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University Moxjfilms International 300 N. Zeeb Road Ann Arbor, Mi 48106 8411133 A ra n e ta , Jose Camus HIGH TEMPERATURE MICROWAVE HEATING AND CHARACTERIZATION OF DIELECTRIC RODS N orthw estern University University Microfilms International Ph.D. 300 N. Zeeb Road. Ann Arbor, Ml 48106 Copyright 1984 by Araneta, Jose Camus All Rights Reserved 1984 NORTHWESTERN UNIVERSITY HIGH TEMPERATURE MICROWAVE HEATING AND CHARACTERIZATION OF DIELECTRIC RODS A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Electrical Engineering BY JOSE C. ARANETA EVANSTON, ILLINOIS JUNE 1984 Copyright by Jose C . Araneta 1984 All Rights Reserved ABSTRACT High-Temperature Microwave Heating and Characterization of Dielectric Rods JOSE C. ARANETA A rectangular cavity excited by an iris is used to heat a di electric rod to temperatures higher than 1,000 qC and simultaneously used to measure the complex permittivity of the rod material. Schwinger *s variational formulation is used tc- derive the gene ral expressions for the impedances of the equivalent circuit of the rod regardless of its diameter or whether it is in resonance. This is utilized to calculate the complex permittivity and the amount of heat generated in the rod based upon the measured values of standingwave- ratio and position of the minima of the standing wave outside the cavity. An equivalent exciting wave approach for calculating the total electric field intensity inside the rod is described. Conditions for thermal instability and optimum heating are also derived. Experimental results on optimum heating conditions, temperature profile control, variation of complex permittivity with temperature and transient heating characteristics are presented and explained. iii of /J-A^Og , ZnO and NiO PREFACE The measurement of microwave complex permittivity of materials at elevated temperatures is usually done by heating the sample in a conventional oven. This dissertation presents a method whereby the same microwave energy, used to diagnose the complex permittivity, is also used to elevate the temperature of a dielectric rod sample to incandescent range. The technique does not only simplify the system by removing the oven, but it also conserves energy by utilizing the more effi cient process of microwave heating. The method is implemented by a rectangular cavity, excited by an iris and terminated by a variable short-circuit. Chapter I reviews the literature on related work in using the rectangular waveguide and its variants as a heating device. This research work is the first attempt to treat in depth the application of the rectangular cavity as a high-temperature heating device and diagnostics tool, at the same time. Chapter II presents the theoretical analysis. theoretical analysis is the development model for Central to the of a generalized variational the impedances of the rod based on Schwinger's formulation. Electrical engineers are very conversant with electric circuits. It is therefore natural that the theoretical approach taken models the cavity and rod by equivalent circuits and use circuit theory in the iv analysis. In addition, conditions for optimum heating as well as ther mal instability are given. Chapter III describes in detail the components of the measurement and heating system. Experimental results on yS-A^Og, ZnO and NiO are presented and analyzed. The conclusions and suggestions for future research work are co vered in Chapter IV. It is hoped that this work is the "spark" that starts the widespread exploitation of simultaneous high temperature microwave heating and characterization as a technology. I am grateful to all those who made this work possible. My deepest acknowledgment to the university and my family who gave sup port and encouragement. Special thanks to my adviser, Professor Morris E. Brodwin, who guided me with patience and wisdom. I would like to thank Professor Gregory Kriegsmann for his helpful comments on the mathematical aspects and J. 0. Jung for the rod samples. Finally, I would like to thank my wife, Marilene, without whose patience, sacrifi ces and support this work would not have become a reality. JOSE C. ARANETA 1983 V To ALBERTO and ROSA, my parents TABLE OF CONTENTS Page ABSTRACT ............................................................ i i i P R E F A C E ......................... iv LIST OF FIGURES ................................................... X Chapter I. II. I N T R O D U C T I O N ............................................... T H E O R Y ...................................................... 1 12 II—1. Variational Model for the Dielectric Rod in a Waveguide II-2. The Equivalent Exciting Field and Electric Field Intensity Profile ............................... 3 2 II-2-a. Using the Galerkin method II-2-b. Equivalent exciting standing wave II-3. Heat II-3-a. II—3—b. II-3-c. Generated .................................. Thin rod in a lossless cavity Large rod in a lossless cavity Effect of losses in the iris and shortcircuit II-4. Optimum Heating Conditions .................... II-4-a. Optimum values of ^ an<* b^ for thin rods II—4-b. Maximum P^r and E ^ >eq 5^ II-5. Thermal Stability and Temperature Control ...... 6 5 II-5-a. Thermal instability II-5-b. Stable temperature II-5-c. Temperature profile control II- 6 . Materials Characterization ..................... II- 6 -a. Thin rod case II- 6 -b. The general case II— 6-c. Numerically finding II— 6 —d. Numerical evaluation of the Bessel functions vii 7^ Page III. 88 EXPERIMENTAL SYSTEM AND RESULTS III-l.Components of the Experimental System ........... III-1-a. Rectangular cavity applicator III-1-b. The iris III-l-c. Calorimeter III-l—d . Microwave generator III-1-e. Pyrometers III-1-f. The temperature control system III-1-g. Four-probe and single-probe standing wave measurement 91 III-2.Experimental Results of /J-A^O^, ZnO and NiO . . . 1 3 1 III-2-a. Optimum heating conditions III-2-b. Equivalent exciting field III-2-c. Characterization III-2-d. Transient heating response IV. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ............ 1 8 5 REFERENCES .......................................................... 1 9 9 APPENDIX A. Derivation of the Marcuvitz Model for Z ^ + B. Derivation of the Marcuvitz Model for Z ^ — C. Evaluating the Limit of Lim x-*x z->o ° 204 Z ^ 218 Lim f ......... 2 2 8 x!>x I ) ~ » V z b-o v k/ ' <z z' 33 D. Second and Third-order Approximations for Z^1 and Z^2 • 2 3 5 E. Derivation of the Expressions for D ^ , D ^ and Cm of a Centered R o d ............... *.................... 2 4 0 P. On the Limit of the Derivatives of r' as r, r'->o and the Series £ e 3 ns ............................. *....... 2 5 5 G. Shorted Waveguide Integral Equation .................... 2 7 1 H. Infinite Waveguide Equivalent to the Shorted Waveguide. I. Integral Forms of ^ and t0 viii 277 ..................... 2 8 0 Page J. Heat Generated with Lossy Iris and Short-Circuit .... 28 3 K. Expressions for tq , pQ , L. Transient Temperature Profile of the Hottest Zone ... 29 2 M. Implemented Flow Chart for Bisection Technique ...... 2 9 7 N. Operating Instructions for Temperature Controller ...299 0. Four-Probe Detection Signals ................ P. Photovoltaic Detector Response Characteristics ...... 3 0 5 Q. Electric Field Profile and the Impedances of the Rod in a Shorted Waveguide ................................3 0 8 t ' , p' and p ^ ...... 290 302 VITA. ............................................................. 3 1 1 ix LIST OF FIGURES Figure 1. Equivalent Circuit of the Rod ............................ . 2. Co-ordinates Relative to the Waveguide .................... 3a. Equivalent Circuit of the Rod in a Waveguide ............ . 3b. Magnitude of reflection coefficient as a function of dielectric constant ....................................... 4. Rectangular cavity applicator ............................ 5. Thin-Rod C a s e ........... ................................. 6. Thin-Rod with Cavity Losses .............................. 7. Normalized generated heat in the rod as a function of rod resistance and reactance ................................ 8. Normalized distance of the peak of the exciting field from the rod as a function of rod resistance and reactance .... 9. Normalized optimum distance of the short-circuit from rod as a function of rod resistance and reactance ........... 10. The experimental s y s t e m ....... ........................... 11. Rectangular cavity applicator ....................... . 12. Normalized susceptance and conductance vs. normalized iris diameter ............................................. 13a. Side view and split front view of variable iris ......... 13b. Seetional side view and half-front view of variable iris housing and chokes .................................. 14a. Normalized susceptance of variable iris vs. gate excur sion into the waveguide ................................ 14b. Normalized conductance of variable iris vs. gate excur sion into the waveguide Page 15a. Microwave output power calibration scheme ................ 10^ 15b. Microwave output power vs. magnetron anode current of PGM 100 microwave generator .............................. 105 16. True temperature H I 17. True emissivity vs. true temperature .................... 112 18. Control system interconnections .......................... 