuation constants for group (2) are very small in the high frequency region. As shown in Figure 4.10a, at xoa > 1.6, group (1) is in slow wave region, while group (2) is in fast wave region. Again, the cutoff frequency is increased with increasing a". Figure 4.11a and 4.11b display the m—B and m-a diagrams for a waveguide loaded with a nylon rod for various rod radii. The dispersion curves shown in Figure 4.11a point out that the intersections with the free space light line increase as a result of an increase in rod radius. All the 0) - [3 curves cross the free space light line before xoa = 2.3 and become asymptotic to the nylon light line at high frequency. As dep- icted in Figure 4.11b, in the frequency range of iron < 2.0, attenuation increases as a result of an increase in rod radius. Note that when Koa > 2.0, xoa - aa curves con- verges. It implies that at high frequency the attenuation constant a corresponding to larger rod radius is smaller. Also seen is that the larger the rod radius, the higher the normalized cutoff wavenumber. All the normalized cutoff wavenumbers are shifted down from the empty 1E0, normalized wavenumber. (B) Field patterns The changes of the electric field of TE.” mode as a function of e: are displayed in Figures 4.12a through 4.12f when 1:41:25. If the rod radius is 1.25 cm and the waveguide radius is 7 .62 cm, the corresponding operating frequency is 9.4 61-12. 11501 has only one electric field component E,. Thus only IE,I2 is shown in the following examples and its magnitude is normalized with respect to IE,I},u. As is depicted in Figures 4.123, 4.12b, and 4.12c, IE,I2 is confined in the rod and changes only slightly as e," is increased from 0.039 to 0.5. Figure 4.12d displays the distribution of IE,I2 for e," = 1.0. In comparison with Figure 4.12c, it is found that IE,I,E,“ and most of the 140 terdslspf terderlsdf — terdsr08.sdf 7/26/89 4 ' l ' l 1. Fast wave Region light line _ o/b=S/6 o/b=2/3 "_ o/b=1/2 o/b=1/3 _ o/b=1/4 o/b=1/5 o/b=1/6 o/b=1/12 I I 4 5 IIEIEIII! 60 Figure 4.118 w-B diagram for TEoi mode for various a/b, 8=3. 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F c Q _ _ V . _ . _ . _ p L p L b r0.0 le0 [die lmfio Iumgu _ _ _ h _ . _ P _ . P h . mw\:\n commemomzc Emmoozc N A 146 .0.nhlflo.fiuu —P~ .0005 «ON..- .m.unoax .ouuxn Lo» mCLouuoa 0H0“; OaLuoon ONH.§.0L30HK A no; 05266 V0. .6“ch of E9: V Eu Q n _ . o _ V p m _ - ¢ r — _ N _ IIOAU IINAU late 1.0.9 _ _ _ _ _ . 3);} endows? 33 Page. N.F qb ZI 3! 147 .o.o«nnno.nuu .oeoe a.u» to» actouuoa aged» o«guoo~m. eufl.e otsoflm A US 05202.0 .6 Lmycoo of Set V 80 Q. o _ _ m _ .m.uuo.x .euoxo 4 n p — h — p N _ IIOgu IINgu lags IIm.o IImAu IIO.P _ _ mm\¢_\m . _ _ _ _ $9313: E3. 5;: N.P 95 z' 3| 148 electric energy have been shifted into the air-filled region due to an increase in e," from ' 0.5 to 1.0. The redistribution of IE,I results in not only a fast wave TEO, mode, as shown in Figure 4.10a, but also a large reduction in attenuation, as depicted in 4.10b. Further increases in 5,", as shown in Figures 4.12e and 4.12f, will reduce the amount of IE,I2 confined in rod and almost all of IE,I is sinusoidally distributed in the air-filled region. Note that this behavior is similar to that discovered for the TM 0, mode, i.e. as a: increases the electromagnetic field is shielded from the material rod. Figure 4.13 diSplays the distributions of IE,I2 of a waveguide loaded with a nylon rod for xoa = 0.65, 1.2, and 2.5. For waveguide dimensions of a --= 1.27 cm and b = 7.62 cm, Koa = 0.65, 1.2 and 2.5 correspond to operating frequencies of 2.45 6112, 4.523 GHz and 9.423 GI-Iz, respectively. At xoa = 0.65, most of IE,I2 is distributed in the air-filled region and IE,I}m is also located in this region. As xoa is increased to 1.2, IE, I3“, moves towards rod surface but is still positioned in the air-filled region. It is of interest to see that although most of the electric energy is still confined in the air-filled region, IE,I2 in red has grown as much as 2.2 times larger as compared with xoa = 0.65. When rod is 2.5, almost all of the electric energy is restained in the rod. Note that for the case of e,"=0.039, Figure 4.10a indicates that the TED, mode is near cutoff at Koa=0.65, a fast wave at tcoa=1.2 and a slow wave at xoa=2.5. The influence of rod radius on the electric field distribution is demonstrated in Figures 4.14a through 4.14c for an excitation of 2.45 GHz in a waveguide of 7.62 cm radius which is loaded with a nylon rod with radii of a = 6.35 cm, 2.54 cm and 0.635 cm, respectively. Figure 4.14a shows the field pattern for a = 6.35 cm. Most of the electric energy is distributed in the rod. As the rod radius is reduced to 2.54 cm, the field distributions shown in Figure 4.14b indicates that the energy distributed in rod is about the same as that in the air-filled region, but IE,I,fiu is still located in the rod. For a = 0.635 cm, as can seen in Figure 4.14c, the energy stained in the air-filled region is dominant over that in the rod. 149 .On0\fl .GNO .0“.an Inuu .eox msowLo> Lo» 0005 «cub Lo» mCLoquQ Uan» OwLuoon nH.v 0L30«m A6139: of to 5:36 $5 EotV Eu Q n m w v m N _ Q ..... §%€-<§¢e¢£§.§-fe§et§£¢ebe n _ . c . c T [film IA 1 '4 T- ..H rue r A y I . .5 r-vAV r 79 r r z . «m a lad Mr. '4 r 7L I 4 5 I I . a _ Vnmvc e r4 _ . .I I _ A amp _ II 0.. _ II. C — mVNHGQQ aid _ I N Auoov BIO _ I mcduooe ole _ . _ _ t . _ - . . _ .. _ . _ . _ . _ r r . Em mocaect Vennmcoamét.Vemdcomect Em.vomct N F 150 .ON.nu0o¥ .EO No.5un .EO mWIOnO .ono.ohuno.nuu .ooos .ow» Lo» acrouuoa ego“; oaruoodm b F _ AUS 2.50230 do ..Scmo of 80.5 Eu Q m m Q n N“ _ _ P _ _ . _.. _ oefi.v orsoaa loé _ _ _ _ _ . P . mmENE VB. 5% 63 an. :5 53 N.— 151 .ono.0nnno.nuu .oeoe «cm» to» merouuoa ego“; ofiruoofim 64H.4 orsouu Ana; 2.302026 .6 L350 65 60.5 Eu Q \I _ w . _ _ _ _ _ .nIHuflov. .EO NOINNO .EO <0.Nufl n .v n _ _ _ r V _ _ . N _ mmV¢wa _ . _ . 03.8353 330.303 .emn.ono.x .eo mo.nun .eo mne.ono .omo.0wtno.nuu .ooos «cup Lo» nCLOuuod Dav“; OwLuoo~m. ov«.v oLBOfiu Ana; 2.50220 do .528. of 80.5 Eu Q 152 m m — h F - _ _ _ m t n _ _ r . _ _ N _ —‘“*“‘~___*_—fl__“_flfl _ . mm\¢N\n _ t. _ _ _ _ unmfioao «03 Emfido F03 153 According to the dispersion characteristics and field patterns for TM), and TEO, modes illustrated above, it is found that TM.” mode is a better mode for microwave heating than 77501 mode. First of all, TMO, mode stays in the slow wave region for a larger range of e: than TEO, mode. As noted earlier, most of the electromagnetic energy of the slow wave is confined in the dielectric material. In other words, TM 01 mode can be used for microwave heating of materials for a wider range of the dielec- tric losses. Secondly, the large difference between the relative magnitude of the E, and 5,, for TMO, mode provides an easy way of coupling electromagnetic energy into the dielectric rod. That is, for TM.” mode, only very small E, been excited will give rise to relative large E, in the dielectric rod. 4.3.2.1.5 m — x, diagrams and field patterns for HEu mode (A) or - x. diagrams Figures 4.15a through 4.15c depict the m- B and co- a diagrams for (In?ll mode for various 2:. The waveguide has a radius of 7 .62 cm radius and is loaded with a dielectric rod of 1.27 cm radius and dielectric constant of a; = 3.03. As the cases for TM.” and TE],l shown in Figure 4.6a and 4.10a, two groups of dispersion curves can be identified in Figure 4.15a. Group (1) is the family of curves which has a: s 2.0. The to - [3 curves for this group cross the free space light line before Koa < 1.4. Group (2) is the family of curves which has a: 2 2.5 and stays in the fast wave region for Koo < 3.0. At high frequency, the dispersion curves of group (1) approach the nylon light line asymptotically, while the dispersion curves of group (2) converge to the free space light line. As can be seen in Figure 4.15b, the icon crossings for group (1) increase with increasing 2:. The m-a diagram displayed in Figure 4.15c shows that above K00 =1.0 the attenuation for group (1) increases as e," is increased decrease in a is observed when a," is increased from 2.0 to 2.5. For group (2), at high frequency an increase in e," may K%Q 154 fig4-15o.spf hbdzr101.sdf — hbdzr117.sdf 7/26/89 3.0 T T T I I l l /1 I 1. Fast wave Region /2. Slow wave RegiOv I 5,6,7,8 / 2.54 / _ .— 2.0- - 1.5— a 1.0— - 8 — e"=10.0 7 — e"=5.0 ,, 0.5— 6 — e"=3.9 - 5 -— e"=2.‘5 4 — a"=2.0 3 — c"=1.0 ‘ 2 — e"=0.39 1 — e"=0.039 0-0 I I I ' I F I ' I ' O ‘l 2 3 4 5 I30 Figrue 4.15a c’=3.03. u-B diagram for HErr mode for various c", b/a=6, [COO 1 5 fig4-15b.Spf d1.sdf — d17.sdf 7/26/89 . I I r I I I I I I I I /l T I _ 1. Fast Wave Region _ 2 6’7 / _ 5 //// ‘ 2. Slow Wave Region 1.0---1 I _- / ’/ ‘ / /’/ fl / _ I / e " / //<;9 / v" - / o {/qu _ /~$ §~ l‘? ///6P / Q / C15- / Q? //’ 1‘ \ / 00 / 0 .. // .3 / _ / a: // 3 —" ’3"=10'0 :7 / K __ N... W / , 7 e -5.0 .. / / 6 — s"=3.9 r / I 5-—— e"=2£5 / / 4 -— e"=2.0 - / .' 3—- c"=1.0 ‘ /{/ 2 — e"=0.39 0.0 l l l I l I I I I I [1 — cl =O;039l 0.0 0.5 1.0 1.5 2.0 50 Figure 4.15b Low frequency portion of the w-B diagrm for HErr mode for various e", b/a=6, 8=3.03 156 DUI. o.o NI eI of mI 07 87.5.7 SI m. OAUV _ 71 V . V L V b V . V m V . P I? / mnoduzw olo mn.on..u I .... 3a..» mIm V « AYNuzw .1. m . a / man.» «In .3 .. mar...» I .53 / o.ml a I a“, . I.. . £ , . “:0 I 2 s... V.. .\ r h’ g“ u u g / m._‘l.,..... / h x..- p u 1‘ x a V 1., e . VV... /.“\o. r 5 I... I. i. /‘M\ '4 —. w 5 3 / 5 z. / ... O.” c _ V a /_ V L V .- V r — p h _ V V .Oflfl\fl .gU mJOHLC> LOV OUOE admI LOF EULOOHU 5l3 Ino.nu.u OWHIfi OLJOwl .... I mmkmh set :82 I 530232 V8333 157 result in a reduction or an increase in or, depending on the frequency. The field pat- terns given in section (B) will shed light on the changes of a due to the changes in BM field distributions resulting from an increase in a," at high frequency. By compar- ing Figures 4.15a and Figure 4.15c, it is seen that group (1) is in slow wave region and group (2) is in fast wave region, provided K00 > 1.4. The a) - on curves at low fre- quency, i.e. Koa < 0.35, are barely altered as e," is changed. There are two cutoff fre- quencies for each of a," = 0.39, 1.0 and 2.0 curves. Only one cutoff frequency exists for a: = 0.039, 2.5, 3.9, 5.0 and 10.0. The cutoff frequency in the low frequency range, i.e. K00 < 0.5, is slightly increased as e," is increased. The a) - B and a) - on curves as a function of rod radius are shown in Figures 4.16a and 4.16b. It is of interest to see that the intersecrions of the a) - [5 curves with the free space light line, as shown in Figure 4.16a, increase with increasing rod radius. All to - [3 curves cross the free space light line before Icoa = 1.25 and become asymptotic to the nylon light line at high frequency. As depicted in Figure 4.16b. the normalized cutoff wave number for each co - a curve is increased as rod radius is increased and is smaller than the empty TE11 cutoff wave number. Above cutoff, i.e. at Koa > 1.5, an converges for all a/b ratios. Below cutoff, increasing 2: yields an increase in on at a fixed Icon . (B) Field patterns Figures 4.17a through 4.17h show the electric field patterns of HIE:u mode for a variety of a," at Koa = 2.0. In these examples, the radii of rod and the waveguide are 1.27 cm and 7.62 cm, respectively. Accordingly, the operating frequency correspond- ing to W = 2.0 is 7.52 GHz. The e,’ of dielectric rod is 3.03. HEll has three electric field components, E, , 1?:9 and E,. The field patterns shown in Figures 4.17a through 4.17d are the field patterns for group (1), i.e. e," s 2.0, depicted in Figure 4.17a. A close examination of the field dis- tributions discloses that most of the electric energy is distributed within the dielectric 158 herdslspf herderIsdf — herderBsdf 7/26/89 4 I 7 I T 1 I I A 1 1. 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Only minor changes in the fields are observed as a: varies. By comparing Figure 4.17a and 4.17d, it is seen that IE,I2 and IE,I2 in rod has drOpped as much as 0.85 times due to an increase in e: from 0.039 to 2.0. In Figure 4.17d, IE, I2 is still the dominant field component, but IE, l3”, has slightly moved from the dielectric surface toward the center of dielectric rod. Note that IE,I,L, in air-filled region grows to 0.04 from zero, IE,I}m drops from 0.25 to 0.2 and IE,I2 in the air-filled region becomes more tightly bounded to the rod surface as a result of the increase in 2,". Apparently, the energy stored in the dielectric rod does not undergo much change as e," varies from 0.039 to 2.0. Again, the field patterns show why the attenuation for group (1) shown in Figure 4.15c increases as e: is increased at high frequency. The field patterns for group (2) are presented in Figures 4.17e through 4.17h. Figure 4.17e depicts the electric fields corresponding to e," = 2.5. In comparison with Figure 4.17d, it is found not only almost of the EM energy is now confined in the air- filled region, but the dominant field component is IE,I instead of IE, I. Obviously, the cause of the enormous reduction in or displayed in Figure 4.15c as e," is changed from 2.0 to 2.5 is due to the EM energy redistribution within the waveguide as It: varies. Also, the transition of most of the EM energy from rod into air-filled region moves the HE" slow wave into fast wave region, as can be seen in Figure 4.15a. When a," is increased to 3.9, the field patterns shown in Figure 4.17f indicates that the distributions of IE,I and IE,I have changed and the energy of E, restrained in rod has also increased. However, most of the EM energy is still confined in the air—filled region. Figure 4.17g shows the field patterns of the case where e," = 5.0. When compared with Figure 4.17f, it is found that Ep is the dominant component and IE,I,L, is almost ten times smaller as a result of the increase in 8,". Although the distributions of IE,I are not altered much but IE, I3“, is increased from 0.1 in Figure 4.17e to 0.15. It is interesting to see that when a," = 10.0, which is shown in Figure 4.17h, IE,I becomes the dominant field and gains as much as 67 times due to the increase in a," from 5.0 to 169 10.0. In the mean time, IE, I3“, reduces from 0.155to 0.0165. The field distributions displayed in Figures 4.17e through 4.17h do not indicate a general pattern of the change of the EM fields as e: is increased, as was seen for TMOI and 11501. It may be because of the existence of all the six field components, any change in a," gives rise to much more complicated perturbation to HE" mode than to T1501 and TMOI modes. The next three Figures, Figure 4.18a, 4.18b and 4.18c, depict the field patterns of a waveguide loaded with a nylon rod for different Koa . The waveguide dimensions are identical to the examples given in Figure 4.17. As shown in Figure 4.18a, where Koa = 1.8, IE, I is the largest field component and most of the EM energy is distributed in the nylon rod. Apparently, this is a slow wave. In Figure 4.18b, the field patterns correspond to xoa = 0.65. It is seen that IE,I is the dominant field and the EM energy distributed in the air-filled region is much larger than that in the nylon rod. Obviously, a slow wave has been transformed into a fast wave due to the reduction of the operat- ing fiequency from Koa = 1.8 to xoa = 0.65. When xoa = 0.26, the field patterns given in Figure 4.18c show that the EM energy confined in nylon rod is increased, however, the majority of the EM energy is still distributed in the air-filled region. The changes of the field distributions as a function of rod radius are investigated in Figures 4.19a, 4.1% and 4.19c. In these examples, a cavity of 7.62 cm radius is loaded with a nylon rod and excited at 2.45 GHz. In Figure 4.19a, the rod radius is 6.35 cm, which results in Icoa = 3.25. It is found that IE,I,L, and IE,I,§,, are equal and are located in the center of nylon rod. Also seen in Figure 4.19a is that most of the EM energy is confined in rod. As the rod radius is decreased to 2.54 cm, the filed pat- terns depicted in Figure 4.1% show that the EM energy distributed in air-filled region is well beyond that in the nylon rod. A significant reduction in IE,I2 is observed due to an decrease in rod radius. Figure 4.19c presents the field patterns of a rod of 0.635 cm radius. 1 170 .OWO .Oflan ..nuu m m .OUOE .0.Nuo.x .amr Lop mCLOuuoa Ono“; OwLuooAm 90:03.: 05 .0 ammceo 05 E9: E0 Q .ouo\o omd Iv 0L30wk m m w M N O .IIIIFQFIQIIIlflfilllflliuiiiri- mmuunirllklllrcnv , _ I c _ 5 I _ II _ IIN.O c_ _ '4 = I I- u .. Ire II _ 5 _ . I II _ T s _ e IIwAu _ I... I _ I _ II _ iwd l 16.— Nm .1. _ T 3m Dim “ T Qm oIo IIILIIIIFIIII _ _ _ _ _ I _ r T _ _ I . Emmommct Emdcuoct Emmo~0ct mmmmomct N _ 171 .mmo.ohImo.nuI mm.ou00x .mu0\n 90:30:. 9: 00 03:00 05 E95 80 Q Nw Ole emmlm Q _mQIIO _ I _ _ .0005 —.m= now mcumuuwa 0H0wu owuuo0am nma.v 0usoam _ _ _ _ _ _ 50.noa0ct 50.5005; 000.5085... EmNO0ct NT squauoduuoo 95 ‘d 172 .omo.ohImo.TIu .0005 om.ou0cx .ou0\0 ..mm No“ mah0uuma 0H0wu ewauc0am 9200008 05 00 00:00 05 E95 E0 Q oxa.v mMDUpm . m m o n w n N T o OO 10'. In," I I l a.