116 19. Voltage reference, difference amplifier and voltagecurrent converter 118 vs. IRCON pyrometer temperature ........ 20 . Saturable-core reactor control current vs. reference voltage andinput voltage ................................. 121 21. Magnetron anode current vs. reference voltage and input voltage ................................................... 1 22 22 . Microwave output power vs. reference voltage and input v o l t a g e ................................................... 123 23a. Power supplyconnections for temperature controller ....... 12^ 23b. Effect of chan g ing reflections from source side on the four-probe plot of reflection coefficient of the load 128 24. Normalized generated head in the rod, iris and shortcircuit vs. normalized short-circuit distance from rod (ZnO) ...................................................... 135 25. ZnO under the same conditions as in Figure 24 except <*= 0.1078 ............................................ 136 26. Normalized generated heat in the rod and short-circuit vs. normalized short-circuit distance from rod (/J-A^O^). 1^7 27. Normalized generated•heat in the rod, iris and short-cir cuit vs. iris susceptance ................................. 1^ 0 28. Similar to Figure 27 except 29. Normalized generated heat in the rod vs. iris susceptance at different temperatures ................................ 30. Normalized generated heat in the rod vs.normalized short-circuit distance from rod at differenttemperatures . 01= xi 0.1078 .................. 1^*1 -[Ll C. ^ Page 31. Division of heat generated between short-circuit and rod vs. short-circuit distance from r o d ..... ................. 1*J*7 32. Normalized distance of the peak of the exciting field from rod and normalized generated heat in rod and cavity v s . short-circuit distance from rod (jEi-A^Og ) ............. 1 5 0 33. Same as Figure 32 except: ZnO with (X of 0.1078 at 1300’C ....................................................... 1 5 1 34. Rod front temperature minus back temperature and heat gene rated in rod and cavity vs. short-circuit distance from rod ......................................................... t n;< 35. Normalized distance of peak of exciting field from rod, electric conductivity and dielectric constant vs. nor malized short-circuit distance from rod .................... 15 7 36. Normalized electric field intensity A vs. normalized radial distance ............................................. 159 37. Electric conductivity vs. true temperature ................. 16 2 38. Dielectric constant 39. Effect of short-circuit distance from rod on characterization results ............................................... loy 40. Effect of iris size on characterization results ............ 41. Effect of available microwave power on transient generated heat ........................................................ 42. Effect of available microwave power on transient surface temperature ........... ..................................... 1 7 ° 43. Effect of rod diameter on transient generated heat ......... 180 44. Effect of rod diameter on transient surface temperature vs. true temperature .................. 162 xii 169 181 CHAPTER I INTRODUCTION The simultaneous characterization and heating of circular dielec tric rods to high temperatures, with a single microwave source, is the principal result of this research. The technique utilizes a single-mode rectangular cavity applicator, to heat the rod to high temperatures. The cavity is made from a standard rectangular waveguide of the same size as the waveguide connecting it to the microwave source. It is can- pleted by a variable short-circuit on one end and a wall with an iris on the other. The complex dielectric constant of the rod in the cavity is deduced from measurements on the standing wave outside the cavity and the parameters of the cavity, e.g., admittances of the iris and short-circuit, their respective distances to the rod and the diameter of the rod. A portion of the microwave energy in the cavity is converted to heat in the rod. The amount of heat generated at a given point in the rod is proportional to the "equivalent" electric conductivity and the square of the magnitude of the electric field intensity there. source is therefore distributed throughout the rod. The heat Consequently, m i crowave heating is faster and more selective compared to conventional heating, where the heat source is outside the rod. Microwave heating is more selective because high temperatures exist only in the rod and the applicator remains relatively cool. perature is achieved in less time. It is faster because the same tem 2 Heat is generated from a conversion of electromagnetic energy via two possible mechanisms — one is by collisions between the structure of the material and accelerated free charges and the other is by the cyclic alignment and relaxation of dipoles. Macroscopically, the effect of the two mechanisms may be combined and quantified as the "equivalent" electric conductivity of the material. The speed and selectivity of microwave heating result in a more efficient utilization of available energy and a greater throughput of processed rods, compared to conventional heating. The resonant cavity applicator produces electric field intensities that are much higher than the incident field. Thus, high temperatures can be realized in the rod. The simplicity of the rectangular cavity operating with a single resonant mode helps to simplify the characteri zation procedure and the task of maximizing the efficiency of energy conversion. The rod is mounted parallel to the electric field vector at the middle of the cavity, where the electric field is most intense. A simpler applicator is a rectangular waveguide terminated by a matched load. Some examples where this type has been experimentally used are: drying plywood and wallboard [46] , separating pyrite from runof-mine coal by microwave heating [14] and separating conductive and con vective heat transfer mechanisms in a gas-solid particles system [5 ] . The relatively low electric field intensity produced in this type of applicator limits its usefulness to relatively lower temperatures. In addition, it has a lower efficiency than the cavity applicator because the microwave power absorbed by the matched load is not utilized. Waveguide applicators terminated by a short-circuit can potentially realize higher efficiencies than the one terminated by a matched load. This is because the microwave energy that is initially transmitted to wards the short-circuit, past the rod to be heated, is eventually reflec ted back to re-excite the rod; and, the subsequent multiple reflections between the rod and the short-circuit results in a higher efficiency of energy conversion. The shorted waveguide applicator may be considered as a limiting case of the cavity applicator, where the iris is expanded to the size of the cross-section of the waveguide. This type of appli cator can achieve high temperatures if the rod has a high electric con ductivity. Variations of this type of applicator have been experimen tally used to sinter alumina in a microwave induced plasma [3] and to induce chemical reactions at high temperatures, e.g., produce hydrogen cyanide from ammonia and carbon as well as dissociate molybdenum disul phide to liberate sulfur [37]. Temperatures above 1,000*C were achieved in these examples. Rectangular waveguide applicators were modified by inserting lowloss dielectric or metallic parts into the applicator to realize a more uniform heating of the sample. Some examples are the slotted waveguide applicator lined with a tapered low-loss dielectric for drying sheet materials [22], a waveguide partially-filled with a low-loss dielectric for drying continuous filaments [4 5 ] and ridged waveguide applicators for more uniform heating of sheets or large diameter rods [6,44] . The applicators in C2 2 ] and [45] were also proposed for measuring the moisture content of the samples. They are examples of attempts to simultaneously use the applicator for heating and characterization. However, the dielectric filled applicator has relatively low efficiency due to losses in the dielectric inserts. Ridged waveguides have relative ly more complicated electromagnetic fields than the regular rectangular waveguide. Hence, a ridged waveguide cavity is less atractive as a diag nostics tool than the regular rectangular cavity. The extra complications introduced by the presence of higher order resonant modes,in a multi-mode rectangular cavity,[19,20,31,47,50],makes them.;even less attractive as a diagnostics tool. Moreover, it is much more difficult to control the field in a multi-mode cavity than in a single-mode cavity. gy Thus the task of maximizing the efficiency of ener conversion is more complex. Therefore, among the various types of rectangular waveguide applicators, the single-mode rectangular cavity is the best for the simultaneous characterization and heating of dielectric rods to high temperatures. Some examples where they have been experimentally used as a heating device are extraction of oil from tar sand on a small scale [ 7 land direct sintering of refractory materials [ 4] , [ll]. The latter application was the primary motivation of this research. As results, the diagnostics capability was added and the roles that the electrical and thermal factors play in determining the maximum efficiency of energy conversion and stable temperature were established. The capability of in-situ characterization while sintering is very useful in providing insights into the various dynamic processes associa ted with sintering. In general, in-situ characterization is important in giving insights into the nature of physical properties and pro cesses of the rod being heated. The speed of microwave heating offers a means of discriminating against slow processes that are detrimental to proper sintering. The approach used in the analysis of the cavity and rod is to model them by their equivalent transmission line circuit representations and use circuit theory in deriving the pertinent equations for charac terization and heating calculations. An accurate equivalent circuit model fore necessary. for the rod was there Marcuvitz[28 ] gave a variational model representing the red by a T-equivalent circuit. However, the given expressions for calculating the impedances of the equivalent circuit are accurate only when the diameter of the rod is less than 15% of the width of the wave guide. In addition, it is valid only when the rod is notin "resonance". Nielsen[35 2 described a numerical technique which removes the limitation on the rod diameter. His method calculates the reflection and trans mission coefficients of a rod in an infinite waveguide. The equivalent impedances of the branches of the T—equivalent circuit can be derived from the reflection and transmission coefficients of Nielsen. Hence, Nielsen's technique may be used in the equivalent circuit approach of analyzing the cavity and rod. Although Nielsen's method shows an improved representation near resonance, it still lacks the necessary accuracy there. These models assume that the rod is homogeneous, i.e., the electric conductivity and dielectric out the rod. constant are uniform through Some work has been done to calculate the scattered field in a rectangular waveguide, from a rod whose complex dielectric constant vary parabolically along the radial direction [9 J, [ 18] • The moti vation behind these efforts was the characterization of plasmas. This research confined itself to the complete treatment of homo geneous circular rods. The variational model was adopted and the res triction on rod diameter and the inaccuracy near resonance were removed by deriving a generalized variational model. The derivation was based on Schwinger’s formulation[41]. Note that Marcuvitz's model is actually an approximation to Schwinger’s solution. The generalized equations make it possible to implement an approximation of any order regardless of the rod diameter or resonance. A related work [23] presented a numerical solution to the same inte gral equation for the electric field which Schwinger used to formulate his variational formulas for the impedances of the rod. The results presented indicate that any cross-sectional shape of the rod can be treated. However, the variational model was deemed more useful because its direct compatibility with the equivalent-circuit approach offers more computational efficiency. The detailed derivations of the generalized variational model are covered in Appendix A to F and Section XI— 1. dure of choosing the appropriate Section II-1. In addition, the proce approximation is illustrated in Compared to the numerical technique of Nielsen, the variational model is easier to implement and coverges more rapidly. The problem of calculating the electric field distribution in 7 the rod was resolved by using the Galerkin method to solve the same in tegral equation for the electric field, that Schwinger used. This is made possible upon insertion of the expression :of the '^equivalent exci ting field" in the cavity into the integral equation. The "equivalent exciting field” is a standing wave and it is deduced from the equivalent circuit of the cavity and rod. The procedure bypasses the difficult'* problem of determining the Green's function of the cavity. The detailed derivation’s are found in Section II-2 and Appendix G to I. The amount of heat generated in the rod is calculated from the equivalent circuit model. The same approach is used for the amount of heat generated in the iris and short-circuit when they are lossy. Section II-3 derives the pertinent equations. Maximizing the efficiency of energy conversion is an important practical consideration, specially when the output power of the micro wave source is very limited. The size of the iris and the position of the short-circuit are the two convenient variables to change for maxi mizing the efficiency. these two variables. Section III-4 discusses the optimum values of In addition, the relationship between the position of the peak of the equivalent exciting field and the occurrence of maximum efficiency is examined. The problem of thermal instability is also an important practical consideration. A simplified analysis for the transient temperature profile is given in Section II—5. It shows that the rod diameter, the thermal conductivity of the rod, and how the generated heat varies with temperature, are the primary determinants of whether the temperature is stable or not. In addition it also shows that radiation and convec tion losses on the cylindrical surface of the rod are the limiting fac tors that determine the steady-state temperature. When the complex dielectric constant of the rod varies with tem perature the electromagnetic field and the temperature field in the rod are"coupled", i.e., a rigorous solution for the two fields requires the simultaneous solution of the electromagnetic and thermal problems be cause the electromagnetic field depends upon the temperature field and vice versa. The analysis covered in Section II-l to IIr5 treat the problem as "uncoupled". Hence, they are very accurate when the complex dielectric constant changes very little with temperature and also when the temperature profile at steady-state is uniform. Note that only the calculations for the impedances of the rod as well as for the elec tric and temperature fields may become inaccurate. The expressions on the amount of heat generated, optimum heating conditions and the equi valent exciting field are always valid per se because they are expressed in terms of the impedances. They give the correct results when the correct value of the impedances are used. Finding the solution to the coupled problem is very difficult in view of the non-linearity of the resulting differential equations. The difficulty is aggravated if the thermal conductivity and heat capacity varies with temperature. Only few examples exist where the coupled problem is solved, even in an approximate way. The solution to the half-space problem under adiabatic conditions was approximated by the WKB method [38]. The solution to the radial profile of the electromag netic and temperature fields of circular cylindrical induction-heated plasmas was approximated by using a combination of the variational and finite-element techniques [40 ]. The value of the results presented in Section II-5 is at least in showing what important factors determine thermal instability and the stable temperature of the rod. They provided very useful "rules of thumb" in preventing thermal instability and controlling the stable temperature in conducting heating experiments. The complex dielectric constant of the rod is deduced by equating the measured and theoretical admittances of the cavity and rod. The measured admittance is derived from measurements on the standing wave and the theoretical admittance is derived from the equivalent circuit. This gives rise to a transcendental equation whose roots yield the com plex dielectric constant. determine these roots. A modified bisection method is used Section II-6 to describes the details of the characterization procedure. The experimental set-up utilizes a circulator and water load. In i effect, the microwave generator, circulator and water load, together, act as a source of internal impedance ZQ . presented to the generator is always Zq . Moreover, the impedance The interaction between the microwave generator and the applicator is thereby avoided making it possible for the generator to maintain constant output power regardless of the conditions of the applicator. This greatly simplifies the ope ration and analysis of the system. An automatic feedback control system is incorporated into the sytem to maintain the surface temperature of 10 the rod at a desired value. The detailed description of the experimental system and its components are given in Section III-l. Steady-state type of experiments were conducted with the tempe rature maintained by the control system. The following were investi gated: optimum heating conditions, variation of complex dielectric constant with temperature , temperature profile control by short-circuit translation and the relation of the position of the equivalent exciting field to optimum heating conditions. The relative response in time of generated heat and surface temperature were also experimentally investigated. /3-alumina, zinc oxide and nickel oxide rods were used in various experiments. The rods of /$-alumina, zinc oxide and nickel oxide showed prac tically uniform temperature profiles. geneous even at