“ _ — p — b A _ _ “110.0 _ _ r. m-IaquIII ax I..II0 INWO :— II. _ I _ laio _ I _ _ meo _ _ or. _ _ Imo _ I _ I molmmél _ I I “ Io; NoImwTI em 1. _ m 010 I I Qm Q10 _ “ NOImoN I_I ...T._._. . mwmmomydt EmmoIfict EWmOcht Emmomét _ m _. siuauodwoo 96 ‘0’ 173 . . .c ION.nIOox .80 No.5ufl .EO mn.0u0 Ono ohIno.nu0 .0008 «um: Lo» 0CL0uuoa Uuo«p ONLuOOHM 907.068 05 To 03:00 05 E83 80 Q T . 0 m, m N _ r T _ _ O as 0 —; 0 so that V? is constant. Solving for W vs Ba and xoa vs our , the following conditions were investigated numerically. Two plasma waveguide geometries for the TM 0, mode were considered. The circular waveguide radii are 4.75 cm and 8.89 cm, respectively, and the radius of the uniform, cold, lossy plasma rod is 1.25 cm. The elecrron density and plasma colli- sion frequency that are examined for each waveguide geometry are listed below. (1) N0 = Ion/cm2 and Ion/cm2 V (2) 70- =1.0 and 0.1 The results are displayed in Figures 4.20 through 4.21. This problem has been solved before [88] but is plotted differently here. On the dispersion diagrams, 2.45 GHz (Koa =0.641) is shown as a horizontal line. The points where this line intersects the IcOa-Ba and Icon-Ira curves are the operating points at 2.45 GHz.‘ Note the presence of the backward and slow waves, as reported earlier [88, 90], in these diagrams. KOO [COO 178 1 O mnOldspfanHzr.sdf mflOly.sd18/1/89 - ' I jaflj,,-L-,L«—L* I I l ' //”-— 7 0.8... :1 I// __ I “if r=2.45 GHz / o.s—; — -3 _ i Ou4‘fi __ J x198 _ . . c 0.2— 1‘9“ - - -- N,=1.0E12 - 0.0 . , . , . 1 T I . _ 111N=10E11 0.0 0.1 0.2 0.3 0.4 O. 5 O. 6 0.7 Ba 1'0TTTTTI'T117111111111711/ 0.8— 0.6— _ 0.4— _. 0.2- __ - " No=1.OE12 - 0.0 . r r , I I . 11"]- N..=1.OE]11 Figure 4.20a I -1.2—1l.1.-10-9 -.8 -.7 -].6 -.5 -l.4 -.3 -.2 -].10.0 —CXO mode for a waveguide w-B and w-a diagrams for TMo: b=4.75 cm loaded with plasma rod, where a=l.25 cm. and V/w=0.1 179 meZd.Spf meer.sdf meZZixsdf 8/1/89 I l I I 1 fl'a‘”l I j I j . -- N,=-1.OE12 - — N,=1.OE11 0.0 t I I I I I I l fl r I I [“1 I I r 6] V I ' -1.2 -1.1 -1.0 —.9 —.8 —.7 -.6 —.5 -.4 -.3 —.2 —.1 0.0 -CXO Figure 4.20b w-B and w-a diagrams for TMo: mode for a waveguide loaded with plasma rod, where a-1.25 cm. b-4.75 cm and V/w=1.0 180 me3C.spf me3zr.sdf 8/2/89 Ln III1IfirIrTIfiIIITI I I I I T I I I I I I Ir J _ // 13.-4.45 Gliz - / / O T I I l I I T I 1 I I I r I I I I I l I I I I 7 l I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 30 :5 I I I I I I I I I I I I I I I I T I I l I .1 _ 2_ _. 1— ‘ / f=2.45 632 a 01F‘l'l‘l‘l‘l’rl‘l'lrl' —1.1—-1. -.9 —.8 —.7 —.6 —.5 —.4 —.3 -.2 —.1 0.0 —ao Figure 4.20c (1)—[3 and co—a diagrams for TM.” mode waveguide loaded with isotropic plasma rod, a=l.25 cm, b=4.75 cm, N=10", %’)-=0.1. 181 me4c.spf me4zr.sdf me4zi.sdf 8/3/89 3 I I I f l I I I r r I I I I T I I I I I I I I T I I I I I/ I / _ // T // 2‘ /,/ ._+ // g, _ I/Hog _. // “V. \‘ 'Q / \\ 1— "/0“; A / 9 1 /. ‘(ce '1 // f=2.45 GHZ ‘ / O I I I r I I I I I I I I I I l I I I I l I I I I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 £0 3 I I I I l 2—1 0 o -4 12 1— ‘ / f=2.45 GHZ ‘ O I I I I I l I I I I I l I I I I I I l I —1.1 —.1.0 —.9 -.8 -.7 -.5 -.5 -4 -.3 -.2 -.1 0.0 “'00 Figure 4.20d w—B and m—a diagrams for TMO, mode waveguide loaded With isotropic plasma rod, a=l.25 cm, b=4.75 cm, N=10.12, 1:10. (D [Coo 182 meSd.§pf meSzr.sdf meSzi.sdf 8/2/89 1.0 I I I I I I I I j T I I 1 I j \ 1 Li i\ & ~ f=2.45 GHz lllllllllll 0.9 #0 1.0 1 I T I I T I I ' l l " i 0.8— 1 - f=2.45 GHz 1 {16- ! 04d -a=::::::ZZZZ' _ E 0.2-1 -— ‘ -- N.-1.0E12 " - anI.OEII 0.0 I I I I T I I I I -25 . -=4 -a3 -a2 ‘-J 01) —a0 Figure 4.21a w-B and w-a diagrams for TMoi mode for a waveguide loaded with plasma rod, where a=l.25 cm. b=8.89 cm and U/w=OJ 183 pm06d.spf pm06zr.sdf£m062i.sdf 8/2/89 I I I I I j j i I j T I \ ‘J 1‘ I IJL\ ‘ f=2.45 GHz llllllllLll a -- No=1.0512 — NA=LOE11 00 1 I I I I I I I 1' I T I l 1 I T T 00 01 02 03 04 as 06 07 08 09 30 1.0 III I IIIT l I I ITTII ITIITI I l I II I II I II II I I II III IIIII 1‘ _ '— o=1.OE11 I_ - Mmoaz } 0.8— I- I ‘ i=2.45 an / IIII IIII 0.0 IIII[IIIII'IIIIIIIIIIIIIIIITWTIIIII‘IIIIII -.50 -.45 —.40 -.35 -.30 -.25 -.20 -.15 —.10 -.05 0.00 -ao Figure 4.21b w-B and w-a diagrams for TMOI mode for a waveguide loaded with plasma rod, where a=l.25 cm, b=8.89 cm and v/w=1.0 I600 I600 184 pm07c.spf pm072r.sdf pm07zi.sdf 8/2/89 I I I I I I I I I I I I I I I I I T I x 3.0 I I I I I I I I I I -1111L pa U 7 O lgllJlll in l llllllllllllJl'lllJlllleJlll 1.0: 0.5-3 f=2.45 GHz 0.0 I j I I T I I I I 1 I I I I I I I I I I I I I I I I I I I 9‘ o 0.0 0.5 1.0 1.5 2.0 2.5 to b) «a O ()1 O lllllllllillllllllllLl d 1 q d 1 .J —-1 d d .4 in 'o f=2.45 on: ' j J .0 w l J 00.1 I T 1 I I I I I I II I I I I ft r If 1 1 I 1-1 -25 -20 -.15 -.10 —05 000 Figure 4. 21¢ (1)-[3 and «H: diagrams for TMm mode waveguide loaded with isotropic plasma rod, a=l.25 cm, b=8.89 cm, N=10'2, 73:01. Koo [Coo 185 )m08c.spt meBzr.sdf pm082i.spf 8/2/89 I I I I I I I I I I I I NW.“ OU'lO JLlJllllllllll in I I I I I I I I j I I I I —‘ O JilLIiLitJLLLII IlllllllllillllJllllllLLlJJll“ a” f=2-45 9H2 I o o ()1 o o: N o N ow u o 9‘ 0 go 01 N O AllilllllJllll 'o in llLllL lLJ IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIf‘IIIIIIIIII ”I.“ f=2.45~GHz I F3 (11 l 1 J11 #— 0.0 IIII IITI IIIIIIITIITIIIIIIIIIIIIIIIIIIFIIIIIII -.45 —.40 -.35 -.30 -.25 -.20 -.15 -.10 -.05 0.00 "(XO Figure 4 , 21d m—B and «Hi diagrams for TMO, mode waveguide loaded with isotropic plasma rod, a=l.25 cm, b=8.89 cm, N=10'2, -%=1.0. 186 These curves demonstrate that the numerical technique can be applied to uniform, cold, lossy plasma rod loaded circular waveguide. The numerical routine is available for diamond thin film and microwave electrothermal thruster experimentalists to apply to their particular discharge applications. 4.3.3 Axial boundary value problem (dielectric-loaded cavity problem) 4.3.3.1 Introduction Numerical results of the cavity-short type material load which is described in sec- tion 3.42 are presented. Equations 3.71, 3.106b, 3.107a and 3.107b are solved for the resonant frequency and resonant length for TM 012, HE", and T501 1 modes under different material loading conditions. The changes of the resonant frequency (or resonant length) as a function of 2;, e,” and rod radius are investigated in section 4.3.3.2 below. In addi- tion, the quality factors of the material loaded cavity are calculated for various efi, e: and 4.3.3.2 Resonant frequency and resonant length for mu, HEm and TE,“ modes Figures 4.22 through 4.27 show the change of the resonant frequency (f,) as a func- tion of e,’ for a variety of sf, where the cavity radius is 7.62 cm. Figures 4.22a and 4.22b depict the f, vs. a; diagrams for TMm mode at a fixed cavity length L. = 15.5 cm where the rod radii are 0.635 cm and 1.27 cm, respectively. The 15.5 cm cavity length corresponds to a 2.45296 GI-Iz TMm empty cavity resonance. As is shown in Figure 4.22a, the resonant fi'equency decreases with increasing 2;. It is of interest to see that for e; < 7.0, larger 8,? results in higher resonant frequency at a fixed 8;. When e,’ is constant but > 7.0, the resonant frequency decreases as e," increases. All the resonant frequency curves intersect at e,’ = 7.0. Hence, for the cavity radius and rod radius given in Figure 4.22a, the dielecuic-loaded TMm cavity is resonant at 2.375 GHz at efi=7._0 for any 2,". The low dielectric constant portion (a; < 10) in Resonant frequency (GHz) 187 rr301.spr fr301.sdf — fr308.sdf 8/11/89 2 50‘ I I I I I I I I r I I I I 0—0 c”=01) e 3' B—EJ e"=0.0003 _ K a—a e"=0.039 7.45— i , o—o s“=0.39 _ LN v—v c"=3.9 _ :\ +——+ e"=5.0 _ f; H e"=7.0 2,40- H c”=10.0 _ 2.35— . — \. \. 2.30— ._ 2.25— -1 2.20— _ 2.15— _ 2-10 r I ' l ' l r I r r ' l ' O 5 10 15 20 25 30 35 Figrue 4.22a Resonant frequency vs. :' for TM“: node for various a", a=0.635 out, b=7.62 cm and Lo'15.5 cm. Resonant frequency (CH2) 188 I:'.3IJI'I3.SIII' dpnI'IBO'I $th — (Jenfr3()8,sdt 8/1 1/89 ) 50—7 I I I I I l l I I I 6—0 5’ =0. 0 .I B-EJ 8' '=0. 0003 -* A—A e"=0.039 7 484 \ G—o a"=0.39 A v—v t”=3.9 ‘ \ +——+ c"=5.0 ‘ \< \* x—x s"=7.0 4 ’7 [IR—4 \ \ H CH=IO.O \\ _ _ ”\\x\ x \ 2.44— T \: A 2.42— -1 2.40— - 2.38— — 2.36— _ 2.34- — 2.32 I I I l 1 l I l I O 2 4 6 8 10 Figure 4.22b Low dielectric constant portion of Figure 4.223 189 Figure 4.22a is displayed in Figure 4.22b. Figure 4.22c displays the case of a=l.27 cm. For a," < 7.0, the resonant frequency is reduced as e; is increased. When a," = 10, the resonant frequency increases with increasing 3; until s,’ = 4.1, after the point, an increase in 2; yields a decrease in the resonant frequency. Except for e,” = 3.9 and 5.0, all the resonant frequency are shifted down from the empty cavity resonant frequency, 2.45296 GHz. The resonant frequency for a,” = 3.9 and 5.0 are higher than 2.45296 GHz if e,’ < 1.25. When a; > 1.25, higher a: give rise to a lower resonant frequency at a given a; for all the curves shown. Figure 4.23a shows the case of a=0.634 cm and L, = 6.94 cm for a 2.44994 GHz 1151" mode. The resonant frequency vs. a; are almost identical when a: < 3.9. It is found that for a; < 13.5, increasing a," will decrease f,, and the opposite is true for a; > 13.5. At e,’ = 13.5, the resonant frequency is about 2.4088 GHz for all a," given in this example. The resonant frequency for e," < 3.9 are decreased with increasing a}. When 22 3.9, there is a 2; region where the resonant frequency is increased with increasing 2}. Outside this region, an increase in e,’ will reduce the resonant frequency. All the resonant frequency are shifted down from the 118 m empty cavity resonant frequency. The example shown in Figure 4.23b is the same as Figure 4.23a except that the rod radius is twice as large. Again, the f, vs. 2; curves are almost identical and are a decreasing function of e; if e," < 3.9. For a: = 3.9, 5.0 and 7.0, increasing a; will decrease the resonant frequency. It is interesting to see that when 2.1' = 10.0, an increase in e,’ result in an increase in f,. For a given 2;, increasing dielectric loss may give rise to either an increase or a decrease in the resonant frequency, depending the value of 3;. Except for the case of e: = 10, the resonant frequencies shown in Figure 4.23b are lower than the empty cavity resonance. Figure 4.24a plots the fr vs. a; diagram for T5011 with a = 0.635. As shown, all the resonant frequencies are decreased with increasing 2;. The fr vs. e,’ curves do not vary much for s," < 3.9. At a given 2.1, increasing a," will increase the resonant Resonant frequency (GHz) f\) U" [‘0 (A [\J {\J 20 190 fr311.spf fr311.sdf — fr318.sdf 8/11/89 l I l T I r l I If I l l l f 0—0 e“=OiJ _ a—a e“=a.aoos '“ A—A e"=0.039 o—O e”=0“39 — :: v—v £"=3.9 — +——I- c"=5.0 _ .v X—x c"=7.0 .. H e“=10.0 _ \; e _ \‘ e \is. _l \k‘ _ \K \t‘ “sis. - ‘sig. ~ ‘is: “is... " "TN... ‘ l l I I l l 1 I l I A l I O 5 1O 15 20 25 30 35 8| Figure 4.22c Resonant frequency vs. t‘ for THO]: mode for various z", a=l.27 cm, b=7.62 cm, L.=15.5 cm. Resonant frequency (CH2) 191 fr321.spf fr321.sdf — fr328.sdf 8/11/89 2.46 I I I I T I I I I I I I I G—O e”=0.0 _ B—El e"=0.0003 A—A e"=0.039 2.45— H ”:0” If; H e”=3.9 _ +—+ e”=51) _ x—x e"=7.0 2,44—_ *—* c 2101) _q 2.43— I. . - 2.42— ' ' - . .~ _ '\: € \KQN. ‘ “rig. 2-41 — "s; _ _ ‘k§\b _ fix 2.40— ~ — Q‘s s. _ kg - .\\ ‘\~ .\ 2.39— \~;\ — ‘\"~\ ‘\ _ ‘\; _ 2-38 I I I I I I I I I I I I I O 5 10 15 2O 25 3O 35 8I Figure 4.23a Resonant frequency vs. e' for Hfirrr mode for various a", a=0.635 cm, b=7.62 cn, Ira-6.94 on. Resonant frequency (GHz) 192 fr331.spf fr331.sdf — fr338.sdf 8/11/89 2.5 I I I I I r T I I I I I r i w %figfx)k*gk*a&*flE**Hexsexeexaex _ 2.4— .. ., _ 2.3— — 2.2-I — 2.1— _. 2.0— "\. _ \ "1 ‘\ .1 1.9— \\ ...— . \ _ ~\ 1.8-— \:2\ _ \1 — \ - H e"=10.0 \‘s 17“ x—x "=7.0 \\ -‘ _ +—+ e"=513 \$\ ‘ V—v e"=3£3 ‘Qi ‘ 1.6— 0—9 e"=0.39 §\ — ' H e"=0.039 \IN .- B—a s"=0.0003 \ - H e"=0.0 1'5 I I l I I I f T I I I I 1 O 5 1O 15 20 25 30 35 80 Figure 4.23b Resonant frequency vs. 1' various c". a-1.27 cn. b-7.62 cn. Ira-6.94 for HErrr mode for cm. Resonant frequency (GHZ) 193 fr341.spf f-r341.sdf — fr348.sdf 8/11/89 I I I I I I I I 2.455— I I I l f — ‘ G—O £”=Ol) a x: 9—6 e"=o.aaas : “ .. , A—A 8"=0.039 ~ 2450": 3) 0—0 e“=O.39 _ _ V§xz , v—V e"=3.9 “ q \" ’ +—+ e”=5.0 : 2.445— “5%. H ”=7.0 — : “\I; H e”=l0.0 : 2.440: S 2.435— S .1. Z .I _ 2.430— —' 2.425— .2 2.420— 1 2.4155 - 2.410; 1 2.405 I I r I I I I I I r j I I O 5 1O 15 20 25 30 35 Figure 4.24a Resonant frequency vs. e‘ for TBorr node for various t", a=0.635 cm, b=7.62 on and Ina-30.8 cm. 194 frequency. It is seen for very low a}, the resonant frequency may be higher than the empty cavity resonance due to the increase in 5.7. Figure 4.24b depicts a T150" cavity loaded with a rod of 1.27 cm radius. The variations of the resonant frequency vs. a; are quite different from those seen in Figure 4.24a. For a," < 10.0, the resonant frequency is decreased when e,’ increases. When a: = 10.0, the resonant frequency increases as e,’ increases and is always larger than the T5011 empty cavity resonance. The next three figures, Figures 4.25, 4.26, and 4.27, are the f, vs. a; diagrams for a seven-inch cavity, where a = 0.625 cm and b = 8.9 cm. Each diagram has a family of curves and each curve corresponds to a specific dielectric loss. Figure 4.25 shows TMm mode, where the cavity length is 14.407 cm. H151“ and IE0” modes are shown in Figure 4.26 and 4.27, respectively , and the corresponding cavity lengths are 6.69 cm and 11.255 cm. Note that the a/b ratio associated with the seven-inch cavity is smaller than that shown in Figures 4.22a, 4.23a, and 4.24a. The variations off, as a function of ab ratio are examined in Figure 4.28 for Him, HE,“ and 175011 modes. In this example, the radius of the cavity is 7.62 cm and the load material is nylon, i.e. é,=3.03-j0.039. The cavity lengths associated with TM 012, HE,” and T8011 modes are 15.5, 6.94 and 30.8 cm, respectively. Note that the empty cavity resonant frequency corresponding to the cavity length stated above is 2.45296 GHz for IMO” and 2.45 GHz for TB," and T50" modes. The resonant frequency of the cavity completely filled with materials with é,=3.03-j0 is 1.409 GHz for IMO” and 1.4075 GHz for Him and T50" modes. As shown, the resonant frequency decreases with increasing rod radius for each mode illustrated. Also note that when alb —> 0, the resonant frequency approaches the empty cavity resonance and when a/b —> 1, the resonant frequency approaches the corresponding resonant frequency of the homogeneous cavity that is filled with dielectric material where é,=3.03-j0. Figure 4.29 shows the resonant frequency vs. a/b diagram for a seven inch cavity, i.e. b=8.9 cm, loaded with nylon rod. The cavity lengths for TMm, HE“, and T150“ modes are Resonant frequency (GI-i2) 195 ,) 7 fr351.5pf fr351.sdf - fr358.sdf 8/11/89 4.. I I I I I I T I I I I r I W , - ZiS-Ixeeaexae*46* ._ -I s ”u;- “. _. 2.4“ s '\ '— Ilfy 2.3.“ .\v\ _ — ‘ ‘I\“ _ 2.2— \‘I‘: _ 2.1—- \ ._ _ I. \‘3 " 21)—- ‘\i ._ V\ _ \; _ 1.9— \ _. t. 1.8- \at ._ _ %-* £"=101) ‘\&f x—x e”=71) \Rt ' 1.7 "‘ «l—I- 8":5.0 \{A _ _ “—5 €"=3£3 \Wx, - 0—0 e"=0.39 ‘3. 1-5‘ H e"=0.039 \ r - B—EI c"=0.0003 .- o—e e“=01) 1.5 f I I r fi I I I I I I I I 0 5 10 15 20 25 30 35 8 Figure 4.24b Resonant frequency vs. 2' various a”. a=l.27 cm, for TEorr node for b=7.62 cm and L.-30.8 cm. .y (C ‘14) A Resonant Frequen. ~ 196 " -¢ ° ‘— 8. 2.50 I I yrd«,V11lJ..)pf; buodf Y b ISdf1 I I o—e c”=1.0e—1O H s"=3.0e—4 G—a e"-0.0I’>9 ~ H e”-O.39 3 _ :\\ H C'=3.9 /.4‘_)— ‘§i\x\ H 2:”=5.0 '— ifi§ii e—e £”=7() §5\ x—* c“=10() V39 , ' A " W5- 2 40-— \w ‘— K _ \ 2“35-* '\ ‘— ‘\v \ \\ \§\ 230'—1 \\\\ '— “§\ 1 ‘ * ‘V\\ a W\\ V 2.25# \. 2.20 I I I I I I T I I I I 0.0 ' 5.0 10.0 15.0 20.0 25.0 30.0 Dielectric Constant 5' Figure 4.25a Resonant frequency vs 2' for TMoxz mode for various a", where a=0.635 cm, b=8.89 cm and L.=14.407 cm Resonant Frequency (GHZ) 197 idcvtbbspi dcnbisdi dviibtflmfl 0—0 H G—EJ \ I—I A—A H o—o H m m m m q m m m H 2.410— 2.400— 2390 . 1 . l I 0.0 2.0 4.0 6.0 8.0 10.0 Dielectric Constant 8' Figure 4.25b Low dielectric constant portion of Figure 4.25a Resonant Frequency (CH2) 198 r'd<;vH.t;pl ii.:,: now n\s .u> hocosoouu guacamox mm.v ousofim {0 mgu “no mgu mgu egg mac Ngu _ o n20 h _ _ .— _ — L _ b _ b b b _ h _ ¢.— 1 . .uo._ i ..., 1m; 1 NFOSDW. u I .inam I .am m i is FFOMH XIR ——OMH . l FFPUI I l NPOEH @Io t b it _ _ _ _ a . _ _ _ _ _ _ _ _ ©.N mmxmim .633: .. .633: 339.: (2H9) Aouenbei; tuouosea 201 .mno.ofiino.nuu .EU mm.mln wHQSS .mwmuOE MSOfihmkr HON Q\fl .u> hocoavouu uneconox mm.v assuah p\o mo 50 mo no ...o no we so o.o . _ . _ p _ _ _ . _ _ _ _ _ _ _ . e P 1 n u ...414.... u I. o u ".4 .l©.— i a . 1w; 1 13.. 