high temperatures. They were therefore almost homo Hence, the various equations de rived, e.g., for heat generated and for characterization, were appli cable . The experimental results were consistent with the theoretical re sults even when the complex dielectric constant varied with temperature. This is because the diameter of the rods used in the experiments were small, making the steady-state temperature profile practically uniform and the rod almost homogeneous. Section III-2 presents the experimental results and their analysis. Some of the notable results are: (1) the optimum iris size is larger when the electric conductivity is higher; (2) the distance of the peak of the equivalent exciting field to the rod axis is small when maximum efficiency occurs; (3) the electric conductivity of /3-alumina in creases as temperature increases while that of zinc-oxide and nickel oxide decreases; independent; (4) the dielectric constant of ZnO is temperature- (5) the dielectric constant of /3-alumina decreases while that of NiO increases as the temperature increases; translation affects the temperature profile; (6) short-circuit (7) transient surface temperature lags in time from the generated heat, indicating higher internal temperatures than surface temperatures. Conslusions and suggested topics for future research are given in Chapter IV. Chapter II THEORY The objective of the analysis that follows is to relate the various factors involved in heating and characterizing a circular rod inside a rectangular cavity applicator. The quantitative aspects of the analysis are primarily confined to the treatment of rods that are uniform in cross-section and homogeneous in its electrical and thermal properties. The approach used to calculate the amount of heat generated in the rod and the optimum heating conditions is to represent the cavity and rod by their equivalent transmission-line circuits and use circuit theory in the analysis. Basically, the same approach is used to arrive at a procedure for determining the electric conductivity and dielectric constant of the rod material. Central to the solution of the two problems is the formulation of an accurate equivalent circuit for the rod. Marcuvitz[28 ] cited a variational model that is only useful when the rod is thin, i.e., the radius is less than l/20th the width of the waveguide. The model is actually derived from Schwinger’s variational formulation[41] for the equivalent impedances of the rod. To remove the small rod limitation of the Marcuvitz model, the generalized variational model based on Schwinger’s formulation was derived. The details are shown in Section (II-l) and Appendix A to F. A third problem considered is the determination of the total field inside the rod. The solution adopted in the infinite waveguide 12 13 applicator case utilizes the Galerkin -method for solving the resulting integral equation derived from an application of Green’s theorem. The procedure requires knowledge of the Green's function appropriate to the given applicator geometry. A similar approach appeared to be impractical when the applicator is a rectangular cavity because the Green's function of the cavity excited by an iris is difficult to find. However, as suggested in Section (II-2), the same approach may be utilized if an equivalent infinite waveguide situation can be found that gives the same total field inside the rod as in the cavity. The problem is one of finding the equivalent exciting field. It turns out that the equivalent infinite waveguide case has two microwave sources, one at each end of the waveguide, giving waves that are related to the multiple reflections inside the rectangular cavity. Section (II-2) analytically shows that the equivalent exciting field approach is correct in the shorted-waveguide applicator case and intuitively suggests its applicability to the cavity applicator. The relationships for calculating the amount of heat generated in the rod, the iris and the short circuit are derived in Section (II-3). Optimum heating conditions are discussed in Section (II-4). The practical problem of thermal instability and how to control it is dealt with in Section (II-5). Finally, the procedure for determining the electric conductivity and dielectric constant of the rod material including the specific computer algorithms used are detailed in Section (II-6). 1^ II-l. Variational Model for the Dielectric Rod in a Waveguide The variational model for circular dielectric rods in a rectangular waveguide cited by Marcuvitz[28] is only accurate when: (1) e* is uniform throughout the rod; 2ttR (2) — r— « A 1, where R is the radius of the rod and X is the free- space wavelength; (3") the value of e* is not in the vicinity of a resonance condition of the model. The following development will show why these three restrictions must be observed in using Marcuvitz's model and will indicate how to extend the model when restrictions (2) and (3) are to be removed. The model was derived by applying Schwinger's variational formulation[4l] for the equivalent circuit parameters of a circular dielectric rod in a waveguide mounted parallel to the E-field. It was assumed in the derivation that e* is uniform throughout the rod — this is the first restriction mentioned above. Schwinger's variational formulation expresses the impedance of the branches of the T-equivalent circuit in terms of e*, the radius of the rod and the frequency of the exciting electric field. Schwinger's notation, the equivalent circuit is, Fig. 1. Equivalent Circuit of the Rod Following 15 where T denotes the reference plane passing through the axis of the rod. The pertinent expressions are[41], <e* - l)k2 (Z11 + Z 12>2j S " /<p2 (x ,z )d S - (e* - ( x ,z ) G '( x ,z |x',z')(j> ( x ’ .z ^ d S ’dS 2------------ E---------- ®------2------------ e------------ (1) ( U j edS)2 2j(e* - l)k2 < a(Zn - Z12) /*q (x ,z)dS - (e* - l)k2//<j)0 (x,z)G'(x,z|x,,z,)(J>0 (x,,z,)dS,dS (2) C /W o d s)2 where, the integration is over the cross-section of the rod and, G! — s i n ~ sin— 1 tea + a a ■ (sin k Iz - z 'I . ~ s i n ^ ^ . s i n "7r-— 1 I ---- 2 _ ----- 1---n=2 n 1 1 ± sin k |z + z 1 '|) 1 j k |z-z'| jic |z+zT| (e n ‘ ± e n ) where, the plus sign is used when i = e and the minus sign is used when i = 0; and, (f>(x,z) = total electric field intensity i{i(x,z) = incident electric field intensity k = 2ir/X 2 ,2 . n* ,2 Kn = - (~ > > *1 “ K s* = complex dielectric constant of the rod 2 tt k = j— , Xg = wavelength in the waveguide (3) 16 The subscripts e and 0, respectively, denote even and odd symmetry about the reference plane T located at z = 0. are as shown in Figure 2. The co-ordinates used The width of the broad side of the waveguide is "a" and the axis of the rod is the line (xn , y t0). R Fig. 2. Co-ordinates Relative to the Waveguide Eq. (3) actually expresses the real part of the Green’s function of an infinite waveguide with even and odd symmetries about the reference plane (x,y,0), respectively factored in. The Green's function of an infinite rectangular waveguide is [41], „ sin S22L sin S H l I ------- £--------G(x,z|x',z') * ^ a n=l n j< |z-z’| e ^ and the real part of the Green’s function is, G'(x,z|x*,z*) = - — K3 . sin — - sin — — - sin a a k Iz - z niTx s i .n_ mrx* 3. ln_ _ jk ;n e- +in=2i, when the time dependence e”'*a)t is used. ' » ,i |Z-Z'| (5) The use of G ’ in Eq. (1) and (2), followed from the definition of an equivalent voltage and current of appropriate symmetry about the reference plane T. It was suggested 17 that the factor 2 on the left-hand side of Eq. (1) and (2) may be removed by using G' in both Eq. (1) and (2) instead of and G q , respectively. Eq. (1) and (2) are evaluated in conjunction with the solution of the wave equation, (V2 + e*k2)<|> - 0 (6) inside the rod. When e*(x,z) is constant, the solutions to (6) take the form[41], 00 *e (*,e) = I {A2m cos m0J2 m (k»r) + A 1sin(2nri-l)0J2m+1(klr)}, EVEN m=0 (7-a) 00 <j>o (r,0) = {B cos(2nri-l)eJ2 jCk'r) + B2msin 2m0J2m(k’r)}, ODD m=0 (7-b) where, (k’)2 = e* k2 . When the rod is at the middle of the waveguide, its axis is at x = a/2, z = 0. In this case, Eq. (7) may be simplified by symmetry considerations about the x = a/2 plane. ij>e (r,0) <|>o(r,0) Thus, CO = £ A 2mcos02m J^Ck'r) m=0 CO = I B2m+1cos(2m+l)0J2nH_1(k,r) m=0 (8-a) (8-b) Note that Eq. (7) and (8) may be considered exact solutions of Eq. (6). In the derivation of the Marcuvitz model, ( Z ^ + Z 12> and - Z^2) were approximated by using only the first terms of 18 <J» and <|)q , in Eq. (7), respectively; hence, what was used in Eq. (1) and (2), respectively, were, (9-a) £ A 0J0 (k’r) <f>Q = B^cos 0 J^(k'r) (9-b) Also note that Eq. (9) express the first terms of Eq. (8) as well. With this observation, we may add a fourth restriction on the accuracy of the Marcuvitz model — it is only accurate when the rod is located at the middle of the waveguide. However, this restriction depends on how significant the neglected terms are, relative to the first term. terms are not significant. When kR is small the neglected Thus, the fourth restriction is really covered by the second, i.e., kR << 1. This is referred to by Marcuvitz's statement[28] that the model is within a few percent error when x0 d/a < 0.15 and 0.2 < — < 0.8; where, X q is the location of the rod axis Using Eq. (5) and (9-a) in (1) gives (Appendix A) nux, (10) where, a = kR * 8 = ^ *" a r In C = 0.57721566490 19 Using Eq. (5) and (9-b) in (2) gives (Appendix B) , . . k 3 a S 2 — <Z U - Lim 3r' 8 r '->0 3z* r 2= (— \ Lim f 8z 2 + ‘ -JL 8 6J0 (e)Y1 (o) - a J 1 (0)Yo (a) a J 1 (e)J0 (a) - BJ q CBJJ j Co ) Z 12> (11) where, r* (x,z|x' ,z') = G ’ (x;,z |x',z') + — Y0 (k|r-r' |) (12) is the difference between the real parts of the Green's function of' the infinite rectangular'waveguide and the free-space Green's function. Eq. (10) and (11) may be called the 1 x 1 approximation. Imposing the restriction kR = a << 1 on Eq. Marcuvitz model for (Zj^ + Z^2) > i*e *» Eq. 7TX^ i 7TX (Z H + Z 12) = -j( 2a + fin( — — sin ttR )csc2 nx 0 7 2 a— ) - 2sin — a (10) results in the (A-42); -h 00 2 I sin n=2 a 2 - ( 2 ) + j ^ O l n 1 CaJ0 (B)J1 (a)-BJ1 (6)J0 (a)] (13) Using the same a << 1 restriction on Eq. (11), the first term of the right-hand side of (11) was dropped, and the second term was approxi mated with a « 1; thus, (Eq. B-22), ., a w 2irR .2 . 2, , N J ( j g ) ( - g ~ ) sin (^xQ /a) (Z11 " Z 12^ (14) 1 2 J l<e> 1 2 a J.(a) 6tJ1(0)J(J(a) - 0Jo (e)J1(a)3 20 This is because normally, the first term is of order a 2 while the second is of order unity when a << 1 (Appendix B and C ) . Let us now evaluate the relative accuracy of the l x l mation. the An idea of. how much was neglected may be had contributions of the next significant approxi by calculating term in the expansion * of and respectively. Hence, for a rod at the middle of the waveguide, cf>e (r,0) - A 0J0 (k’r) + A 2c o s 20J2 (klr) (15) <f>Q(r,0) - B^cos 0J^(k’r) + B^cos 30J^(k’r) (16) and using Eq. (15) in (1) and (16) in (2) (Appendix D ) ; (G* “ l)k Ka ( Z n + z 12) - J (17) D00 “ (D02D20/D22) (e*- l ) k ( Z n - Z i 2) = j C0 ~ C0C2 ^D02 + D2(P ^P22 + C2D00^D22 c J - c ic 3 (D13 + D 3 1 )/D33 + 4 W D 33 (18) <a When, (19-a) C0 >:> I”C0C2 ^ 0 2 + D2 0 ^ D22 + C2D0(/D 22 ^ (19-b) D 00 >:> lD02D20/D22l and a << 1, Eq. (17) reduces to the Marcuvitz model for + When, C2 » '11 |-C1C3 (D13 + D31)/D33 + C3D 11/D3 3 | » |Di3D 31/D3 3 | (20-a) (20-b) 21 Eq. (18) reduces to, (e*-l)k2 C2 (21) 11 Eq. (21) is more accurate than the Marcuvitz model — the term of order a 2 which dropped in D jj (Appendix B and E ) . The general expressions for D mm , D „ and C due to the higher mJo m order terms of Eq. (8) (Appendix E), indicate that their contributions decrease monotonically as the order increases when a << 1. Hence, the accuracy of the Marcuvitz model may be good when a << 1 and Eq. (19) and (20) are true and provided the value of 8 is such that the retained term in is not zero or less significant than the dropped term. The detailed expressions for D q q , d q 2* D20* °22’ C0 ’ C2* D ll* D 135 D31’ D33* C1 and C3 in APPendix E suggest the following when a << 1 (^-UlDoohi/a2 » 0 = * - l ) b o 2 D 20/D22l * ^ ( e * - ! ) 2! ^ ! * 1 / (k a )2 » (22-a) (22“ (s* - 1) 2 | C2D00^D22 “ C0C2 (DQ2 + D2Q) /D22 | (22-b) (e*- 1)|DU | 'x* 1/a2 » ( e * - l ) 2 |c2 | 'x* (1 /k)2 » (e*- 1) ID13D 31 /D3 3 I (e*- l)2 |Dn C2 /D33 - (23-a) (23(D13 + D31 )D33 | (23-b) 0 22 Therefore, when a << 1, the only possible inaccuracy in using the Marcuvitz model occurs when the value of g is such that, 2 "00' >111 £ o' (e*- D | C 0 I =£ a 2/k2 (24-a,b) (e*- D l c J (25-a,b) ^ a 4/k2 C25rc) £ a The extreme implications of Eq. (24) on the l x l ^C2D00^r)22^ approximation are, , c0 = 0 » D 00 " ° * D oo (26-a) D00 ~ (D02D20/D22) (ica)C(e* - D k 2]” 1 CQ = ^D02D 20^D22^ J(Zll + Zl2> while that of Eq. C0C2^P0 2 + P2 0 ^ P 22 (26-b) = Cn = 0 (26-c) (25-a,b) are, CC^ U / D 33) (27-a) Ka(z u - Z 12) j ( e * - l)k2 C1 C lC3^D 13 + D3 1 ^ D 33 (D13D 3 1 /D33 ) » Eq. (25-c) suggests that the term dropped from should be included in calculating (27-b) D 11 " 0 DU = C1 = 0 (27-c) in Marcuvitz's model The conditions D q q = 0, 23 CQ = 0, D jj = 0, = 0, D q q = C q = 0 and = 0, define the conditions of resonance for the 1 x 1. approximation. When terms in = Cj = 0 or D q q = C q = 0, the contributions of the third and <J>q , respectively, will have to be taken into account. Nevertheless, when Eq. (24) and/or (25) are true, the l x l approxi mation and the Marcuvitz model are not accurate. To cite an example, Nielsen[35] compared the results of his numerical approach in determining the reflection (p) and transmission (t) coefficient of a dielectric rod in a waveguide with the results using the Marcuvitz model. The example has (R/l) = 0.0357, (R/a) = 0.05 or a = 0.2243 which is also one of the three numerical examples that Marcuvitz gave [28]. Figure 3 in[35] shows a dip in |p| vs. er in the range 115 < er < 120; Marcuvitz's result exhibited a minimum of [p| ~ 0.51 at w 120 while Nielsen's result showed a minimum |p| ~ 0.45 at er ~ 115. The dip in |p| occurs in the vicinity of the value of 8 that makes Dj^ = 0. Note that = 0 results in ( Z ^ - Z-^) "» thus, the use of the term "anti-resonance” in z n ~ % i 2 ' From Appendix C and E, = 0 implies that, |C8J0(8)Y1(o) - aJ1(3)YQ (a)3 nirx, - oJ1(8)J(J(a)D = 0 (28) * The resonant value of f that makes Z,_- Z,_ infinite satisfies r 11 12 Eq. (28) when nator the l x l 24 approximation is used while it makes the denomi of Eq. (14) zero when the Marcuvitz approximation is used. In general, the two values of e* are not the same. Note that the resonance condition for the Marcuvitz approximation may be expressed to the same degree of accuracy by, aJ1 (8)Y0 (a) - 8J(J(e)Y1(a) = 0 In Nielsen's example, e 'ft r is real. A value of e r (29) that satisfies Eq. (29) is 119, in close agreement with the dip in J p | of the Marcuvitz curve in Figure 3 of[35]. On the other hand, e = 108.9 is a root of Eq. (28). Let us now determine | p | using and (18), and the l x l the more accurate 2 x 2, Eq. (17) approximations and compare the results with Nielsen’s results. The equivalent circuit of the rod in an infinite waveguide is shown in Figure 3. When Z - z12 — > 00 (Z1 i -Z i 2)20 » Figure 3a suggests that Z — >■ oo (z h “z 12^Z0 TO 12 0 Figure 3a. Equivalent Circuit of the Rod in a Waveguide and this implies that \ p \ — >1. However, a more detailed examination of 25 l> gives the correct result. In the example, * Z is real. Kence, Z and Z 12 4- Z 11 are 12 purely imaginary, i.e., reactive. Let, (30) Z 11 + Z12 “ (31) j (x2/Z0) Using Eq. (30) and (31) in, (32) = (ZQ - Z„)/(Z0 + Z J gives, x2 + zo X lJ (33) Ip I = 2-,)l/2 *(S)] The value of when X 2 and X-^ IP I when X^ is infinite is less than those values have the same sign. In addition, the minimum value of |p j occurs in the region where X 2 ral, the value of minimum and have opposite signs. In gene er at resonance is not the same as the one at which j p | occurs. Before proceeding with the calculations, let us determine the accu racy of the l x l approximation relative to the 2 x 2 approximation. Since a — 0.2243, Eq.(22) and (23) suggest that the l x l is accurate enough when the rod is not near resonance. the only condition not satisfied is Eq.(23-a) because = 108.9 . Thus the l x l approximation In this example, 0 at approximation'is sufficiently accurate 26 for calculating X 2 . The only requirement for using the 2 x 2 approximation is the calculation of X^. Eq. (27-b) shows that the correction term for calculating X^ is D of order a £ However, the contribution of this term is only, as shown in Eq. (23-a). This means that the contri- bution of the (ZQ /X^) term to J p | is of order a only. At resonance of X^ of the l x l approximation, i.e., using the l x l approximation is 0.331. Therefore, even at resonance, the 1 X 1 er = 108.9, the value of (X2 /Zq ) approximation is accurate in determining [ p |. The resulting value of [p| at er = 108.9 Is 0.296. The results of the calculation for | p | and X^ using the l x l approximation compared to the results when X^ was corrected by using the 2 X 2 approximation agree to within four significant figures. This means that the variational solution has converged beginning with the l x l solution. The convergent solution gives a minimum |p| of zero at er = 110.015. In addition, the values of | p | at er = 115 and 120 are 0.554 and 0.788, respectively while the Nielsen results are 0.45 and 0.73, respectively. Figure 3b graphically compares the results. The expression for the total electric field intensity inside the rod used by Nielsen is the same as Eq. (8) combined. For the example reviewed above, he utilized ten terms in his calculations while the l x l approximation used a total of two only. These facts as well as the convergence of the variational solution beginning with the lxl approximation suggest that the variational solution converges to the exact answer more rapidly and is more efficient computationally com pared to the Nielsen numerical technique. 27 REFLECTION COEFFICIENT •Ot 0.8- 0.6- 0.4- 0.2- NIELSEN — MARCUVITZ 0 40 80 120 DIELECTRIC CONSTANT Fig. 3t>- Magnitude of reflection coefficient as a function of dielectric constant. 28 Knowing the values of at which the "resonance conditions", D00 = 0 c0 = 0 (34-a,b) D 11 = 0 Cx = 0 (35-a,b) occur, is useful in avoiding the inaccuracies of using the l x l appro ximation in these cases. In general, the value of e at which resonance occurs, increases as the diameter of the rod decreases. a = 0.1 and of e r For example, e = 574 when = 141 when a = 0.2 in Eq. (29); the respective values at which D , , = 0 are near these values, 11 In addition, for a given rod diameter, there is an infinite number of values of e illustrated by r that satisfies the resonance condition. the fact that Eq. (29) has an infinite number of The values mentioned above correspond to the zero of Eq. (29) gives This is zeros. first zero.The second = 758 when a = 0 . 