1 «Na i [am FPOUP fllh . Eu: aim - NPOEH QIO _ _ _ _ p _ b b . P _ _ _ _ b _ ©.N ...8. P 6.3 552.89% 33.95: (2H0) KouenbeJ; IUDUOSGJ 202 14.407, 6.69 and 11.255 cm, respectively. Perturbation technique has been widely used to determine the change in the eigenlength (or eigenfrequency) of a material loaded cavity, due to a change in the dielectric property of the material load. Based on the eigenlength shifts (or eigenfre- quency shifts), the dielectric property of the load can be predicted. Often the fields within the loaded cavity are approximated by the empty cavity fields in the perturba- tion formula. If the field patterns within the loaded cavity are not altered much from those of the empty cavity, the approximations can give reasonably accurate results. On the other hand, if the the material load causes great changes of the empty cavity field patterns, perturbation technique may yield incorrect results. Comparisons of the exact solution and perturbation formula are given in Figures 4.30a and 4.3%. The resonant length perttn'bation formula is similar to the resonant frequency per- turbation formula given in equation 5.15a. The only differences are for the resonant length perturbation formula f, is the constant operating frequency and f,o is replaced by 3 108 f r0 = J—(b— Inifiglr): where L, is the resonant length, b is the cavity radius and npq is the mode index. In this example, TM”, is excited at 2.45 GHz in a seven inch cavity and the rod radius is 0.635 cm, i.e. x01=2.405, f, =2.45 GHz, b=8.9 cm and a=0.635 cm. As displayed in Figure 4.30a, the straight line represents perturbation solutions. The other curves are calculated by the exact formulation. It is interesting to see that for a; larger than 38, the perturbation formula deviates away from the exact solutions. In between e,’ = 10.0 and 38, larger 8: gives rise to greater difference between the per- turbation and exact solutions at a given 2;. Figure 4.30b shows the low dielectric con- stant portion of Figure 4.30a. In this particular example, it is found that perturbation formula only works for very small ranges of s,’ and 5,". Note that the rod radius is very 203 .Nzo mv.mlou can E0 mm.mla .30 23.9.3 ...u msofiua> now 0006 «.928 now .u .u> spaced usasouou son.v enough Ogu IIIIIIfIIIIIIIIIIIIIIIIIIIIII .' ...O_._.<,mm3_.mua 03 r") O 0 ll - _ _ _ — r _ ____ _ b - can? WINS: meme: momma. was. memo? Woman, I mmnni mmi. mmlo Cu 3.. .. a 9: Ema 30: b b (uuo) uibuel lUDUOSGJ 204 .won.v ousowm uo :o«uuoa .u sad as» no usofioouuacm non.v mascam mmwemio event .. Emfickmfissoc .w 0.0 0.0 _ NBS IVNNHF lmhohe i /- $3.: 792mm3mma I ..,/ .. o_ M :o XIX // r W N :w I ..o T I 0.0 N ..Q Ulla / l mnoaunuzo Ola _ _ r t _ . . _ . _ 03.: (we) uibuel tuouoseu 205 small compared to the cavity radius. If the rod radius increases and hence larger pertur- bation is caused to the empty cavity resonance, the perturbation formula may yield erroneous results even for very small a; and 2,". By comparing the f, vs. e,’ curves among the exact solutions shown in Figure 4.30a, it is observed that the resonant length decreases as 2; increases. In the range of s; < 7.0, at a given 8;, an increase in 8,? gives rise to an increase in the resonant length. For 60 > a; > 7 .0, the resonant length is reduced with increasing 2: at a specified 2;. As e,’ > 60, all the resonant length vs. 8; curves converge. Figure 43% presents a close look at the low dielectric constant range. Figure 4.31 depicts the resonant length vs. a/b for TMm, H151“ and T50" modes. The radius of the cavity is 7.62 cm and the dielectric material is nylon. The cavity is excited at 2.45 01-12. At the 2.45 GHz resonant frequency, the resonant length of the empty cavity is 15.530 cm for TMm, 6.94 cm for TE”; and 30.8 cm for 725011, and the resonant length of the homogeneous cavity that is filled with material of é,=3.03-j0 is 7.52 cm for TMm, 3.653 cm for TE," and 4.255 cm for TEou. As shown, the resonant length decreases with increasing rod radius. It is of interest to see that at a/b=0.25, IMO” and 1750“ degenerate. Also seen is that as a/b —> 0, the resonant length approaches the empty cavity resonant length and when a/b -) l, the resonant length approaches the resonant length of the homogeneous cavity filled with material with €,=3.03—j0. 4.3.3.3 Quality factor In this section, the quality factors of TM)”, HE“, and T50" modes are studied as a function of 8;, e," and rod radius. As mentioned in chapter 3, the quality factor Q can be expressed in terms of the quality factor due to the conducting wall losses and the quality factor due to the dielecuic losses. Hence, the quality factor is written as 206 .Nmo mv.muou use mno.0nlmo.nuu .Eo mm.>un .moGOE u50wua> haw nxs .u> :uosoa ususouom an.v assuam p\o we we no we no so no No no o.o _ _ _ _ . _ _ m . _ . _ . _ _. _ . O eEflEO i ..m I 000000)! I I (6000000000!) [OF I {3000000000 ImP i iom I fimN T :pr XIIX .. 99.: one a _ _ _ _ _ _ _ . _ . _ . _ _ _ . swig anew—e: 5843.: .533... mm (qu) uibuel iuouosea 207 g=m+§3+0§3 (4.15) where Q, : the quality factor due to dielectric losses, Q“ : the quality factor due to the lateral conducting walls, ch : the quality factor due to the end conducting walls. The quality factors due to the conducting wall losses are obtained by perturbation theory [13]. In the following examples, Q , Q4, Q“ and ch are examined to see the influence of the dielectric losses and conducting wall losses on the overall quality factor. Figures 4.32a through 4.32d show Q, Q, ch and Q; for TMm mode, respectively, as a function of e; for some selected sf. The excitation frequency is 2.45 GHz in this example and in the later cases. The radii of the cavity and dielectric rod are 7.62 and 0.635 cm, respectively. Note that in order to keep the cavity at resonance at 2.45 GHz for various loading, the resonant length must vary for different loading conditions, i.e. for different 2;. As shown in Figure 4.323, the overall quality factor Q increases gradually as 2; increases. For a given e,', increasing 2,? yields a decrease in Q. In comparison with Figure 4.32b, Q, is approximately the same as Q. As depicted in Figures 4.32a and 4.32d, Q“ and 9.2 do not change much for all the losses illustrated. Q“ increases with increasing 2;, while Q; is constant for all 2;. Obviously, in this example Q... is the dominant quality factor that determines the Q . Q decreases with increasing a," is because of the increase of power dissipation in the dielecuic rod as a result of an increase in :5, provided the shielding effect, as noted ear- lier in section 4.3.3.2, does not occur as e," increases. The increase of Q as 2; increases is most likely due to the redistribution of the field patterns, which gives rise to more stored energy as e,’ increases. As mentioned earlier, the cavity length must vary as 3; changes to maintain the cavity at resonance. Since the cavity length 208 .N20 mv.mnou 6:0 EU do.hun .su mne.oua muons .=u unawus> uOu woos «.eza saw .u .a> o onn.v «sauna .Q mm on mu ON m— or m o _ _ _ _ . _ . _ _ _ . _ mnoduzo ole aflouiwalh I m.m”":Q I l 1 .lo 1 .I0— I .IOO I 12: .T . _ _ m _ _ _ _ 2.6.003 .6933 am; 3 209 on .N26 mv.mlcu can E0 «0.7-n. .Eo mnw.o-.u b _ g _ _ ohms: .zu macaua> Man «005 ..eza new .u .u> so nmm.¢ mnsmfim .Q on mm ON or OF 0 _ _ _ _ _ P r _ _ — mmoduzo olo Opiouzu E . manauolo - I .Im €398 .. $328 33% a .umo mq.mu°u can so mo.pua .so mnm.oua 9323 ...u «~5qu, new 000.: «328 uOu .u .u> .90 021v 93on on mm ON m— 0— m o _ _ L r . r b _ _ b I _ r mmodu...» olo . _ _. mndlzu «In maul...Q I 210 I'I' '11] I ‘111 l I H.oo. n... com l 000 w IIIT I IIT1 I .1 000m I. ¢O+we ..l ¢O+mo .l mo+mp l no+uo — p — — b _ . T . hvméopoa I Endopoa mméFoc 90+”: 211 N30 mv.Nuou ”an EU mm.bun .Eo mmm.oua mumnz .:u msofium> new 0005 «doze hen .u m> «no vmm.v musofim w on on mm ow my OF m o L _ _ _ _ F r _ _ _ _ L P mmoduzu elb r on.ou..w «la. 1 r. m.nfl:w I 1 m we I IS W ”:00 I 109 pl Woow I loco. fl 1 w. .188 I 13$, fl- . _- .I -- ai-.¢f¢éélaz§§ W¢o+mo .T Imo+m: fl .. WI _ _ _ _ _ F Wnofia L r 8.3980 I @3903 5.285 8+”: Z30 212 decreases with increasing 2;, the lateral conducting wall losses reduces as 8; increases. This explans why Q“ increases with increasing 2;, as shown in Figure 4.32c. Figures 4.33a through 4.33d display the Q, Q, Q, and Q; of HE,“ mode, respec- tively. The cavity geometry is the same as the case of Figure 4.32. Similar to the case of TM)” mode, the Q shown in Figure 4.33a increases with increasing a}. For a; > 2.5, an increase in 5," yields a decrease in Q. When 2} < 2.5, the Q corresponding to 23:39 is less than the Q corresponding to 25:0.039 but larger than that corresponding to £:=0.39. Explanation for this phenomenon is that as a," is increased from 0.39 to 3.9, EM fields are shielded from the dielecuic rod, as shown in Figures 4.17a, through 4.17h, which causes the decrease of the power dissipated in the dielectric rod and hence the increase of Q. Again, the Q, and Q; shown in Figures 4.330 and 4.33d are almost constant for all the e,’ and e," illustrated. Note that Q“ is much larger than ch. This indicates that the losses due to the end plates are much larger than those due to the lateral conducting walls. The quality factors of mm are shown in Figures 4.34a through 4.34d. As shown, Q decreases with increasing 2;. For a given 2;, increasing dielecuic loss e: results in an decrease in Q. As depicted in Figure 4.34b, the changes in Q, as a function of e," are similar to those of Q. The QC, shown in Figure 4.34c is identical for any a," given in this example and is barely increased due to an increase in a}. This must be due to the fact that the field patterns near cavity walls do not change much as the material load changes. In Figure 4.34d, it is seen that Q; decreases slightly as 2; increases. In comparison with Figure 4.34d, ch is much higher than Q“, which indicates that most of the conducting wall losses are due to the lateral conducting walls. Close examination of Figure 4.34b and 4.34c finds that Q, and QC, corresponding to 8:=0.039 are of the same order. While for e,"=0.39 and 3.9, Q, is much lower than Q“. Apparently, T50“ is not a good mode to heat low loss material, say 83:0.039, because the EM energy coupled into the dielectric material and lateral conducting walls is almost equal. 213 .Nmo mv.~u°u can so «o.hun .EU MMW.OIG when: .zu macauo> uOu 0005 ...m: new .u .u> a onm.v annuah .o mm on mm ON 9 OF 0 o P p _ _ _ _ _ _ e _ . _ _ F $0.01....» I I 30"..» To . mafia. To 1 it «Lula: I alone) 0 O o o o o o o I . 3+3 T p _ L _ _ _ F — _ y b _ _ “0+”— .Beomv u 398% 33% 214 mm 0% .Nzo mv.mlou can EU aw.hun .EO mam.OIu mums: .=u u:0«ua> you 0602 ..awz no mm .Q u .0 ON 0— .u> so or amm.v magnum -I.D I 1 $00.0“..Q 0'0 OM.O":Q I momuocu I _ _ P T P 909v 831.. Sam owva .amémn a .Ivo+wm mo+mp 215 mm, on _ .Nmo mv.mueu 6cm E0 «m.hua .EO mnm.ouo mums; ...u mSOMua> MON 060.: ~._MS hOu .u .u> .00 mm .Q ON me o— omm.v magnum —tn I IIII I [III I [III I IIII I [III l'l1 III — mmoduzu olo mndflb «In m.nu..u I _ _ _ p _ auméomoa .. 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OF ovm.v mMSme I l'l‘ I III I l‘l' [III I I IIII I IIII III _ mnodflzu Ole mnouiwala inawolb _ p b _ _ V0. ”too [00. m. 000 I 000 _ .1 0000 l ¢o+mp nu¢o+mm .lmO+wP n. 00+“: T almo+uo » 00.3. 0000 I EQNOnoo Ewénoa ho+wp L30 l mo+wm 220 mm Nmo mv.muou can Eu no.0un .Eo mam. mums: .:u mzofium> Ham ovoe . 0% mm _ some “cu .u m> Q ON mP _ . _ OHM «co 000.0 musufim OF _ l—ln I III I I IIII Vl‘ III I_I Vl‘ I I—IIII III mmoduzu Ola mn.ou:~ alb m.mfl..w I , a a aletalclalelclalalalaIOIalol . ulclelclclclcielclalele k _ P . _ _ _ L :Vméomo mc .. 630080 a + Qpnoo U 221 The influence of e," on Q, Q, Q“ and 9,; is investigated for TMm, HE“, and TEm for a wide range of a: in the next eight figures. The radius and a; of the dielectric rod are 0.635 cm and 3.03. As shown in Figures 4.35a and 4.35b, Q and Q; for 115111 decrease with increasing e5. While for TM 012 and T50”, Q and Q4 decrease with increasing a,” in the low e,” range, but when a," gets larger, Q and Q4 increase as 8," increases. It is interesting to see that the Q for TM”; is the lowest among the three modes when a: < 70. However, when a,” > 70, the Q for H5,“ becomes the lowest. It implies that for the particular cavity geometry given in this example, TM 012 is the best mode to heat materials with a: < 70 and HE,“ will achieve better microwave heating when a: > 70. The Q“, as shown in Figure 4.35c, for T50“ and H5111 modes are almost constant over the range of 0 < e: < 1000. For TMm mode, Q“ is constant only when a: > 100. In Figure 4.35d, it is seen that 0:2 for IMO]; and HE", modes change only slightly as a: increases. While the Q; for TEon mode increases with increasing 2:. The low dielecuic loss ranges of e: < 15 shown in Figures 4.35a through 4.35d are displayed in Figures 4.36a through 4.36d. Close examination of these figures reveals that for very low e,", i.e. a," —> 0, Q is determined by the conducting wall losses. When a: becomes larger, Q; is dominant and hence Q is primarily determined by Q4. The changes in the quality factors as a function of rod radius are investigated in Fig- ures 4.37a through 4.37d. The dielectric rod is nylon rod, i.e. €,=3.03-j0.039, of 0.635 cm radius. As is seen in Figures 4.37a and 4.37b, the Q and Q; for TM”; and T50“ modes decrease as rod radius increases. The Q and Q, for HEm mode increase with increasing rod radius when a/b < 0.24. As a/b > 0.24, increasing rod radius causes a decrease in Q and Q; for HEm. The field patterns for HE,“ displayed in Figure 4.18b and Figures 4.19a through 4.19c explans why the changes in Q for 118111 modes are different from those for TM”; and mm. As shown in Figures 4.19c and 4.18b, when a/b increases from 0.0833 to 0.167, more EM energy is distributed in the 222 .Nmo me.mnou 0:0 EU um.ban .Eo mmm.0uo .mo.mu.u muons .uocoe usowua> uOu :u .n> o amn.v unauflm ..Q 00: COOP com com 00h 00m 00m 00¢ com o _ _ _ p _ lL _ _ _ _ _ _ _ c _ . _ lL lg l nuuuuuunnunnunuuuunuucto .. H _ 1 «FFMII .. 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P02 P ".89 f T r n O n. [3+3 m, V p.91”: v “.03”: w H n. ”4.0+”: fl _ r _ _ L . P r _ _ _ g _ . n » . h I 538% .. $388 ”—3.38 8+2 233 .Nmu mv.~uou can u mm.hun .mmo.on|mo.mnu mumn3 .mvaE macaum> How n\a .m> «no chm.v wusoflm n\o dd md md dd md ¢d md Nd Pd dd _ _ _ — p — _ — P — _ — _ _ _ — . N—CEF I 1 : ’mI Old I r. Y 1 T m _ _ _ b r k L _ _ _ _ _ _ _ _ _ H Eahomooc I Eméomood Em._nooc mo+m— 234 air-filled region as a result of the increase in rod radius, which gives rise to an increase in Q and Q4. However, when a/b > 0.24, the field patterns shown in Figures 4.19a and 4.1% indicate that more and more EM energy is redistributed in the rod as rod radius increases, which yields a decrease in Q and Q4. Comparing Figures 4.37a through 4.37d, it is found that Q, is dominant for most of the a/b range. Hence, Q is again basically due to the dielectric losses. Chapter 5 EXPERIMENTAL SYSTEMS AND TECHNIQUES 5.1 Introduction The experimental system described in this chapter was used to study the dielectric properties, the absorbed power characteristics and the field distributions of dielectric- loaded cavity applicators using either high power single-frequency or low power swept-frequency techniques. Experiments were conducted to compare the measured field patterns, resonant frequencies, absorbed power and complex dielectric constant with theoretical results given in Chapter 4. The experimental system consists of: (1) a microwave circuit for generating, transmission and measurement of microwave power, and (2) a cylindrical cavity loaded with different shapes of lossless or lossy materials. An equivalent circuit is presented to explain the coupling mechanism of the microwave energy, the quality factor of the loaded cavity and the experimental techniques from the view point of circuit theory. Descriptions of the cavity applicator, the experimental system and its equivalent circuit are given in the next two sections. In addition, the diagnostic measurements and associated measurement procedures are described. 