2 and er = 3,042 when a = 0.1. The condition a J 1 (8)J0 (a) - 8J0 (g)J1 (a) = 0 (36) m akes, D 11 = c i = 0 When this happens, Eq. (37) (18) is not accurate anymore — of the third term of <j>g in the contributions bave to be taken into account. Eq. (D-12) shows the contributions of the third term and it becomes, (e* — l)k2 (Z,. j J11 -Z.,) "12' = J C3D51D15 ~ C3C4^P31P15 ***D51^°13^ + C4D31D13 Ka - D3 1 (D13D55 _ D 53D15) “ D51 (D13D35 “ D33D15) _ (3 8 ) 29 when = 0. The first zero of Eq. (36) corresponds to e = 1 when a << 1; the second zero gives e = 659 when a = 0.2 and e = 2,637 when r r a = 0.1 while the third zero has e = 7,085 at a = 0.1. = 1,771 at a = 0.2 and Thus, smaller rod diameters correspond to larger values of e at resonance. r The condition, (39) aJ0 (8)J1 (a) - 8J1 (B)J(J(a) = 0 makes. (40) When this happens, Eq. (D-ll) gives 1 (Z 11 + 212> (41) The first zero of Eq. (39) corresponds to er = the second zero gives 1; = 368 when a = 0.2 and e^ = 1,469 when ct = 0.1 while the third zero has e when a, = 0.1. 1 when a « = 1,231 at a = 0.2 and = 4,923 Again, smaller diameter rods have larger values of £r at resonance. Finally D 00 0 (42) 30 when, -h CeJ1(6)Y0 (ot) - a J 0 (3)Y1(a)] + y I j- . sin — n=2 2a „ . mt0 -t— C sin --A a + The l x l 1 n 2 nnX0 — 3 “M * ) CaJ0 (e)J1.(o) - pJ1(g)J0 (a)I = 0 - sin approximation gives Z ^ + Z ^ (43) = 0 and thus becomes erroneous when the value of t is such that Eq. (43) is satisfied; when this occurs r at least Eq. (26-b) or a better approximation will have to be used instead. Therefore, in addition to satisfying a « l,the 1 x.'1 approximation is accurate when the values of er and a are such that Eq. (22) and (23) are true. Otherwise, a more accurate model must be used, e.g., Eq. (17) and (18). The general procedure for calculating for any number of terms is shown in Appendix D. expressions for D|imt> and z n + z i 2 m a y t*1®11 The corresponding necessary to implement the calculation procedure are derived in Appendix E. Z^ —Z^ and Zll + Z12 The appropriate values of calculated for any rod diameter by using a sufficient number of terms dictated by the required accuracy. The degree of accuracy may be estimated by calculating the contribution of the next higher order terms in Eq. (8) that were dropped. The more terms considered the less error incurred because the contributions of the higher order terms decrease monotonically. It may be emphasized at this point that in using any of the approxi mations to z n ~ z i 2 and Zil + Z 12* mentioned above and in Appendix A to F, 31 in conjunction with the transmission line impedance equations, the imaginary number j should be replaced by -j. This will make the above formulas consistent with the transmission line equations since the above formulas were derived with an exp(-juit) time dependence while the transmission line equations were derived using an exp(+ju)t) dependence. Note that the results by Marcuvitz [28] already have this adjustment in the sign. Eq. (22) and (23) may be used to decide on the maximum diameter of the rod for a given required accuracy of the l x l For example, making a 6 =10 “"3 approximation. as the limit gives a maximum a = 0.316; at 2.45 GHz in a WR430 waveguide, this translates to a maximum rod diameter of 1.23 cm. or 0.485 inches. Note that increasing the number of terms in <t>e and <(>q, results in a larger a for a given required accuracy. 2x2 For instance, using the approximation in Eq. (17) and (18) for calculating Z-q + Z^2 and Z H ~ Z 1 2 » respectively, means that dropping the contributions of the -3 third terms when they are of order 10 , will or maximum a = 0.562. give a 12 =10 -3 Again at 2.45 GHz in a WR340 waveguide, this translates to a maximum diameter of 0.863 inches for Eq. (17) and (18) to be acceptably accurate. These guides for the maximum applicable diameter are only valid when the calculations are not made near a "resonance condition". 32 II-2. The Equivalent Exciting Field and Electric Field Intensity Profile The total electric field intensity in the rod may be calculated using the Galerkin method; this requires knowing the Green's function of tlje specific applicator geometry used. The Green's function of a rectangular cavity excited by an iris is not known. Nevertheless, the total electric field intensity in the tod may still be calculated using the Galerkin method, by transforming the cavity problem into an equivalent infinite waveguide problem. The first step of the transformation is the determination of the equivalent exciting field in an infinite waveguide that gives the same total field in the rod when it is inside the cavity. When this is done the Green's function of the infinite waveguide is used with the equivalent exciting field to calculate the total electric field intensity in the rod. The feasibility of using the equivalent exciting field approach in solving the cavity problem is demonstrated in Section II-2-b. There , the conventional approach using the actual Green's function and the equivalent exciting field approach are shown to give the same result in the shorted waveguide applicator. The shorted waveguide applicator is a special case of the rectangular cavity applicator — of the iris is the cross-section of the waveguide. the size The Green's function for this case is known thus making it possible to compare the two approaches. The accuracy of the equivalent exciting field approach is restricted by the condition that the iris and short-circuit are far enough from the 33 rod to make the contributions of higher order mode interactions negli gible. Hence the higher order mode components in the vicinity of the rod will be highly localized. The propagating mode component of the total electric field intensity inside the cavity, when the rod is present, may be calculated by summing the multiple partial reflections in the region between the rod and short-circuit and in the region between the iris and the rod. The sum of all the fundamental wave components propagating towards the rod is the equivalent exciting electric field. It is essentially the sum of all the reflections going towards the rod and the component of the wave coming from the microwave generator that initially arrives at the rod. Utilizing the variational model of the rod and the consequent equivalent circuit approach makes it possible to determine the magnitudes and phases of all the partial multiple reflections inside the cavity without exactly knowing the total electric intensity in advance. This is because the variational model gives the reflection and transmission coefficients of the rod by simply knowing the form of the field inside the rod, i.e., these quantities are independent of the coefficients A m Il-2-a. and B of Eq. (7) and (8). m Using the Galerkin Method In the Infinite waveguide case, the total field intensity, E(r,0), in the rod may be calculated from [41], <j.(r,0) = Ei (r,0) + (e * - l)k2/<f>(r',0 ')G(r,0/r',8 f)dSr (44) 3^ where, TKZ sin(irx/a)eJ E^(r,0) = TE^ q incident wave, E(r,0) = Ay <j>(r,0) G(r,0/r',0') = Green's function dS' = differential cross-sectional area of the rod Ay = unit vector along the y-axis Note that the Green's function is as expressed in Eq. (4) and <p(r,d) is as shown in Eq. (7), i.e., CO <j>(r,0) = \ (a cos n0 + b sin n0)J (k’r) (45) n=0 where, a = A- and b = B_ when n is even and a = A . and ’ n 2m n 2m n 2nt4-l b = B0 .. when n is odd. n 2ntf1 The problem is how to evaluate a n and b using Eq. (44); Galerkin's n method may be used to evaluate a and b . J n n To illustrate, when the rod is at the middle of the waveguide, Eq. (45) reduces to, 00 <j>(r,0) = \ a cos n0J (k'r) (46) n=0 Let, f (r,0) = cos n0J (k'r); Eq. (44) may be multiplied by f 0 and then n ix & integrated over the cross-section of the rod. CO I an/fn M n=0 Hence, CO S = /Ei (r,0)fAdS + I an (e* - l)k2//fnGf£dS'dS (47) n=0 The actual calculation will have to be done for a finite n; since the contribution of the higher the amount order termsdecreasemonotonically, oferror incurredbecomes considered is increased. Let, less as thenumber of terms 35 Fn* ' / W (48) S (49) P* ' /EifJldS hm - (ef - D ^ Z / f ^ d S ' d S Using Eq. (50) (48), (49) and (50) in (47) for an M + 1 number of terms result in. M y n=0 M 7 aH L n n n£ : n=0 a F „ = P. + n nil Z (51) Since f and f. are orthonormal, F „ = 0; hence, n Z 9 nZ ’ ([V - (52) a - p where, CF ] = diagonal ( M + 1) x (M + 1) matrix nn CH 3 => (M + 1) x (M + 1) matrix nJt r a0“ r po i • • p = (53) X. pT i i " a = The coefficients a^ to aM may then be evaluated from Eq. (52). In the shorted waveguide applicator, it is shown in Appendix G that, E..(x,z) = sin(irx/a)Le^tCZ - e jK (z+2b)-j (54) and the Green's function is, G0 (x,z|x',z') - (j/a)I(1/Kn ) sin (n T r x /a ) sin (n iT x ’ /a) -jKn (z+z'+2b) r jKn |z-z’ (55) 36 where the short circuit is at z = -b and exp(+jwt) time dependence was assumed for consistency with the transmission line impedance usage. Note that the first part of Eq. (55) is exactly the Green's function for the infinite waveguide case when exp(+jwt) time depend ence is assumed. Eq. (52) may be used to evaluate the coefficients ag to a^ while Eq. (54) and (55) in evaluating P^, For the more general case of Eq. F' ‘ n£, nn Hn& and H ^ . (45), Eq. H' “1 nil a (52) becomes, n " V (56) > nn J where, Kn£ K’ _ n£_ b n __ - pi_ g = sin n0J (k'r) n n F' nfi. G nn Jen V S "nn - C ^ - U l ' V / g ^ d S - d S = fg2 dS J&n ‘kn - ( * i - m 2 /Jfn G SidS-dS = diagonal sub-matrix P^ “ / fA (57) = /E^g^dS ds The coefficients aR and b^ may then be evaluated directly from Eq. (56). Note that the matrices are 2(M+1) x 2(M+1) in size when the upper limit of n is M. II-2-b. Equivalent Exciting Standing Wave Consider the rectangular cavity applicator of Figure 4. The fundamental mode component of the total electric field intensity may be calculated by replacing the rod and the iris by their equivalent ROD IRIS SHORT z=- z=o 2= 1 Fig. a k- Rectangular cavity applicator. waveguide is "a". The width of the 38 circuits and summing the multiple partial reflections between the iris and the rod and between the rod and the short-circuit, respec tively. Note that the variational model of the rod enables us to calculate the multiple reflections because the impedances of the equivalent T-circuit of the rod are independent of the coefficients a n and b . n With phase taken into account, the sum of all the multiple reflections that are propagating towards the rod may be considered as an equivalent exciting wave analogous to the incident TE^ q wave of the infinite waveguide applicator. In the shorted waveguide applicator, the equivalent exciting standing wave consists of the incident TE^ q wave from the source and the sum of the infinite number of multiple reflections from the short-circuit towards the rod. This is because replacing the short circuit by a matched source producing a wave of the same amplitude and phase as the combined reflections from the short circuit, gives the same excitation on the rod. Note that the calculation for the total field <{>(r,0) in the rod inside the equivalent applicator composed of an infinite wave guide with two different sources at each end, involves the Green's function of the infinite waveguide. In effect, the calculation bypasses the determination of the Green's function appropriate to the given appli cator structure by knowing the equivalent sources in the infinite waveguide structure; and the variational model of the rod enables us to determine the waves from the two equivalent sources as the oppositely directed wave components of the equivalent exciting standing wave in the actual structure. Hence, for the shorted waveguide with co-ordinates shown in Figure 4, and exp(+jmt) time dependence, the equivalent incident wave is, -JK (z+2b) ip(x,z) = sinCnrx/a) e^KZ - 0 ( where (58) -i2KbN PQe > J and P q are the transmission and reflection coefficients, respectively, of the rod. The second term inside the brackets is the sum of the multiple reflections from the short-circuit evaluated at the axis of the rod. For convenience, the amplitude of the wave from the source is unity. As shown in Appendix H, the equivalent Eq. (x,z) to be used in (44) when the Green's function of the infinite waveguide is utilized is, (59) where A is the amplitude and phase of the combined reflections from the short circuit towards the rod. As in Eq. (58), (60) The equivalence between using Eq. (54) and (55) in Eq. using Eq. Eq. (44) and (59) and the Green's function for the infinite waveguide in (44), may be shown by assuming the same <{>(x,z) in either case and 40 then after subtracting the two equations, determining whether Eq. (59) comes out from the combined equation. Hence, <f>(x,z) “ sin(irx/a)£e^KZ - e J|CCz+2b)j + (e* - l)k2/<{>GQdS' (61) <p(x,z) = E! (62) X J6(J« (x,z) + (e*-l)k2/(j>G dS' i where G is as in Eq. (4) with a negative sign on the argument of the exponent and G q is as in Eq. (55). Subtracting Eq. (62) from (61) and rearranging, Ei,eq(x,z) = sin(ux/a)[e3 K Z - e “jK(z+2b)] + (e* - l)k2/<j>(GQ - G)dS’ (63) If E' (x,z) = E. (x,z) of Eq. (59), then the equivalence is i,eq i>aq demonstrated. From Eq. (55), CO Gq - G = -(j/a)^(l/Kn )sin(mrx/a)sin(mrxl/a)e 3Kn(z+z + 2b) (64) To localize the effects of the higher order modes the short circuit will have to be placed far enough from the rod, e.g., b plus several wavelengths away. In this case, Eq. (64) simplifies to, G q - G z (-j/Ka)sin(Trx/a)sin(Trx'/a)e 3K ^Z+Z + 2b) Using Eq. (65) in (63) results in, E! (x,z) k s i n ( W a ) [ e jKZ-e“jK(z+2b)l - (e*-l)k2 (j/i<a)J<{>sin(iTx/z) sin(uxT/a)e 3K (Z+Z + ^ ^ d x ,(jz | (66) 4-1 If is shown in Appendix I that starting with Eq. (44), for an infinite waveguide applicator with source at z = + 00 of the form E^e sin(nx/a) and producing a total field in the rod <j>^, E i CTq -I) ® k2 (e* - 1) (j/ica)/^ ^<Z sin(Trxf/a)dx’dz * M «4 When the source is at z = - “ of the form E2e (67) sin(irx/a) and producing a field cj>2 in the rod, E2p q = k 2 (e* - 1) (j/<a)/<f)2e“^KZ sin(irxf/a)dx'dz ' (68) The equivalent source at z = + 00 for the shorted waveguide appli cator is the actual source and 2 = _ oo has E2 = A. = 1; the equivalent source at Since, <j> = <|>^ + (J>2 (69) the second term in Eq. (66) becomes, - sin(TTx/a)e = (z+2b) (£^ _ i)k^(j /<a)J(<f>^ + <j>2)sin(TTx’/a)e ^ KZ dx'dz' - sin(Trx/a>e“jK(z+2b):(x0 - 1) + APg] (70) However, -j2(cb T0e T0 + Inserting Ap0 = T0 “ 7 7 1 p0 (l + pQe J Eq. (70) T0 -j2 ic b . = 7 7 1 ) (1 + P0e -j2 ic b . > /-7i\ (7 1 ) and (71) in Eq. (66) result in, -jx(z+2b) E! (x,z) x ,eq ’ k sin(irx/a) _ 0 , , -j<2b p0e « sin(trx/a) Ce^KZ + A e (72) kz where Eq. (60) has been used. Comparing Eq. (59) and (72), we find, Ei,eq(x’z) Ei,eq<Z'Z) ■ <73) Moreover, the total field at z = -b in the equivalent infinite waveguide applicator is zero; from Eq. (H-2), (H-6) and (H-12), the total field at z = -b is, -jKb <j>(x,-b) = ^2 (x ,-b) + (xQ + Ap0)sin(irx/a)ea J -j2 < b -jKb ~T0e i j. 1 + P 0e Topoe s in (iT x /a ) + T0 “ -j2<h -j2<b sin(7rx/a)e -jK b ( i + P 0e (74) This should be the case because the total field in the actual applicator at z = -b is zero, being the plane of the short-circuit. Thus the validity of using the equivalent exciting field approach has been shown for the shorted waveguide applicator structure. It is expected that the same approach will yield valid results in the rectan gular cavity applicator case. The equivalent exciting field in the more general case is, -jK (z+2«,2 ) E i,eq ’ n (x z ) 5 _____ , -j2KJl.. (1 - p p .e J 1) 1KZ eJ , + p sp0e (7 5 ) U - P o P . ^ ^ J where the component of the wave from the source transmitted by the iris is made unity for convenience; and where, T q and P q are the transmission and reflection coefficients of the rod, p^ is the reflection coefficient of the iris, p T is the equivalent reflection coefficient of the rod and ^3 short-circuit evaluated at the rod-axis plane, P g is the reflection coefficient of the non-ideal short circuit at the shorting plane, is the distance between the iris plane and the rod-axis and the distance between the shorting plane and the rod axis. The expressions for T q , p^, p' and in terms of the equivalent circuit impedances of the rod, iris, waveguide and short-circuit are derived in Appendix K. In the derivation of Eq. (75), the actual microwave source is at z = +°° and a time dependence of exp(+jwt) was assumed. on the right-hand side of Eq. The first part (75) is the sum of all the reflections from the iris towards the rod and the component of the wave from the source transmitted through the iris and arriving initially at the rod; the second part is the sum of all the reflections from the shortcircuit towards the rod. Eq. (75) reduces to Eq. (59) when there is no iris and when the short-circuit is ideal since the transmission coefficient of the iris becomes unity making p^ = 0 while = b. In the equivalent infinite waveguide representation of the cavity and rod, the source at z = +°° provides the wave, E^ *» ( 1 - P'P^e j ^ K ^l) ^ sin(irx/a)e^K:Z (76-a) while the source at z = - 00 provides the w a v e , E (76-b) When the rod is very thin,making the integral in E q . (44) negligible compared to E^, the total field <{> becomes approximately equal to E^. Therefore, for very thin rods <Kr,e) a E x9 _,n (r,e) (77-a) In effect, Eq.(77-a)states that the total field inside the cavity is approximately equal to the empty cavity case. p* p and This is because P q -»■ 0, -> 1 when the rod diameter approaches zero making ^i eq^X,Z^ aPProach the expression of the empty cavity standing wave. In general, Eq. (75) is used to calculate P^ and while the. infinite waveguide Green's function is used to calculate H ^ , k 0 and Xl)u k ' HX« of Eq. (56). from the solution of Eq. and The total field <J>(r,0) is then determined (56) for the coefficients a^ and bn * In particular, for the shorted waveguide applicator with the rod at the center of the waveguide, Eq. (59) is used in Eq. (49) to evaluate Pj^. Eq. (52) is then used to evaluate the constants an . It is shown in Appendix Q that if the l x l approximation is used, the coefficients and a-j_ may be expressed in terms of the rod impedances. The total field is then given as, r f ( r . S ) ----------------[ (Z1;L + Z12) + (Z1X - Z12)/16 ] <77-b > [ (1 + A) + (Zu - z 1 2 > ( 1 ~ A)/4 1 Jo (k’r) /caQ0 + j ka (ZX1 - Z12) [ (1 - A) <Z1;L+ z12) ~ (1+ A)/4]cos9 Jx (fcfr)j k5 where the amplitude of the wave from the microwave generator is assumed unity for convenience. Eq. (77-b) suggests that if Z - Z the field distribution is dominated by of the heating is confined to much smaller than Z ^ + Z.