5.2 Experimental cavity design considerations and description 5.2.1 Experimental cavity design considerations The microwave cavity applicator is a section of a right cylindrical waveguide with conducting shorting plates at each end. Due to the presence of the two shorting plates, the field distributions in the cavity are standing wave patterns. The empty cav- ity modal solutions can be obtained by modifying the homogeneous circular waveguide problem presented in Chapter 3 to satisfy the additional boundary conditions of zero tangential E’ on the two terminating conducting plans. The solutions can be divided 235 236 into TM and TE modes. TM modes are the solutions with no longitudinal magnetic field components while the TB modes are the solutions with no longitudinal electric field components. The resonant frequencies for these two modes are given by, respectively, ,’:=bje—u\fi2+<1fi1x (5.1) ( m. )3; = 3718—: x..." + < 33—} )2 (5.2) where b and L, are the radius and length of the cylindrical cavity, respectively, x", and x,’ correspond to the pth zeros of the Bessel functions 1,,(2) and J,’(z) respectively. For TM modes, the indices n, p and q may be any integer value and only It and q are allowed to be zero. The indices )1, p and q for TB modes may be any integer value but only It is allowed to be zero. If, as is the case in some of the experiments described in this dissertation, the Operating frequency is constant at 2.45 GHz, then each resonant mode can be excited by adjusting the cavity length L, (resonant length) according to equations 5.1 and 5.2 where to, = 21c(2.45x10’). There are a number of factors need to be considered in the design of a cavity applicator that is utilized for microwave material processing using single mode excita— tion. Among these are the cavity dimension for each mode, coupling probe position and dimension, and the effect of material permittivity a, size, shape and location within the cavity on the excited mode frequency and field patterns. These considerations are described below. The material shapes considered in this dissertation are either cylindrical rods or slabs. Appropriate cavity modes can be selected for each material shape if the material load can be placed in the proper position within the single mode exciting fields. Materials having different shapes and sizes need to be processed with different 237 cavity resonant modes. Appropriate mode implies that most of the microwave energy is focused on the material and hence can heat material efficiently and uniformly throughout the power cycle. However, the presence of material load will alter the EM fields and the resonant frequencies from those of the empty cavity in a complex manner depending on the material a, shape and location in the cavity. Due to the changes of the resonant frequencies as the material is heated, it is necessary to adjust the cavity length and the coupling probe position in the cavity to keep the mode in resonance at 2.45 GHz. Figure 5.1 displays the transverse electric field patterns for various empty cavity modes. The type and quantity of modes present depends on the cavity dimensions as well as the frequency of operation. Two commercially available brass tubes, 17.8 - cm - id. and 15.24 - cm - i.d., were used to build two experimental cavities. These two cavities are named seven inch and six inch cavities respectively according to their radii. In the experiments considered in this dissertation, the excitation frequency is fixed around 2.45 GHz, while the cavity length is adjustable. Therefore, the range of cavity lengths must be determined for a 2.45 GHz excitation operating frequency such that the cavity modes best for the experimental purpose can be selected by length tun- ing. For instance, a cavity applicator operating at 2.45 GHz can process materials with different electromagnetic modes by adjusting the sliding short position relative to the base plate. The resonant fi'equency versus cavity length diagrams, i.e. plots of equa- tions 5.1 and 5.2, for these two cavities are displayed in Figures 5.2 and 5.3. Exami- nation of Figure 5.2b indicates that, at 2.45 GHz, seven cavity modes exist for a range of cavity lengths of O to 16 cm for the seven inch cavity. Since these modes included those which were suitable for the experimental purpose, it was appropriate to build a seven inch cavity with variable cavity lengths within this range. In a similar manner, as shown in Figure 5.3b, the range of the cavity lengths for the six inch cavity were 238 b ‘9; .2- fl {-5. 4“; f ) .., :2 132535.21. ""1 {anti-1,454,) ; e 5:: t .. v I m 0‘ Figure 5.1 Transverse electric and magnetic field patterns for various empty cavity modes 239 all i d 10 I I I I I l ' I TMorr TMorz TEort « TEoiz TMitr TMttz T8111 T3112 T3211 T8212 TMzrt TMzra T8311 T8313 T8113 T3114 —‘ \OmdeIIbUNI-J fO GHz. (resonant frequncy) l l ' T ' I ' l ' 0 5 10 15 20 25 30 L cm. (resonant length) 5: Figure 5.2a Resonant frequency vs. resonant length for seven inch cavity 240 am 592 Eoc0woc m> socodcot “cocomoc nmh 230E Eo mm.m H 360.. 330.0 26:2 “cocommcv 80 J om 2 Ne w a L P bl » b b p P n — b b P 00F 1 10:0. 1 tam .. tma N6 (KouenbeJ; lUOUOSGJ) 2H9 °4 f0 GHz. (resonant frequncy) 241 ‘0 ' 7 ' ‘1 ' I Tj l I I j 1 : TMoti 2 : TMotz 3 : TEort " 4 : TEoiz " 5 : TM111 6 : TM11: 7 : TEttr 8— 8 : TE“: - 9 : T3211 10 : T8212 11 : TMzii a 12 : TMarz - 13 : T3311 14 : T3312 15 : TEria 16 : T8114 _ 6— 4.... .. l ‘\\g ‘<=:ff==:: . V 2“ VK. 0 fl T l I I If I l T I O 5 10 15 20 22> 30 L3, cm. (resonant length) Figure 5.3a cavity Resonant frequency vs. resonant length for six inch £92 20:88 m> 6:03am: “cocomoc and 839d Eo No.5 u 360.. £30 26:2 «cocommcv Eo . ON on. N_. m b _ — _ P P 242 (Kauenbelj iuouosel) 2H5) 01 243 determined to be 0 to 20 cm. In reality, the actual ranges of cavity lengths for these two cavities are shorter than those specified above because the presence of the cou- pling probes in the cavities limits the movement of the sliding shorts from 6 to 16 cm for seven inch cavity and 5 to 20 cm for six inch cavity. The resonant frequency vs resonant length curves shown in Figures 5.2 and 5.3 will shift when the material load is placed into the cavity. Curves corresponding to different modes will move differently. Some of them may intersect with each other at the operating frequency as a result of the shifts. As Figure 5.2b shows, at 2.45 GHz the resonant modes adjacent to Tum are not so close as those adjacent to TMou. This may provide a wider tuning range of single mode operation for TMm mode when the cavity is loaded. with material. The reason to locate the coupling probe close to one end of the cavity is to allow a larger range of movement for the sliding short. Intuitively, the size of the probe should be small to minimize the perturbation of the electromagnetic fields in the cavity due to the presence of the probe. However, if the size of the probe is too small, it becomes difficult to convey large amount of power into the cavity. Unexpected arcs may occur at the tip of the coupling probe. The coaxial probe must have a charac- teristic impedance of 50 (2 so as to match the characteristic impedance of the external feed transmission line. Nor until the losses in the coupling probe are carefully studied, can the optimum size of the coupling probe be determined. This is an area for interested researchers to explore. 5.2.2 Experimental cavity description The two cavity applicators, i.e. a seven inch (17.8-cm-i.d.) cavity with variable cavity length from O to 16 cm and a six inch (15.24 - cm - i.d.) cavity with a range of cavity lengths of O to 20 cm, are described in detail in this section. A photograph of the seven inch cavity is shown in Figure 5.4. This cavity was made of a cylindrical Figure 5.4 Picture of a seven-inch cavity 245 brass tube (17.8 cm ID, 25.4 cm long and 0.318 cm thick) covered each end with two transverse circular brass short plates. One of the shorts is adjustable to provide a vari- able cavity length of 6 - 16 cm and as shown in Figure 5.5, the cavity length is denoted by L,. Adjustment of the cavity length is made by manual rotation of the knob located just above the cavity top, as indicated in Figure 5.4. A micrometer indi- cator on the sliding shorts is used to measure L, to within 0.01 mm. There is an access hole 2.25 cm in diameter at the center of the sliding short to allow the place- ment of loading materials into the cavity interior. Two sets of brass collars are avail- able to fill the access hole allowing the hole diameter to be reduced to 1.27 cm or to be completely closed. The bottom shorting plate is fixed in position during operation and is either soldered to the brass shell as shown in Figure 5.5 or is removable for inserting large radius materials as shown in Figure 5.6. As depicted in Figure 5.5, the cylindrically shaped material load was placed in the cavity through the hole located at the center of the sliding short and suspended in the center of the cavity by a cotton thread. The soldered bottom plate also has a hole of the same size as that in the slid- ing short so that a long rod can be inserted into the cavity coaxially filling the entire length of the cavity. The removable bottom plate was designed to hold disk-shaped material loads which may be processed with the TE," cavity mode. In order to reduce the disturbance to the TE resonant modes, there was no hole in this plate to change the surface currents and the electromagnetic field. If uniform heating was required, the material load dimensions were limited to lengths of less than one half a free space wave length. Silver-plated finger stock is soldered to the movable short plates to pro- vide good electrical contact with the cavity walls. The gears and handles are commer- cially available and constructed of stainless steal and aluminum respectively. As can be seen from Figures 5.4 to 5.6, a rectangular brass block with 16, 2.26 mm diameter, diagnostic holes 0.922 cm apart was soldered outside the cavity wall. 246 32g 838g eons A»; 3360 £05 Gabon no $25103» ad Paw?— H - .L a L iiuoo mummy—cm Oceania +5ch make? H \ / denounce _ .llll. 1. , . - —l 0 one. 1:533: _ll Cr! Em .. / , a. a 30: 033.336 _ H _ _ ton 25980 P835 .55» N — \ toga gnu onoun // WM J « musuauaaeau \ a _ _ a _ d. emmngu _ cauuoo n. . d f s _ 247 363 50.309 09—3583 53 533 A05 :26.“ go Soglogm 90 9:85 ssa 833.. .2939. 289's . , _ _ _ £0: , - canon—no? _ i r a. d 248 The diagnostic holes were drilled through the cavity wall and the lowest hole was located 0.635 cm above the cavity bottom plate. A 2 - mm microcoax probe, called an E-field probe, could be inserted into the cavity applicator through the diagnostic holes to measure the square of the radial electric field IE,I2 on the inside surface of the cavity. The penetration of the center conductor of the diagnostic probe is limited to 1.5 mm so that the probe causes minimal perturbation to the resonant mode field pattems and resonant frequency during measurement. Thus, an accurate radial standing wave pat- tern of a cavity mode can be determined from the diagnostic E—field probe measure- ments. A copper-screened viewing window on the cavity wall, as shown in Figure 5.5, allowed visual viewing of the material load. Two coupling ports, which were 4.72 cm above the inside surface of the base plate, were located at 90' from each other. The port Openings were 2.77 cm in diame- ter. Coupling of microwave power into the cylindrical cavity was accomplished with an adjustable coaxial probe inserted through the coupling port. Cavity modes were excited by the EM energy fed from the coupling probe and various coupling conditions could be achieved by the adjustments of the cavity length and the coupling probe posi- tion within the cavity. Only one of the two coupling port was used in this dissertation. The 50 - a coupling probe has an outer conductor 2.52 cm in diameter and an inner conductor 0.965 cm in diameter. The inner conductor of the coupling probe was extended into the cavity. The depth, denOted by L,, could be manually varied from 0 to 40 cm. Silver finger stock attached to the center conductor provided excellent electrical contact with the inside of coupling port of the cavity. A reducer (Andrew type 2260B E507C) was adapted to fit the end of the coaxial section providing a means to connect the coupling probe to the external network. Adjustment of the cou- pling probe depth L, in the cavity is made by manual rotation of the knobs of the rack and pinion systems shown in Figure 5.4. Similar to the sliding short, a micrometer dial indicator on the coupling probe was used to measure the depth of the probe to 249 within 0.01 mm. Figures 5.7 and 5.8 shows the photograph and cutaway view of a six inch (15.24 - cm - i.d.) cavity. A detailed description of this cavity was given in [91]. The basic design is the same as described for the seven inch cavity but the diameter of the brass tube has been reduced and the other dimensions have been scaled appropriately. Only the parts which were made differently from the seven inch cavity will be described in the following. The access hole on top of the cavity was 2.54 cm in diameter. Two sets of brass collars, having holes of 1.27 and 0.635 cm in diameter respectively, and one set of solid brass collar were available to plug the access hole. Several small con- centric depressions were located on the inside surface of the fixed shorting plate to allow for the alignment and positioning of cylindrical rod loading materials. The coaxial coupling probe was made smaller to satisfy the electrically thin requirement [91]. The diameters of the inner and outer conductors were 0.20445 and 0.635 cm respectively. As shown in Figure 5.8, over 500, 2 mm diameter diagnostic probe holes were drilled around the cavity walls and on the bottom shorting plate. The holes were dis- tributed among 16 vertical layers on the walls and five concentric rings on the bottom plate. In addition, a hole was made at the center of the bottom plate. Detailed field patterns along the wall and bottom plate of the cavity were determined by inserting the E-field probe through these diagnostic holes to measure the square of the normal com- ponent of the electric field. 5.3 Experimental systems and circuits 5.3.1 Experimental system The two different microwave systems used in the experiments described in this thesis are depicted in Figure 5.9 and 5.10. As shown they are a low power swept- frequency system and a high power CW single frequency system. The low power 250 noumozmmm ‘3?ch 20:“ 63m caansawmmmwc an» no masonic s.m enemas 251 Cutaway View of the six inch cavity Figure 5.8 252 89826 832? 0608 hoaoavgulamokm .3309 #3 ad 03mg ? 3832333 - , 28a ? 833:8. 83 causes: «warms cam:— sts «calm. 832? he 03 u 05 crease “HM Toni—ss— — L5.— aura u3£=oao # _ .3398 362.39» _ 19563:. 73030th 333 _ _ 253 8.0.820 8000.? 0008 hon—0:008 0380 .8300 nmE 3.0 0.83.," 000000980008 elm:— eoetoafl maul”— . 008.038 «mula : - 830». 030002“— _ 000— _ 0308 .0308 “UNLIM— 5800 .530.— 00.5.— 00380 \ 1 -l :1 2 1° = Rf“ “+th (5.8) where P, is the total power coupled into the cavity in the steady state. In other words, P, consists of metal wall losses as well as the power delivered to the material. W... and W, are the time average magnetic and electric energies stored in the cavity fields and 10 is the input current associated on the coupling probe at the reference plane a.» The terms R,-,. and 1X... are the cavity input impedance seen by the feed transmission line from reference plane 0,. 259 As can be seen from equation 5.8, the input impedance is purely resistive if the time average magnetic and electric energies are equal and purely reactive if there are no losses in the cavity. The reactance is inductive if the time average magnetic energy is greater than the time average electric energy and capacitive for the opposite case. In order to have maximum power transferred into the cavity, it is required that the reactance and the resistance of the input impedance of the material loaded cavity be equal to zero and the characteristic impedance of the feed transmission line, respec- tively. This is the critical coupling condition. Under this condition, half of the power generated from the oscillator (microwave power source) is delivered into the cavity. According to circuit theory, the input impedance Z,- looking into a0 is the total impedance of the secondary circuit transformed to the primary circuit, as depicted in Figure 5.11. That is 2,. =2,, _ 1 . mL031. . —m{(G,+GL)+J(X+BL+0L,+02L,C,BL)} (5'9) Close examination of equation 5.9 reveals that the critical coupling condition is achieved by a combination of two adjustments : (1) adjustment of the cavity length to detune the cavity (changing resonant frequency), which results in the changes of L,, C,, G, and 1X; (2) adjustment of the depth of the coupling probe in the cavity, which results in the changes of m and 1X. Incorporation of these two adjustments will cancel the cavity reactance, i.e., 1X, = 0, and adjust the cavity resistance to be equal to the impedance of the feed transmission system, i.e., Ri, = 20. Introduction of the material load into the empty cavity will decrease the resonant frequency and change cavity Q. Consequently, the presence of the material load then changes L,, G, and C, and adds additional material conductance GL and susceptance 13,, to the circuit. 6,, and jBL are functions of material load placement, material volume 260 and shape, and material properties. During processing, material properties may change, which gives rise to the changes of the resonant frequency and cavity Q. Because of this, L,, 0,, C,, CL, 13,, and fit all change during microwave heating process. So, the continuously variable probe and cavity end plate tunings of the cavity applica- tor are necessary during processing to cancel the reactance of the input impedance and to adjust the material loaded cavity input resistance to be equal to the characteristic impedance of the feed transmission line. The definition of the quality factor is a) (total electromagnetic stored energy ) total power loss Q: From this definition, quality factors associated with different power losses can be defined. The quality factor obtained directly from experiment is called "loaded quality factor" Q, which takes into account both the cavity applicator and the external circuit shown in Figure 5.11. The quality factor of the cavity itself (with or without material load) is called "unloaded quality factor Q," since it does not account for the loading of the feed transmission line. The third quality factor, "external quality factor Qw", can be interpreted as the Q corresponding to the power loss due to the external circuit. Note that it is assumed here that the generator (microwave power source) is matched so that the impedance looking from the cavity toward the microwave power source is the characteristic impedance 20. The relation between these quality factors is described by [93] _1-=_+; (5.10) If Q,” is smaller than Q.” the power loss due to the external circuit is larger than the power consumed inside the cavity, and hence the cavity is overcoupled. On the other hand, if Q," is larger than Q,” the cavity is undercoupled to the external circuit. Only 261 when Q“, is equal to Q., is the power dissipated in the external circuit the same as that dissipated in the cavity. Then the cavity is critically coupled. As can be easily seen from equation 5.10, under critical coupling condition, Q, = 2QL. The quality factor is useful in describing the bandwidth of a cavity mode and is an indication of the fre- quency selectivity of coupling power into the cavity. This is an extremely important factor in designing microwave circuit resonator. 5.4 Experimental procedures and techniques 5.4.1 Definitions Before introducing the experimental procedures and techniques, some fi'equently used definitions in this dissertation are described briefly below. A more detailed description can be found in reference [94]. ( l) Q-curve (or resonant 'curve) - a plot of power absorbed by the cavity (with or without material load) as a function of frequency (for a fixed cavity length). (2) Unloaded quality factor (Q,) - an indication of the magnitude of the losses in the conducting cavity walls and in the material load. Q. may also be thought of an indication of the bandwidth of a cavity mode. In this definition, the cavity is not connected to the transmission line. fl = — . 1 1 where f, is the resonant frequency, Af is the distance between the half power points on the Q-curve. (3) Loaded Q (Q) - quality factor of the cavity (with or without material load) coupled to the external circuit. Q = L (5.12) (4) (5) (6) (7) (8) (9) 262 Q, is normally measured from experimental data and can be related to Q,, as discussed in section (5.2.2), under critical coupling conditions by the formula Qu : ZQL (5.13) Critically coupled - the impedance looking into the cavity (with or without material load) is matched to the external feed transmission line at resonant frequency. Under this condition, the total input power into the cavity is absorbed and dissipated in the conducting walls and material load, and the reflected power vanishes. P} - incident power measured at the output of the microwave power source, i.e., at the reference point b shown in Figures 5.9 and 5.10. P, - input power measured at the input of the coupling probe, i.e., at the reference point a, shown in Figures 5.9 and 5.10. P, - reflected power measured at the reference point a, as indicated in Fig- ures 5.9 and 5.10. P, - total power absorbed by the cavity includes cavity wall losses as well as dielectric losses. P, - power absorbed by the dielectric. (10) P, - power absorbed by cavity walls. (11) P, - power which is measured by E-field probe is proportional to the square of the radial electric field components near the cavity walls. Note that the generic meanings of the words "loaded" and "unloaded" in the definitions of Q are not referring to the presence or absence of material load in the cavity. The word "loaded" or "unloaded" describes if the cavity is or is not connected to the external circuit. 263 5.4.2 Low power swept-frequency measurements Low power swept-frequency measurements were conducted at power levels of few milliwatts and measured the characteristics of both empty and material-loaded cav- ities. The experimental measurements included input power P,-, reflected power P,, resonant frequency f, bandwidth of the resonant mode Af, coupling probe depth L, and cavity length L, Based on these measurements, the quality factors and the dielectric properties of the material load can be determined. The complex dielectric consrant were evaluated by perturbation techniques or theoretical formula. Form these experi— ments the effect of the material loadings on the cavity Operating parameters such as the resonant fiequency (resonant length), coupling probe depth, cavity Q and field distribu- tions were investigated. As stated in Section 5.3, a resonant cavity is excited in discrete cavity modes. When the excitation frequency is fixed, the resonant modes of a cylindrical cavity correspond to distinct resonant lengths. The sweep oscillator was operated in the swept mode at 2.45 GHz center frequency. A cavity mode appropriate for microwave material processing was excited by adjusting L, and L, to the correct positions for reso- nance and critically coupling the cavity. Since the access holes, viewing window, cou- pling ports and coupling probe caused perturbation to the ideal cavity modes, the experimental eigenlengths of the resonant modes differed from exact theory. Calibration of the feed transmission line shown in Figure 5.11 is necessary for accurate measurements of the input power level P,- and reflected power level P,. As indicated in Figure 5.9 to 5.10, points a and b are the reference points used in calibrat- ing the system. Calibration involved supplying a known amount of incident power at 120 and measuring the power level at no. Then the power ratio, m, between P,- and P; was established. Using this ratio, the actual available power at the input of the cou- pling probe can be determined. That is 264 P,- = "11’; The power readings read from the power meters were corrected by taken into account the attenuations on the directional coupler and attenuators and the loss in the connect- ing cables. For example, if a 30-dB directional coupler is used for measuring incident power, as was the case for swept-frequency experiment, a 5 milliwatts reading from the incident power meter corresponds to 5 watts at the reference point b and the actual input power P,- is m x 5 watts. The total absorbed power by the cavity is given by P, = P,- - P, (5.14) Since the coupling probe is not perfectly conducting, microwave power traveling on it is subject to attenuation. The incident power present at the input of the coupling probe is different fi'om that fed into the cavity. Similarly, the reflected power measured at reference point a is somewhat smaller than that reflected from the cavity. However, the coupling probe loss is very small compared to the input and reflected powers. The assumption of lossless coupling probe gave no significant drawbacks on the experimen- tal results. The incident power, reflected power and the total absorbed power for the empty cavity are represented by P”, P,o and P”. Cavity Q for any mode is measured by sweeping the cavity over a frequency band and displaying the Q-curve on the oscilloscope. The Q-curve represents the power reflected by the system as a function of frequency for a fixed eigenlength. Figure 5.12 displays such a curve as it would appear on the oscillosc0pe. The horizontal line is the zero reflected power line. Consequently, the area under the bell-shaped curve is the power absorbed by the cavity in the vicinity of the resonant frequency. Three markers generated by the sweep oscillator are shown on the oscilloscope. Markers one, two and three (m1, m2 and m3) are pointed at the resonant frequency, two half power points, respectively. The distance between markers m2 and m3 is readily read from the sweep oscillator and denoted by Af. The loaded Q of the empty cavity is obtained using definition (3), i.e., Q“, = f0 If the cavity is critically coupled, E. 265 zero reflected power line r m1 = f0 Q—curve h—Af————- 1/2 power point Figure 5.12 Q measuremnt using Q- curve 266 Q... = 2Q“. The quantities subscripted by 0 refer to empty cavity measurements. When the material load is placed in the empty cavity, it perturbs the electromag- netic resonant modes excited in the cavity. In general, the resonant frequency decreases and the loaded quality factor changes. The changes in these cavity parame- ters are complicated functions of material volume, shape, properties and position within the cavity. For small perturbation, the spatial distribution of the fields of the material-loaded cavity is not greatly altered from those of the empty cavity. Only the magnitude of the fields changes. Under this condition, perturbation technique using a quasi-static approximation to the field inside the material can be employed to evaluate the complex dielectric constant of the material [89]. For TM," and TM,” modes, the perturbation formulas for cylindrically shaped material are 8': 1- 123—:mf-0f’0) 8: - 23b}; — 51-00 where At: J. 15,012dv' t = I 151’ dv f, = resonant frequency of the material-loaded cavity f,,, = resonant frequency of the empty cavity v’ = material volume v = entire empty cavity volume 5,0 = 2 component of the empty cavity electric field E; = empty cavity electric field (5.15a) (5.15b) . 267 For large perturbation, the resonant modes excited in the cavity may become hybrid ones instead of pure TE or TM modes. The theoretical formula given in Chapter 3 is required to determine the dielectric permittivity of the material load. The resonant frequency and the loaded Q of the loaded cavity can easily be obtained using the oscillator and Q-curve technique. By pointing markers m1, m2 and m3 at the proper positions as shown in Figure 5.13, the new resonant frequency and bandwidth are readily read from the oscillator digital displays. Since the introduction of the material into the empty cavity detunes the cavity resonance, two adjustments are needed to retune the loaded cavity to critical coupling condition at the resonant frequency. The adjustment of the coupling probe position in the cavity will match the loaded cavity to the external circuit and the adjustment of the cavity length will bring the resonance back at the 2.45 GHz excitation fiequency. The cavity length and the coupling probe depth are recorded and compared with empty cavity measurements, AL, = L, - L‘o AL, = L, — L,0 where the subscript "O" are referring to empty cavity measurements. 5.4.3 High power single-frequency measurements Though experimental results show that microwave material heating and process- ing can be achieved using swept-frequency microwave cavity, it is a very inefficient method [95]. Since the microwave source is operated in swept-mode, only small por- tion of the total incident power in the neighborhood of the cavity resonance is coupled into the cavity applicator. Most of the incident power is reflected and dissipated in the transmission network and the dummy load. In addition, on- line diagnosis of the 268 material complex dielectric properties during processing is very difficult using this technique. As mentioned in Section 5.4.2, the resonant frequency and quality factor of the material loaded cavity are required for calculating the complex dielectric constant of the load material using perturbation formula. However, material properties may change rapidly during microwave heating. It makes the on-line measurement of the quality factor by Q-curve method almost impossible due to the fast change of Q-curve. As a result, it is evident that the development of a different microwave heating tech- nique is necessary to achieve more efficient heating result and on-line diagnosis of material complex dielectric constant during heating process. High power single frequency technique has demonstrated high microwave cou- pling efficiencies of 70% - 95% and on-line process diagnosis [96]. In this method, both heating and diagnosis were accomplished with the same frequency. The operat- ing frequency was constant at 2.45 GHz during the experiments. Since material com- plex dielectric properties changed during heating, constant adjustment of L, and L, were required to maintain the material loaded cavity resonance at. 2.45 GHz such that most of the incident power was coupled into the cavity applicator under critical cou- pling condition. If the exact theory of the loaded cavity were available, one would solve for the complex dielectric constant a with the knowledge of L,, f, and cavity Q. However, because of the unavailability of the theory, perturbation formulas, equations 5.15a and 5.15b, were used to evaluate 2. Therefore, measurements of the resonant frequency and cavity Q during processing are the key data needed to achieve on-line diagnosis. Due to the single frequency operation, conventional Q-curve technique of quality factor measurement is not applicable. An alternative method for determining Q, of a material loaded cavity was developed by Rogers [94]. The unloaded quality factor was calculated from the following expression: 269 P IE 2 ' 9,: Q; ,1: 72' (5.16) where Qw is the empty cavity unloaded quality factor, P, and P0 are the total powers absorbed in the loaded and empty cavities, respectively, E, and 5,0 are the radial elec- tric fields at a fixed position within the loaded and empty cavities, respectively. This equation is an approximation based on two assumptions: (1) the presence of load materials in cavity does not alter the field distributions significantly from those of the empty cavity, only the magnitude of the electromagnetic fields change, (2) the relation between the energies stored in the loaded and empty cavities can be expressed as H,., 15,012 — = — 5.17 U» IEJZ ( ) Rogers showed the evidences of supporting assumption (1). Assumption (2) is care- fully examined in the following. The total energy stored in cavity is defined as U = [ 03(7):de (5.18) If assumption (1) is indeed true, the relation can be established lid??? 1301’ = = 5.19 15.06,)? IE.(r1)|2 c0) ( ) where Em and I???) are the electric fields in the empty and loaded cavities, respec- tively, cm is a constant depending on the position within the cavity, r1 is an arbitrary position within the cavity. According to equations 5.18 and 5.19, the ratio between the total energies store in the empty and loaded cavities is 0.. lEro(”r)'2 Ko U» |Er(r1)|2 76' (5.20) 270 where [(0 = l c(?) dv (521) K, = 1 c0) dv + e,[ we dv (5.22) V—V‘ V' Above equations indicate that equation 5.17 is true if K, = X, Close examination of equations 5.21 and 5.22 shows that assumption (2) may not hold for materials of large size, large dielectric constant and lossy materials. In order to use equation 5.16 to calculate Q, the ability of measuring the square of the radial electric field at a fixed position within the cavity is necessary. Since the ratio of the E-field probe measurement to the square of the radial electric field at a point within the cavity is a constant, i.e., Pb «IE—- = constant (5.23) the following expression can be established under assumption (1), 2 7% = 11%? (5.24) where P, is the E-field probe measurement for the material loaded cavity and P,o is the E-field probe measurement for the empty cavity. Equation 5.24 is valid if P, and P,o are measured at the same point near cavity walL Accordingly, it is important when measuring Q by this method not to move E-field probe between reading. According to equation 5.23, the radial electric field patterns near cavity wall were obtained using E-field probe inserted through the diagnostic holes shown in Figure 5.5. Note that when measuring electric field patterns using this technique, the microcoax probe depth within the cavity has to be remained constant throughout the experiments. 271 Any change of this depth may result in inaccurate electric filed patterns. Detailed description of the determination of an appropriate probe depth is given in [91]. Inspection of equations 5.15 and 5.16 reveals that measurements of the cavity quality factor, P,, P,, P,, L, and L, need to be taken in the high power single frequency experiments. Before heating started, the empty cavity quality factor Q“, was made by Q—curve technique, and then the single-frequency low power empty cavity measure- ments of P,,, P,,, P,,, L,o and L,o were performed. For low power measurements, the input power was typically a few rrrilliwatts. The material temperature was monitored by Fluoroptic Temperature Measurement System. After the initial measurements were recorded, input power was raised to the prescribed level and heating cycle began. During each heating experiment, the input power was maintained constant within the range of 0.5 to 10 W and the cavity was continuously tuned to critically coupled by manually adjusting L, and L, to minimum reflected power. Data points of P,, P,, P,, L, ,L, and material temperature were taken periodically during heating. As is shown in Figure 5.10, the temperature and power measurements were taken and recorded by a microcomputer at selected data rate which was set by the temperature update rate of the Luxtron 750 Fluoroptic Thermometer. Data points were read every one second or two. After the heating cycle was complete, the material load was removed from the cavity. Then the empty cavity resonant fi'equency, f,o, corresponding to each data point position for L, and L, measured during processing was determined by swept-frequency method. Cavity Q was calculated using equation 5.16. With the measurements of 1;, and cavity Q, the complex dielectric constant of the material for each data point recorded was evaluated using equations 5.15a and 5.15b where f, is 2.45 GHz. Chapter 6 EXPERIMENTAL RESULTS 6.1 Introduction The objectives of the experiments presented in this chapter are (1) to verify the theoretical results given in Chapter 4 and (2) to investigate the microwave heating of solid materials with various dielectric pr0perties using single-mode microwave cavity applicators. The low power experimental measurements of the resonant frequency, resonant length, quality factor and field patterns for various loads were performed and are compared with the theoretical computations of Chapter 4. Load materials included nylon and teflon rods with radii of 0.635 cm and 1.27 cm, and a 0.2 cm inside radius ethylene glycol quartz tube. Microwave heating experiments were performed to carefully investigate elec- tromagnetic interactions with selected materials and to develop microwave processing and on-line diagnostic techniques. Load materials employed in the heating experi- ments consisted of lossy materials such as nylon 66 and water-soaked wood cylinders, conducting materials such as three-inch silicon wafers, and anisotropic composite materials such as graphite reinforced epoxy slabs. On-line measured complex dielec- tric constants of nylon and wet wood were compared with publish data. In addition, the coupling efficiency vs. time was determined in the experiments with wet wood cylinders. The surface temperature of silicon wafers and graphite-epoxy slabs was measured as a function of time during microwave excitation. The influence of the orientation of the graphite fibers on the field patterns and the radial electric field strength near the conducting walls was examined in the low power graphite-epoxy composite experiments. 6.2 Low power measurements (verification of theory) 272 273 6.2.1 Introduction The six-inch cavity described in section 5.2.2 was employed to experimentally verify the theoretical analysis of Chapter 4. As already noted in Chapter 5 this cavity was carefully designed and constructed for precision experimental measurement. Two cavity modes, e.g. TM,” and HE”, modes, were investigated using various material loading conditions. Experimental measurements included the measurement of resonant frequency, resonant length, quality factor and axial field patterns of the empty and material-loaded cavities excited with the TM,” and HE,“ (or T5,“) modes. Cavity loads were finite radius rod materials that uniformly filled the cavity in the axial direction. The TM”, and HE”, modes were chosen as representative examples of a 0 independent and a hybrid mode cavity excitation. The methods used to measure resonant frequency, resonant length, quality factor and cavity field patterns are reviewed in sections 5.4.2 and 5.4.3. All measurements were performed at low power of less than 12 mW in order to avoid microwave material heating and other nonlinear field material interactions. To ensure that the cry- stal detector was in the linear operation range, quality factors measurements using Q- curve technique were made with input powers of less than 1.9 mW. The radial field patterns at the cavity end plate were measured by inserting a rrricrocoax probe into the diagnostic holes along the angles 0f 0", 90', 180’ and 270°. The configuration of the base plate is displayed in Figure 6.1. As shown, the coupling probe is positioned in the angle of 0° and the distance of each of the diagnostic holes to the center of the base plate is identified by a ring. The radii of the rings, r,, r,, r,, r, and r, are 2.225, 3.294, 4.445, 5.3975 and 6.35 cm, respectively. The method of determining the optimum rrricrocoax probe depth within the cavity without major perturbation to the cavity resonance was also described in Chapter 5. As shown in Figure 6.2, two microcoax probes of different dimensions were employed for field patterns measurements. Probe "a" was used for TM,” mode and probe "b" 275 (a) probe 0. F (1:) probe "/ (e) h C—-—. H ~ bat-o plate Figure 6.2 Microcoax probes and base plate dimensions 276 was used for HE," mode. Figure 6.2c depicts the exact probe position in a diagnostic hole on the base plate. Since the thickness of the base plate is 6.35 mm, the actual microcoax probe depths within the cavity were 3.65 and 0.25 mm, respectively, for proms "a" and "b". 6.2.2 TMm mode The theoretical and measured results for TM,” mode are listed in Table 6.1. The resonant frequency, resonant length, coupling probe depth within the cavity, and the quality factor are denoted by f0, L,, L, and Q, for the empty cavity, and by fm, L,,, L,1 and Q, for material-loaded cavity. Af and AL, are defined as (y: f, — fm, AL, = L, - L,,. The resonant frequencies and quality factors given in Table 6.1a were measured with the swept frequency technique described in section 5.4.2, where the cavity length was kept at a constant length of 15.5 cm. In order to minimize measurement errors the experimental quality factors were the average of at least three measurements. The resonant lengths listed in Table 6.1b were measured at a constant frequency of 2.45 GHz. Each experimental measurement was taken under critical coupling conditions. In the constant cavity resonant length experiments, cavity was critically coupled by adjusting the operating frequency and the coupling probe depth. In the constant resonant frequency experiments, cavity was matched by adjusting the cavity length and coupling probe depth. A quartz tube of 0.6 cm radius and 0.1 cm thick was used to contain ethylene glycol for the ethylene glycol experiments. Note that the Af and AL, in the ethylene glycol experiments were the differences of the resonant frequency and resonant length between the cavity loaded with ethylene glycol and the cavity loaded with empty quartz tube. As listed in Table 6.1, the theoretical results and experimental measurements of the resonant frequency and resonant length show excellent agreement to within one 277 0:0 nvfi .0 302.000.. .0208. .0528 A8 ....0 Wm. .0 59.0. .5003. 20.2.8 3 .000... «.02... .0. 0:30.. 3:00.500... 05 70.00.00... 0003.00 000009.000 .0 030... .5 3.“. u 8.5 803. 0.3... ... ......s .38. .8... 86.0.0 ...- ...»8 .83. 2.3 0.3.. 5.3.3 88...... 5.8. 0282 a. 0.0 :03 10.5 3.3. 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However the experimental quality factors are quite different from the theoretical results. For the empty cavity, the error percentage in Q, is 49.8%. While the error percentages in Q for the material loaded cavity vary from 5.6% to 38.0%. In the cases of 0.635 cm and 1.27 cm nylon rods, and ethylene glycol, the experimental quality factors are larger than the theoretical quality factors. The field patterns of TM” mode shown in Figures 6.3 through 6.6 were measured using single frequency at 2.45 GHz. Figures 6.3a, 6.3b and 6.3c shows the actual magnitude of the measured field strength. Comparison of the experimental and theoretical results is displayed in Figures 6.4b, 6.5b and 6.6b. In these Figures, the experimental field measurements are normalized by a constant such that the theoretical field curve is best fit among the experimental measurements. As shown, experimental and theoretical results have good agreement for all the load materials. 6.2.3 HE,11 mode Table 6.2 presents the theoretical computations and the experimental measure- ments for HE," mode. The experimental results given in Table 6.2a and 6.2b were, respectively, measured at a constant resonant length of 6.94 cm and a constant resonant frequency of 2.45 GHz. As listed in Table 6.2a, the experimental resonant frequencies for all load materials match with theoretical results to within one percent. For 0.635 cm nylon and teflon rods and ethylene glycol, the error percentages in Af are less than 8%. For the 1.27 cm nylon rod, the error percentage in Af is 32%. Also seen in Table 6.2a is the very large difference between the theoretical and experimen- tal quality factors. The error percentages in Q, are less than 58%. For the material loaded cavity the error percentages in Q are between 34.3% and 81.0%. Again the largest error percentage occurs in the experiment with 1.27 cm nylon rod. Note that the coupling probe depth in this experiment was 20.37 mm, which is much longer than the probe depths in other experiments. 279 80 b I I I T I I - 0—0 0 - _. H 1880 .. - 8—9 90 - 50“ H 2700 " 4o- — 20— '- 0 I I I I I I I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 6.33 1'2 I I I I I CI 00 I - o _ o 180 10— a o - I _ T I l I l l l 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 p cm (from the center of cavity) 6.3b Figure 6.3 Field patterns for 0.635 cm teflon rod. TM018 mode . (a) actual magnitude, (b) comparison between theoretical and experimental results p Wott lEzl2 .I G—O 0 1 4 _. H 1880 3—0 90 o ' I I I I - I I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 p cm (from the center of cavity) 6.4b Figure 6.4 Field patterns for 0.635 cm nylon rod. We 1 a mode . (a) actual magnitude. (b) comparison between theoretical and experimental results. IF: I' M Watt 0—0 H I 80° 90° 270° I l J i l l. l I _——--—__ I Figure 6.5 ”I I I I I 21) 2&0 ‘4I3 513 €i0 '71) 81) p cm (from the center of cavity) Field patterns for 1.27 cm nylon rod. TMoxa mode . (a) actual magnitude, (b) comparison between theoretical and experimental results. Is. I' uWott IEzI’ ’0.6-- 282 L4 I L2-‘ L0—‘ (18- (14— (I2- 01) 6.6a 2n? - thaxy _ fii l i O 1 2 3 4 5 6 7 8 6.61) p cm (from the center of cavity) Figure 6'6 Field patterns f°r 0-2 cm ethylene glycol. TMoxz mode . (a) actual magnitude. (b) comparison between theoretical and experimental results. 283 ":0 nv.~ no 65:3...— .5388. .5238 3. ...:o 3.0 ..o 598— .533.. .9538 An. .38:— ...m2 8.. 8.32 3%.... v:- 3.082.. 59.83 53.5.68 «6 03:. .8 8% "$88 .38. 3... 9.8... 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X83.“ 9a 18.8.8... 28.2... .8... .88 .. .. .. .. .88 .8... £3 .28: m .n 9.8. 8,8 8... Xe Xe 2e .2 83.. 1888.8 .8 a. an. X..." .- .- Xe Xe ..n... 883 9a 3.82.. ...... 8...z .8... 8.: -- .- -. .. an: as... ...... 88¢ 8% ...: 8X Xe Xe Xe. a." 88“ XX.“ 3.8.8.8 .83.... :8. 83 -. .. Xe Xe 5X. «8...... 9a .8889 ...... 8.»: .83 8.: -. .. . .. .. an... #8.... ...... 88¢ 88. :8. N... a... . Xe Xe 2. .33 X3." 3888....» .8 e8... :8. Xi .. .. .. Xe Xe e... 89.. 9a .888... 8.. 8...... 3...... ha...» 38..— 89.... .838 has! .818. EB .o .o .8... .... .8... .... .8... .... .8. .... ..8. ... 3.... ... 38. ... .8... 5.... 3...... 284 The experimental and theoretical resonant lengths given in Table 6.2b show excellent agreement to less than 1.18%. The experimental AL,’s in the experiments with 0.635 cm teflon and nylon rods as well as 0.2 cm ethylene glycol match with theoretical AL, to within 6.25%. In the case of 1.27 cm nylon rod, AL, has an error percentage of 16% between the experimental and theoretical results. The field patterns of HE,” are displayed in Figures 6.7 through 6.10. As shown, for 0.635 cm teflon and nylon rods and 0.2 cm ethylene glycol rod, the field measure- ments along 0" and 180' have good agreement with theoretical results. In these cases, the fields along 90" and 270° were zero. This is quite consistent with theory, since theoretically HE“, mode exhibits cosq) angular variations. As can be seen in Figure 6.9, the experimental field patterns of the cavity loaded with 1.27 cm nylon rod do not agree with theory. In addition, very small fields were sensed along 90’ and 270’. 6.2.4 Discussion The mismatch between the theoretical and experimental quality factors listed in Tables 6.1 and 6.2 is probably because (1) the exclusion of the losses due to the cou- pling probe in the theoretical formulation, (2) the nonideal structure of the cavity applicator and (3) the measurement errors. The effects of the nonideal structure include the electromagnetic radiation caused by the presence of the access hole and the losses due to the finger stock of the sliding short. The bad contact between the finger stock of the coupling probe and the inside walls of the coupling port would also give rise to inaccurate experimental results. The measurement errors are minimized by tak- ing the average of more than two measurements. In some cases, the experimental quality factors were larger than the theoretical results. Explanation for this is that the actual field patterns within the cavity are altered from those in theory. If the changes of the field patterns result in either less power losses or more stored energy than it would be in the theoretical model, it is '31” [1' Watt 285 l l 1.0 2.0 3.0 ' 4.0 5.0 6.0 7.0 8.0 0.0 O o o 180 —— theory I n 4J__4 4 I 0.0 I I l l 1T0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 5.71, p cm (from the center of cavity) 3Figure 6.7 Field patterns for 0.635 cm teflon rod, HExxx mode , (a) actual magnitude, (b) comparison between theoretical and experimental results. '32 |' I“ won o — — ..- — .-— — —— — al.:— a. . - ' I ' _a;_ CD I I I I . ' I I I 0.0 '110 2.0 3.0 4.0 5.0 5.0 7.0 8.0 5.31, p cm (from the center of cavity) Figure 6.8 Field patterns for 0.635 cm nylon rod, H611: mode . (a) actual magnitude. (b) comparison between theoretical and experimental results. 1880 90 o 270 1312 .000. d oo oo o thl‘ .0 O) l c .0 q 0-0 I I' It fl 1 0.0 1.0 2.0 3.0 4.0 5.0 5.0 7.0 8.0 5.913 p cm (from the center of cavity) Figure 6.9 Field patterns for 1.27 cm nylon rod, HExxx mode . (a) actual magnitude, (b) comparison between theoretical and experimental results. 288 0.9 I 0.8 - 0.7 — 1 0.5 - 0.5-— 0.4- |Ez I' M watt O.3- 0.2 _ 0.1 d o—o nu? 0.8 — 0.6- \ 0.4— \ _ \ 0.2 - \ 0'0 l I l I O I 2 3 4 5 6 5.101, p cm (from the center of cavity) \j— on Figure 6.10 Field patterns for 0.2 cm ethylene glycol, H8111 mode . (a) actual magnitude. (b) comparison between theoretical and experimental results. 289 possible that experimental quality factors are larger than theoretical quality factors. The possible causes of the changes of the experimental field patterns from theory include the perturbation due to the excessive coupling probe depth within the material-loaded cavity, had contact between the dielectric rod end and the base plate and the bad con- tact of the fingers around coupling probe with the coupling port. In the HE”, mode experiments with the 1.27 cm nylon rod, the experimental and theoretical results show much differences in Afs , AL,’s quality factors and field pat- terns. Two explanations may account for the discrepancy. The first is that the cou- pling probe was extended too deep into the cavity. For instance, as listed in Table 6.23, L, in case of 1.27 cm nylon rod was 20.37 mm, which was much longer than the L,’s in the other three cases. The second is the larger access hole on the siding short. In the 1.27 cm nylon rod experiments, the radius of the access hole was 1.27 cm, com- pared to 0.635 and 0.3 cm in other experiments. 6.3 High power heating measurements 6.3.1 Heating of nylon 66 cylinders The high power single-frequency measurement technique described in section 5.4.3 was used to heat and diagnose the nylon 66 cylinders of length 3.1 cm and radius 0.65 cm. As is shown in Figure 5.6, the nylon cylinders were placed exactly in the center of a seven-inch cavity excited in TM”; mode. In order to measure the inter- nal temperature of the nylon, a fiber-optic temperature probe was inserted axially through a small tight-fitting hole shown in Figure 5.6 into the center of the nylon cylinder. During heating process, each data point of L,, L,, P,, P, and P,, was taken at one minute intervals. From the measurements, the complex dielectric constant were calculated as described in section 5.4.2 by means of the cavity perturbation formula expressed in equation 5.15. 290 Variations of the complex dielectric constant as a function of time and tempera- ture are displayed in Figures 6.11 and 6.12, respectively. As shown, the different input powers of 3 and 5 W resulted in different heating cycles. That is, during heating process the complex dielecuic constant changes differently for different input powers. As is depicted in Figures 6.11b and 6.12b, nylon changed from a low loss to a very lossy material as heating processed. Also seen is that higher input power gave rise to higher a; at a given time or at a given temperature. While, as shown in Figure 6.11a and 6.12a, at high temperatures a; dropped and higher input power resulted in smaller 5;. The drop of e; at high temperatures during processing was found to be associated with the coupling probe depth within the cavity [97]. The coupling probe was adjusted inward in order to maintain critical coupling as the nylon became lossier dur- ing heating. The excessive coupling probe depth within the cavity at high tempera- tures not only perturbed the nylon loaded cavity but had a nonnegligible influence on the measurements of the empty cavity resonant frequency. Consequently, inaccurate e', was obtained at high temperatures by using the perturbation formula expressed in equation 5.15a. For comparison, published data [98] is displayed in Figure 6.12a and 6.12b. As shown, below 65° the measured data agrees with published data to within several per- cent. The decrease of the accuracy at higher temperatures is probably due to the ther- mal expansion and deformation of the nylon cylinder, changes of the cavity wall tem- perature, cavity air temperature and humidity, in addition to the effects of the exces- sive coupling probe depth within the cavity. To fully understand the causes of the inaccuracy of the measured 8,, a careful study of each of the possible factors men- tioned above is required. This study is not within the scape of this dissertation. The heating experiments with the nylon rod demonstrated that the complex dielectric constant can be approximately measured during processing using single mode excitation and cavity perturbation theory. By observing the changes in e, or just 291 gm Em asap nuance no cash: uo ..a eHH.m ousofim 33 BE. em mm om QH 0H fig Nd . H o m w N O .r..._._._._._t_._._.___. mm. mgomemeH 0 mmgomebmzm a I Imam << . cram <44 «as 00¢ a a co 00 a lad o a o a < - 000 00 a o a o 000 . 4 tom Imd can“ 292 So go [I H mafia asana> mm scams «0 a .....u nan.» wagons AEEV m2; vm mm om m." @H a; N." CH G “v N O wit _ . _ . _ . _ . _ t _ . _ t _ _ _ _ a _ Aggy ~mmspo;w.bhrm7:. 0 Au.u.w mmgom BDmZH a O 4 4 ,. o . noneno 4. .amwAu O F no nononeno Ilvgu none - nonoxu a ..©.O a a. Imwgu a . l: a . Imw.n a. < . 1: : Lu 293 km N Em. n... 494 .ansuouonsou aaauo> om scahs no .tu 6v amusemmmzme ana.o ouaohm 2: 9: c2 m8 o8 2: 8 E. 8 3 8 3 a Pt-n—n-pp—nn.bbbnnb—npnn—nnbb—b-b——-hhl—-h---n~bhb-—--p mam wagon BDmZH o mhwzzumnpkumze_ 4. . Q mozmmmmmm X igwxw S _ o no tom <<< ( I C > > om soak: uo ..tu nm«.o assuam Q ambimmmzme c2 9: a: a: o3 2: 8 E. 8 9. on 2 o Ehhhb.:_p.t:b-.tbtg._k.ss—u:.F...u-b.bkb:~.~»...__..h 0.0 Sm n mason .szH o 0 3m n mason 9?? j $3.9 mozmmmhmm Imd e AUG 1-«Xnu . 8 . red 4 I: q T a. r04 < 1 < [ma < 4 I -3 :Lw 29S knowing from the experimental measurements of L,, L,, P,, the heating process can be controlled and repeated. This method can be then adopted to build feedback control of the heating process. 6.3.2 Heating of wet wood Figure 6.13 displays the heating results of water-soaked wood rods of radius 0.455 cm and 3 cm length. By means of the experiment technique described in section 5.4.3, the wet wood heating experiments were conducted with input powers of 5 and 8 W. As is shown, in contrast to the case of nylon rod, 3', and e: of wet wood decreased as heating went on. The decrease of the complex dielectric constant is due to the removal of water from wood while heating. As more water was evaporated, the changes in e, leveled out. Hence it is possible to estimate the time when the wet wood is dried by noting the changes in 9, during processing. As shown, for 5 W input power the wood was dry after 12 minutes. For 8 W input power the wood was dry after 6 minutes and the e, is approximately 3.5-1'03. Again, it is seen different input powers resulted in different heating cycles. The coupling efficiency of the material loaded cavity is defined as the percent of the input power that is coupled into material. As a result, the coupling efficiency can be expressed as following [97]: P Eff = 100%x-Fd- = 100%x[l — EQ-j] (6'1) Where P, = power absorbed by the dielectric, P,- = total power absorbed by the material loaded cavity, Q, = unloaded quality factor of the material loaded cavity, Q... = unloaded quality factor of the empty cavity. 296 mafia usuuo> coo: no: no .tu and.m ausuwh , AEEV 688. . wamfimfiafiofimmfimmfimm « o FLPLIPL_LL_._L2L 0 En H mgom 99mg 0 f. EmumgomSm—E a TN_ 4 4 1 <.< a < a It a a T nu filmy O f G o E O 1 o o I: nu o . a TB nv .U f r3 297 asap asauo> poo: 063 no ..t» ama.w assuam 325 can? .vH m.“ NH «HOH m. m h. m n .v m H o r, b b _ _ hiilh.tg _ _ h _ .IH Augu an a mason Sea a E n makes .53: a f 4 4 4 4 4 4 4 red X 4 o 4 190 O 1 . INA o O to H O 10 a o a o a Tom : kw mafia manua> usauaa: 0008 yes no hocafiaauua osdaasou ona.o ausaam AEEV 05E. 298 Emfimfififiofimwbmmemmfio __L___L__u__bu cm 30 n mmeom 5%: o . ... Sm u... mmgom BDmZH 4 u now mos A4 4. m «aaa Inns .4 m wow f mom 4 .. 4 n o . mom aooaoaoo... wmm m H CD CD HE fimtdnoo fiouetot 299 Figure 6.13c shows the coupling efficiency versus time for the wet wood samples. Initially the efficiency was about 90% and then draps to about 70% as heating process continued. It is clear that at constant input power, as drying progressed, the power dissipated in the wood became less, and hence more power was delivered to cavity walls. 6.3.3 Heating of silicon wafer Three-inch silicon wafers were heated with TE?” mode for a variety of input powers. Note that the superscript ’m’ used in the mode designation here and in the later sections is to distinguish the hybrid mode excited in the cavity loaded with slab- shaped materials from the hybrid mode of the cavity loaded with rod-shaped materials and the empty cavity mode, e.g. HE,ll and TIE111 in this case. The experimental setup was similar to the one shown in Figure 5.6. The silicon wafer was placed on a teflon ring 1/8 inches above the bottom of the cavity. The center of the wafer was on the cavity axis. The teflon ring caused negligible perturbation to the empty cavity TE,“ resonance, however, a major perturbation to the cavity was caused by the presence of the wafer. During heating temperatures at two locations were measured by fiber-optic temperature probes. As is shown in Figure 6.14, probe :1 was on a diameter perpen- dicular to the axis of the coupling probe and probe :2 was along the axis of the cou- pling probe. The temperature probes were placed on the top of the silicon wafer such that the temperatures obtained were the surface temperatures of the wafer. However, these temperature readings probably can represent the "inner" wafer temperature because the wafer is thin and has relatively high thermal conductivity. Figure 6.14 displays the heating and cooling of the wafer with temperatures at the two locations. Cooling temperatures were obtained by turning off the input power as the heating temperatures reached equilibrium temperatures. As shown, both heating and cooling curves show exponential rise and decay. The rapid heating rate for rela- tively low power, 5, 7 and 10 W, is significant. Higher input power gives higher 300 onoua apnea an“: made a« I «u .anoua used“ ou .Quaa I “u .uauas soowdaa noswln uo oaaucam «a.» ausoflm L802, .52.? soc_....n co 953; 053822 A335Ev acct. . .4 N o .V—F . N—W - . 0—F p w . w p r P — _ L 1.2?!” ill .1; I , uuimm; u . . III. a» «noun unannsoo wanna ouaa Naygm SHED “Bugs Zugh Nu £50? 30209 213111 ION (3.) a; maJed we; 301 heating and cooling temperatures. Also seen is that the heating rate is greater on the diameter colinear with the coupling probe. It indicates the heating of the wafer was not uniform. The nonuniformity of the heating is because the field patterns of T573, mode is not uniform and the possible near zone effects of the coupling probe. Although the heating is nonuniform, it is not as nonuniform as the impressed fields because the good thermal conductivity of the silicon wafer (0.2 cal/cm.sec.degree) helps smooth out this nonuniform heating. The equilibrium temperatures for l, 3, 5, 7, 10 and 12 W are shown in Figure 6.15. It appears that the equilibrium temperature of the wafer is linearly proportional to the input power. As expected, higher input power resulted in greater difference between :1 and (2. These experiments demonstrate that sin gle-mode microwave cavity applicator can effectively and rapidly heat semiconducting materials. An equilibrium temperature of 170 C was achieved using moderate 12 W input power in less than 6 minutes. The experiments also demonstrates that the inherent nonuniform heating of the single mode cavity fields can be smoothed out if the processed materials have good thermal con- ductivity. 6.3.4 Low power and high power heating of graphite reinforced epoxy fiber slabs These experiments were conducted to study the effects of the graphite reinforced epoxy fiber slab on the field patterns in the cavity applicator as a function of fiber orientation relative to the coupling probe. The microwave heating of the fully cured epoxy-graphite composite materials was also examined. The single-ply graphite rein- forced epoxy samples were 8.15x7.5 cm2 slabs of 2.08 mm thick. An altemating-layer slab was made by gluing two identical single-ply slabs together with fiber orientations perpendicular to each other. 302 anoua usdca nu“: used c« I «u .anoun uanca ou .anoa I .u .uoson panda asnuo> ousuauonfiau asuundaasvm ma.m ousoflh. Loco; cooEm :och. .o 9500: 0539.28 A225 .332. 3a.... .. ... m. ... ._. ... m o . S 0 cm I Na 0 I I a :8 .I . I O r . ... I. O '00—. I . . I O I I03 I O I I O I _ — n — b F b — h — b — - omF (0.) alnlmadwel wnuqmnba 303 A six-inch cavity, as depicted in Figure 6.16, was used in the experiments. As shown, with the diagnostic holes, the radial electric field pattern IE,I2 on the cavity walls can be measured around the cavity circumference and along the cavity axis. T5711, T1553, and TE?” modes were excited at 2.45 GHz in these experiments. The epoxy-graphite composite was placed either near the top or on the bottom of the cavity for a variety of fiber orientations with respect to the coupling probe. A 1/8 inch teflon ring was used to support the composite sample such that the sample did not contact the conducting surface. Again, the teflon ring itself caused negligible field perturbation to the cavity. Any perturbation to the empty cavity was then considered mainly due to the presence of the composite sample. During heating, temperatures at different loca- tions were measured using a fluoroptic thermometry. The temperature probes were placed underneath the sample such that the temperature readings obtained were the sur- face temperature of the composite. Field pattern measurements around the circumfer- ence of the cavity were made from 0" to 180° to study the field patterns in the cavity and the microwave heating of the composite sample as a function of o. The radial field patterns were measured by inserting a microcoax probe (E-probe) described in section 5.2.2 into the diagnostic holes on the cavity walls. The coupling probe was located at o = 0°. Figure 6.17a shows the comparison of 1155,!2 of TEfiz mode around cavity circumfer- ence between empty cavity and graphite-epoxy loaded cavity. The sample was placed on the bottom of the cavity and the input power was kept constant at 20 mW. Meas- urements of the fields were made under critical coupling conditions and at z=2.8 cm height relative to the base plate. As shown, the field patterns for all cases show cos¢ dependence. Due to the material losses, field intensity in the cavities loaded with com- posite sample was lower than that of the empty cavity. The field intensity correspond- ing to the fibers aligned perpendicular to the coupling probe was reduced much more than that corresponding to fibers aligned to the coupling probe. 304 Six-inch cavity loaded with graphite-epoxy slab Figure 6.16 305 .hua>co no eouuon so nude mXOQOIouucaauo . .26.: «22“.: .50 o.« I N as a «some» paced oauuooao Heaven use museum Fomcmouv a of 8 P o... _ on F 2: on on 9 8 o o _ r _ _ . . F _ 4.. _ p _ _ _ t _ m «Tillie! . Iii III... . < < . I. 4 I a On] 1 I u e . I u 00 w .I II I a on _ II I I c I a oowl I o I a 0mm! I 9.26. 0:39:33 I I 38a Sac, 2 dead ole I 2.0.5 «:9: 3 3:93 film I 338 ‘9an ole oonI _ _ _ r _ P _ I _ . _ u _ _ . _ . l. lo ion loop H d 0 I M a low... 1 . m - w . loom Toma I. 00m. 306 It indicates that more microwave power was absorbed by the sample with fibers per- pendicular to the coupling probe. Hence, for constant input power, the quality factor of the graphite-epoxy loaded cavity with fibers perpendicular to the coupling probe is the lowest among all cases shown in Figure 6.17a. In other words, better microwave energy coupling into the graphite-epoxy composite can be achieved when the fibers are oriented perpendicular to the coupling probe. Figure 6.17b depicts the field patterns of the T3112 mode measured at axial posi- tion of z=4.65 cm, with the sample positioned on the top of the cavity. As shown, the field patterns still have cos¢ dependence, and the power readings are larger with fibers aligned to the coupling probe than with fibers perpendicular to the coupling probe. When the composite sample was placed with fiber oriented 45’ relative to the coupling probe, the field patterns were shifted about 45" from those with fibers aligned or per- pendicular to the coupling probe. However, the patterns still show sinusoidal behavior. Obviously, the change of the fiber orientation rotated the field patterns. Low power microwave heating experiments were performed using several hun- dred milliwatts input power. The sample was place on the bottom of the cavity. Fig- ures 6.18a and 6.18b depict the equilibrium temperatures of the graphite-epoxy sample for TE?” and T55}, modes, respectively. As shown in Figure 6.16, equilibrium tempera- tures were measured at fpur different locations on the sample surface. It is seen that the same equilibrium temperatures could be achieved using lower incident power when the fibers were perpendicular to the coupling probe. Apparently, much better EM energy coupling into sample can be obtained as the graphite fibers were oriented 90° to the coupling probe. Also found in Figures 6.18a and 6.18b is that the heating is nonuniform. In addition to the near zone effects of the coupling probe and the non- uniform field pattems, the influence of the anisotropic dielectric properties of the gra- phite composite on the electromagnetic fields may also be one of the causes of the nonuniform heating. 307 .auw>ao no no» no nude axonolauacaauo .ovos «daemh .EU mw.v I N ac t nflwuo> UHOdH OHHHUOHO Hdflvdd nfid.m ohfiuflm A0893 s cm? om. ow? om? amp 0% 0% _ ow . ow . Ilw .Io 01‘ _ _ mI In owl lo— I I d o M m net n—_ml w owl o .Iow mwl Imw . 38a 2 me I I I 398 3 does. I 38.. 2 .283 elo onl _ . _ _ H t _ . _ r _ . _ . _ p * t L Won 308 .hu«>ao uo Eouuon co scan axonanuanncuo .avoe «ensue .coflufimoa ausuauoneou asnuo> ausucuoafiau asduaaawsom amH.m auscum ..onEac omega. .r m . m l org BE com .38.”. 3 oo a low 28 8n .303 8 com a as 8.... .303 2 one x m P I I m . omr I cm x d X G Nu RI 0 o 0 1mm x 4 4 onI F _ _ _ Ion (3°) armoraduraj wngrqugnbg 309 .mu«>ca no Eouuoa co coda hxoaoIauanacuo .aoOE ..uawb .cofiuwnoa ousuauanan asauo> assuanoasou E2auflwawsam an.m ausaam .382... 38a .I n. . x _. oj _ 38 03 .onoa 3 o o Io, :2 on... .303 2 0mm a 36 own .392. 2 "one x 91 Imp omI Iom q 3 x 8.. a m 0 x 1mm 4 001 F _ _ _ Ion d nu nnb (30) arniore wai w . al. 3 310 High power heating experments of the graphite—epoxy composite were performed to demonstrate the variation of heating temperature as a function of time for different input powers and fiber orientations. The graphite reinforced fiber epoxy slab was placed on a teflon ring 1/8 inch above the bottom the cavity such that the sample was not in contact with the conducting bottom plate. Heating was conducted using TE’,"11 mode. Similar to the silicon wafer experiments, temperature readings were taken at two different locations on the top of the sample and recorded at 30 seconds interval. The 5 W experiments is depicted in Figure 6.19a. It is intriguing to see that even at a constant input power, different fiber orientations relative to coupling probe gave rise to different heating cycles. As shown, when fibers were oriented perpendicular to coupling probe, higher heating temperatures were resulted. As disclosed in the low power heating experiments, the heating was not uniform. Interestingly, I, was larger than I; as fibers were aligned with coupling probe, while the opposite was found as fiber orientation was 90’ relative to the coupling probe. Figure 6.1% shows the tem- perature vs time curves for the 5 and 10 W experiments, where the fibers were perpen- dicular to coupling probe. As shown, different input powers resulted in different heat- ing cycles. Consequently, for the graphite reinforced epoxy slabs, the heating cycle is not only determined by the input power level but the fiber orientation relative to cou- pling probe as well. According to the above experiments, it is found that the sinusoidal pattern of IE,I2 along the circumference of the cavity will not change with the fiber orientation. But the rotation of the sample (the fiber orientation) will cause a shift of the radial electric field pattern in II direction. The quality factor of the cavity loaded with the graphite- epoxy composite is lower with fibers perpendicular to the coupling probe than with fibers aligned with the coupling probe. As a result, better microwave energy coupling into the composite sample, and hence more efficient microwave heating, can be achieved when the fibers of the composite are perpendicular to the coupling probe. 311 on! 001 our omL .3 m u nasoa uses“ no.0 .083 008850.. co. A3350: 08:. .oswu.asaua> annuauonfiaa a: to 9.28: 0mH.m auscum o— m m n o n . .v n N P 0 Fl . _ _ _ _ _ _ _ _ _ _ 28a 2:38 u \ «u u n 0 ocean mean u I . H N H N 0 0 U U N U U 1 38a 2 .92:ch ”Nb III 30.6 2 .93 can: u: ole I 393 9.0.0 cont “N. ala 00.3 9.30.. .“ Avie F _ _ _ _ _ _ _ _ n _ _ 0A.... 3 _ ION 1.0... low low Ion low Ion. (3°) arnroredurej 312 .onoun panda cud: mafia c« I «a .oaoua panda ou .canuoa I .u .onoua usnsg ou .coanan a“ coaucucauuo nonuu .ofiau asuuo> onsucuonaoa Ama.m ousuah NP OF _ _ _ ZHZOF.YXU muHZo— 910 2.“;b 010 Nuuzfi Ill 0 no.0 xxomo 009.350.. can: *0 9.500... A3325 2t: 0 o 0 _ _ . _ t —N -—O ION low Fe (3.) armoradtuaj 313 6.3.5 Discussion The heating experiments nwith nylon and wet wood rods demonstrate high efficiency microwave heating and the on-line measurements of the approximate com- plex dielectric constant by means of perturbation formula, using single mode microwave cavity. These experiments are controllable and repeatable based on the empirical measurements or by observing the changes in 2,. In addition to the regular lossy dielectric materials, the silicon wafers and graphite-epoxy slabs heating experiments demonstrate that the cavity applicator can be used for rapid heating of semiconducting and composite materials. The heating nonun- iformity resulted from the nonuniform field patterns can be smoothed out if good ther- mal conducting materials are processed. In the graphite-epoxy slab experiments, it has been found that the heating cycle is determined by the input power as well as the orientation of the graphite fibers. As the fibers were oriented perpendicular to the input probe, better microwave energy cou- pling into the composite samples, and hence better microwave heating, was achieved. The low power experiments with the graphite epoxy composite demonstrate that the shift of the fiber orientation will not change the overall field patterns but causes a shift of the patterns in o direction. Chapter 7 SUMMARY AND RECOMMENDATIONS 7.1 Summary of results 7.1.1 Theoretical formulation and numerical analysis The theoretical formulation of a cylindrical cavity loaded with lossy dielectric material was derived. As a central part of the material loaded cavity problem, the classical EM analysis of the homogeneous and coaxially loaded waveguides was reviewed. By means of the mode-matching method, the cavity characteristic equations for cavity-open, cavity-image and cavity-short types, as shown in Figure 1.1, were derived. In addition, a detailed derivation of the stored energy, power absorbed in the lossy dielectric materials and conducting walls and quality factors was presented. The propagation characteristics, i.e. a - B and co - a diagrams, of a coaxially loaded waveguide were studied numerically as a function of a: for eight selected waveguide modes. In addition to the In - B and to - a diagrams, the electromagnetic field patterns were investigated for a wide range of e}, a: and rod radius for TMOI, T50, and HE“ modes. Both slow waves and fast waves were identified in the (I) - B diagrams. Whether a waveguide mode is a slow wave or a fast wave is determined by the excitation frequency and the complex dielectric constant and radius of the dielectric rod. It has been shown for slow waves most of the electromagnetic energy is confined in the dielectric rod region. Consequently, efficient microwave heating can be achieved if the electromagnetic mode is excited as a slow wave in the coaxially loaded waveguide region. The selection of the operating frequency and size of the cavity radius and the radius of the dielectric material are major factors of determining the efficiency of microwave coupling and hence microwave heating. The resonant frequency, resonant length and quality factors of the cavity loaded with lossy dielectric rods were examined numerically as a function of the complex 314 315 dielecuic constant and the size of the load material for the cavity-short type shown in Figure 3.5. TMm, T80" and HIEm modes were considered in these numerical examples. The perturbation formula using quasi-static approximations to the electromagnetic fields within the load material was compared with the exact solutions for TMm mode. The overall quality factor was expressed in terms of the quality factors due to the dielecuic losses and conducting wall losses. As expected, for very low loss dielecuic material, the overall quality factor is determined by the cavity wall quality factor. As the dielectric rod gets lossier, the quality factor due to the dielectric losses becomes dominant and hence the overall quality factor is determined by the dielectric losses. The study of the quality factors as a function of a: for a given a; for TMm, T50" and HIEm modes has shown that a mode that has lowest Q at small It: and hence is best for mic :owave heating for low loss material may not be adequate for heating as material load becomes very lossy. That indicates mode switching may be required to achieve best microwave heating as e: of the load material changes during processing. 7.1.2 Experimental verification of theory and microwave heating of materials Two experimental systems, low power and high power systems, were introduced in this research work. An automatic data acquisition scheme using a data acquisition board and a microcomputer was developed for the high power heating system. Experi- ments conducted with the low power systems were used to verify the theory and to investigate the effects of the anisotropic dielectric materials, e. g. graphite reinforced epoxy slabs, on the electromagnetic field strength and patterns. The high power heat- ing experiments were performed to demonstrate the use of single-mode circular cylindrical resonant cavity applicator as a microwave heating device. 7.1.2.1 Low power experiments The low power verification measurements of the resonant frequency, resonant length, quality factor and field patterns for TM” and HE,“ modes were compared with 316 the theoretical computations of Chapter 4. Load materials included nylon and tefion rods with radii of 0.635 cm and 1.27 cm, and a 0.2 cm inside radius quartz tube con- taining ethylene glycol. Except for the 1.27 cm nylon rod in HE“, mode, the computed resonant fre- quency, ry, resonant length, AL, and the axial field patterns near the end plate show very good agreement with the experimental measurements with error percentages of less than 10%. The mismatch between the experimental and theoretical results for the 1.27 cm nylon rod in HE", mode was probably due to the perturbation of the coupling probe and the access hole to the cavity resonance. It is seen that the theoretical qual- ity factors do not agree with the experimental measurements. The explanations for the disagreement are the possible measurement errors and the nondieal structure of the cavity applicator such as the presence of the coupling probe and access hole. In the low power experiments with the graphite-epoxy composite, it is found that the rotation of the fiber orientation relative to the coupling probe caused a shift of the radial electric field patterns in a direction but the sinusoidal o variation of the field pat- terns remained unchanged. In addition, at a constant input power, the orientation of the fibers affects the radial electric field strength near the conducting walls, and hence the Q of the material loaded cavity. The electric field intensities measured with fibers aligned perpendicular to the coupling probe were smaller than those measured with fibers alinged and 45" relative to the coupling probe. The experimental results indicate that better microwave coupling into the lossy composite material was achieved with the fibers oriented 90' to the coupling probe. That is better microwave heating was obtained with the fibers perpendicular to the impressed electric field. 7.1.2.2 High power heating experiments The microwave heating experiments demonstrated that single-mode cylindrical cavity applicator can be used for efficient and fast microwave heating of regular 317 dielecuic materials, semiconducting materials and composite materials. Load materials employed in the heating experiments included isotropic materials such as nylon 66 cylinders, water-soaked wood cylinders and three-inch silicon wafers and anisotrOpic materials such as graphite reinforced epoxy slabs. The heating experiments with nylon and wet wood cylinders have demonstrated the ability of on-line diagnosis of the complex dielectric properties of materials during processing. The on-line measurement of the complex dielectric constants [96] provides a method of process monitoring to yield processed materials with desired dielectric properties. The heating of wet wood cylinder experiments have shown high coupling efficiencies in excess of 80% using single-mode cylindrical cavity applicator with 8 W input power. The heatin g experiments with silicon wafer have shown that the equilibrium tem- perattn'e is linearly proportional to the input power. 170°C equilibrium temperature was achieved for 12 W input power. The heating and cooling temperature curves show exponential rise and decay, respectively. The temperature measurements at two different locations on the surface of the silicon wafers reveal that microwave heating using single-mode microwave applicator is not uniform. The temperature differences between the two locations were 20°C (11%) for 12 W input power and about 4°C (10%) for 1 W input power. In the experiments with graphite reinforced epoxy slabs, it is found that the gra- phite fiber orientation has important influence on the heating temperatures and patterns. The heating temperatures obtained with the graphite fibers 90° relative to the coupling probe were higher than the temperature measured with the graphite fibers aligned with the coupling probe. Hence better microwave heating was achieved if the fibers were oriented perpendicular to the coupling probe. The heating results also show nonuni- form patterns. In addition to the input power, it is found that fiber orientation relative to the coupling probe also affects the heating cycle. 318 7.2 Recommendations 7.2.1 Theory In previous research [97] it has been demonstrated that the power losses due to the coupling probe were small compared to the losses due to the load material. Hence the overall quality factor would not be influenced much by the exclusion of the quality factor due to the coupling probe in the theoretical model. However the comparison of the theoretical and experimental results for the HE“, cavity loaded with 1.27 cm nylon rod in section 6.2 discloses that the major disagreement between the theory and experi- ments is due to presence of the coupling probe. Because of the near zone effects of the coupling probe, the actual field patterns deviated from the theoretical patterns and thus inaccurate theoretical results of the resonant frequency, resonant length and qual- ity factors were obtained. Hence, It is important to include the coupling probe in the theoretical analysis to account for the near zone effects on the field patterns to improve the accuracy of the theoretical model. The near zone effects may also be reduced by designing a cavity where the near fields of the coupling probe is far from the load material. The quality factor due to the coupling probe should be included in the overall quality factor to increase the accuracy of the theoretical results. 7.2.2 Experiments Experiments have shown efficient and fast microwave heating of low loss to lossy (conducting) materials. The inherent nonuniform heating using single-mode microwave cavity applicator is the major problem of microwave heating. The nonuni- form heating is due to the nonuniformity of the field patterns of microwave cavity resonant modes. Different resonant modes have distinct field distributions and none of them are uniform everywhere in the cavity. Previous works [97] have shown that heating uniformity can be accomplished by placing small load material in the proper position in the single-mode exciting mode fields. For example, a cylinder-shaped load 319 material placed in the center of a cavity could be heated uniformly in TM012 mode, pro- vided the length of the cylinder is less than one quarter guide wavelength. However this method imposes a limitation on the load size. To process materials of larger size uniformly, the cavity size has to been scaled up and the Operating frequency is decreased, i.e. for example, a 18" - 915 MHz cavity system. That allows heating of larger size of material load. One method is proposed in this dissertation to achieve uniform heating without the limitation on the size of the load material. Mode switching technique The idea of mode switching is to select some resonant modes that their resultant heating patterns are uniform. During microwave heating, each of the selected modes is excited alternately by switching the excitation frequency of the power source to the associated r’esonant frequency at a constant switching rate. The cavity length is kept constant in the heating experiments. Idealy the heating will be uniform if the heating modes are properly selected. Another method of mode switching is to keep the excitation frequency of the power source constant but to change the resonant length from one mode to another alternately while processing. However changing the cavity resonant length mechani- cally at a fast rate is much more difficult than changing the excitation frequency elec- tronically. Mode switching technique by means of changing excitation frequency is more feasible. APPENDIX A FORMULAS FOR EVALUATING THE MATRIX ELEMENTS OF EQUATION 3.102 The formulas for evaluating the the inner products that are defined in equations 3.95a, 3.95b, 3.98c and 3.98d are given in the following equaitons. The matrix ele— ments of equation 3.102 are obtained by substituting these inner products into equa- tions 3.100a through 3.100d. In the following equations the superscript "*" represents complex conjugate. (I) Nonrotationally symmetric modes (I: :0) < EI- . e7“ > = (%V‘)( Wise! )(KIKIQ {[ my - all: + Th0 + ovum-:4 + (—) {7[( Arc. + co — B. = fifi-Xc’m) {[ 7.1-Ia + will-L) - Tina - orifi] 1 ‘9: + (79:) {7 [Arm — 1:.» - B.Ir..I -r..I)] + 11% [Cam-r + 1:.I) " 0.01—I +fur)]}} (A2) phhru =2 211—fl— igr _Et_ < , > 2(prp,){[ I( (”WIN I(+mzuoell 320 321 K91 £2 + (2:) {(E)[An(ml + 1:11) " 8.0.3.1 +f:+r)] _ (:23?!) [Cliff-r - 1:1) - Dr! H ‘flir H} H,.}?! =5 1 - . "Phi-1.1 ._°FI_ < I > 2 = $705? M2 0% + Inn) A A n C l (‘FI‘F>='2'K94(PII+I+Pn-r) 7‘ III III =E“%(pnfl+prl) R l e c < Ii,” . Ii,” > = 31m? “’12 Kp’z (PI+1+ PI-I) (II) Rotationally symmetric modes (I: = 0) 1! Wu! (w )(KIKIQ =2 7:" + «Ex/rat - 8m] (Ei,é;-E>=2 (KIIKI’) - KI, 1‘4. — C ..D jwéIuo 1111 (llzxpzx o’i am < H,- , Ii," > = 21:09:“) I (szxelx III" an] (A3) (A4) (A5) (A6) (A7) (A8) (A9) (A10) (A11) 322 1 who < H, . Ii," > =(21tf)( 1C )Iw‘> [it + (fl-ml: - Dom P2 < a," , a,” > = 21I($)2 Ivl2 x3 p’f' ATE ATE e e' _ 0 c =21trc§p’f '7' “TE 1 I c (hiEIhj >=2fi(m)2M2K92PI where A _ J.(x.,a)~.(x.,b> a - N u(szb)ln(sza) - Jn(szb)Nn(‘cpza) _ J.(x,,av.(x.,b) ' N.(x.,bv.(x,,a) - J.(x.,b)~. C _ J.(Kp,a)N.'(Kp,b) " - N.’(Kp,b)l.(xp,a) - J.’(Kp,b)NII(Kp,a) _ J.(x.,av.'(x.,b) ‘ N.’R.(x;,p) p do "oz = 2Real R.( 'b) -"—-R.(x b>—R.'P.(x.,p> dp + j :93- R.(vc.,p)P.' do a 3 2 1 . - = E [R,,(Ic,2b)P,,(xpzb) - R4193)? .(sza)] 2 b nR.