^, JgCk'r). In this the axial region of the rod and is even ly distributed azimuthally. On the other hand, if Z ^ larger than Z ^ + case, most is muc^ the field distribution is primarily cosO J^( k ’r). In this case, most of the heating is in a region between the axis and the surface. Furthermore, it is mainly confined to the x = a/2 plane be cause of the cosG distribution, which gives zero on the z = 0 plane. The dependence of the field distribution inside the rod on the distance of the short-circuit from the axis of the rod is contained in the term A. Thus Eq. (77-b) clearly shows that the electric field dis tribution, inside the rod, can be changed by translation of the shortcircuit. Finally, the validity of the equivalent exciting field approach is further substantiated by the fact that for both the shorted wave guide and the cavity, the standing waves on either side of the rod, sitting in the equivalent infinite waveguide, are correspondingly the same as those in the respective applicator. h-6 II-3. Heat Generated The total amount of heat generated in the cavity normalized to the available microwave power from the microwave generator, Pn> may be expressed by, Pn = 1 - |p|2 (78) where p is the reflection coefficient of the cavity; and p may be expressed by, 1-yL P = T T ^ <79) where yT is the normalized admittance of the equivalent circuit of the Ij cavity evaluated at the plane of the iris. The available microwave power is the amount of power absorbed by a perfectly matched load, i.e., a load with p = 0. The expression for yT depends upon the appropriate approximation J-i to the T-equivalent circuit impedances of the rod, the size and shape of the iris and the position of the short-circuit. Ideally, no heat is generated in the iris, the cavity walls and I the short-circuit when they are lossless. In this case, Eq. (78) expresses the total amount of heat generated in the rod. II-3-a. Thin Rod in a Lossless Cavity The rod is thin when a << 1 and its T-equivalent circuit can be approximated by a simple shunt element. In particular, Marcuvitz [28] cited that this may be done when R/a < 0.05. 47 The transmission line equivalent circuit of the cavity is shown in Figure 5. The susceptance of the infinitesimally thin iris is ■<------ i ---------> ----- T2 ------- T | §r+Jbr i i • -> T Fig. 5. Thin-Rod Case while the real and imaginary parts of the equivalent shunt element of the rod are gr and br , respectively. When is an odd multiple of a quarter wavelength, the equivalent admittance yL between 0 -0 * (plane of the iris) is, yL = + 1/y (80) where y is the equivalent admittance of the rod and short circuit evaluated at the axis of the rod. Hence, y = gr + jA (81) A = br — coticJ^ (82) where, Substituting Eq. (80), (81) and (82) in Eq. (79) gives, (gr - 1 + Ab±) + j ( A - g rb±) P (83) = (gr + l - A b i) + j ( A + g rb±) 48 Using Eq. (83) in (78) gives, (84) P •n (gr + 1 - Ab±)2 + ( A + g rb ±)2 as the total amount of heat generated in the rod normalized to the available microwave power P . a Note that if i|»(x,z) is the TE^ q wave from the microwave generator, the available microwave power is, (85) where Zq is the characteristic impedance of the waveguide and the integration is over the cross-section of the waveguide. II-3-b. Large Rod in a Lossless Cavity When the rod diameter is large, its equivalent circuit is the Tequivalent circuit shown in Figure 1. Consequently, when is an odd-multiple of a quarter wavelength, 1 I/Z12 + Using Eq. (86) “ Z 12 + jtamci2) (8 6 ) in (79), (Rx - R2) + j (XL - X2) (87) P (R1 + R 2) + j(x1 + x 2) where, R^ = (88 -a) X^ = ImZj^ + tamci2 (88-b) R 2 = r 1 (2 r2 + r 1) - x 2 (x3 + x^) - x ^ (88-c) 49 (88 -d) X2 " r2 (x3 + x4 } + r l(2x2 + x3 + x4) (88 -e) rl = Re(Zll " Zl2) (88 -f) x i " Im(Z H - Z 12) (88 -g) r2 = ReZ12 x 2 = ImZ12 (88 -h) x 3 = xx + bi (88 -i) x^ = x^ + tanK&2 (88-j) Using Eq. (87) in (78) gives, p 4(R,R,+X 1X9) -------- --9 - - ------- j (Rj^ + R ^ ^ + (XX + X 2)Z as the total amount of heat generated in the rod. (89) When &2 an multiple of a quarter wavelength, Eq. (89) simplifies to, 4R. Pn --------- T - 1 2 (R1 + 1)Z + (x2 + x3)Z (90) When Aj is arbitrary, 1 + jztanKJl, y-r = 3 bi + ----------z + jtanKij (91> where z is the equivalent normalized impedance of the rod and shortcircuit at the rod axis and is expressed by, z = (Z11 " Z 12)(Z11 + Z12) + jZ2jtanK&2 (92) + jtan<i2 Consequently, using Eq. (92) in (79) gives, (z + jtanici.) (1 - j b .) - (1 + jztanic£_) p =, 1-------i-------------- — (z + jtanicJlj) (1 + jb^^) + (1 + jztanicX,2) (93) 50 The magnitude of p in Eq. Note that Eq. (93) may then be used in Eq. (78) to get P . (93) can be used for the thin rod case with y in Eq. (81) equated to 1/z. II-3-c. Effect of Losses in the Iris and Short-Circuit In the actual iris and short-circuit,losses occur; these losses become important especially when rods of very low electric conductivi ties are in the cavity. In this case, a significant fraction of the total heat generated in the cavity is actually generated in the iris and the short-circuit. When rods of very high values of electric conductivities are in the cavity, practically all of the heat generated is generated in the rod. The former case manifested in experiments when the external portions of the iris was warm to the touch while the rod was unable to achieve a high enough temperature that it was not incandescent. In the latter case, the iris was relatively cooler to the touch than in the former case even when the surface temperature O of the rod was already above 1,000 C. When the equivalent admittance of the iris is measured it actually has a positive real part. The real part of the normalized iris admittance which may be denoted by g^ is related to the amount of losses that occur in the iris. Similarly, the measured admittance of the actual short circuit has a finite value and has a real part as well. When the rod is thin, the transmission line equivalent circuit of the cavity is as shown in Figure 6 . 51 L ------------ > i < |gr + jbr + ^b i 11 1 Fig. 6 . Thin-Rod with Cavity Losses Hence, y = g +jb (94) +y. where, (gs + jhs) + j tani<&2 (95) 1 + 3 (gs + j b s)tanKA 2 is the equivalent admittance of the short circuit evaluated at the rod axis. Substituting Eq. (95) in (94) gives, (96) y = (gr + g L) + j(br + b 1) where, ggsec k £,2 81 = (97-a) 1 - 2 b tanicJfc + (g2 + b 2 )tan 2 ic£_ 3 _ ^ Zm S S “ b (1 - tan 2 K£0) + (1 - b 2 - g2 )tanKil- £.___________ S____S________ £_ S 1 - 2b tanK£„ + S Z. (97-b) (g2 + b 2 )tan 2 K &9 S S When 2-^ is an odd multiple of a quarter wavelength, yL = g± + 3 ^ + 1 /y (98) 52 Using Eq. (96) and (98) in (79), Cg(l-g,) - (1 - b b .)] + j[b(l - g.) - g b .] p ----------- 1------------i----------------- 1------------------------------ (99 ) Cg(l + g±) + (1 - bb.)] + j[b(l + g±) + gb±: where, g = gr + gj b = br + b^ (100-a,b) Using Eq. (99) in (78) results in, P 4Cg + g. (g2 + b2): i ^ -------------------- a (101) Cg(i+g±) + a - b b ^ r + tbd+g^^) + gb±r as the normalized total amount of heat generated in the cavity. The normalized heat generated in the rod P nr is a fraction of P . n As derived in Appendix J, Pnr = gr |(l + P ,)(l + P i) / ( l - P ,P ie"j2’<Jll)|2 where p ’ and p^ are as expressed in Appendix K. When (102) is an odd- multiple of a quarter wavelength, Eq. (102) reduces to Pnr = 4gr /| (1+ y + yy±) |2 where y. = J2. g.+ ib.. °i J i When the Pnr = where t (103) rod is thick, it is shown in Appendix J that, Cl- |p f I2 - g i l T ' l ^ l d + P ^ ^ / U l - P ’Pi ^ 2^ 1)!2 d04) * is the transmission coefficient of the rod when loaded by the short-circuit. The expression for t ' is derived in Appendix K. The normalized heat generated on the iris P . is derived in m Appendix J as, 53 Pni - Pn<*l'*L> <105> where gT is the real part of yT ; while the normalized heat generated Xj Lj in the short-circuit is derived as, Pns = S i ^ ' C i + P ^ / C i - P ' P i ^ 2^ 1)!2 <106> Finally note that, P = P J +P + P n ni nr ns (107) Inserting Eq. (104) and (106) in (107) give, Pns/(Pn - Pni) = |p'i2) d ° 8) Eq. (107) and (108) are convenient for determining Pnr when P^ and P ^ are known. 5^ II-4. Optimum Heating Conditions in a Lossless Cavity There are two cavity variables which are convenient to change for maximizing Pnr — iris. position of the short-circuit and the size of the The pertinent iris characteristic related to its size and shape is its susceptance since it primarily determines the magnitude of the exciting field inside the cavity. The contribution of the higher-order modes generated by the iris to Pnr is negligible if the iris is far enough from the rod. The following development will show the optimal short-circuit position and iris susceptance when the rod is thin. The relationship between the equivalent exciting standing wave and the occurrence of maximum P nr will also be discussed, In the following analysis, the iris, short-circuit and cavity walls are assumed lossless: hence P II-4-a. Optimum Values of SL^ an^ = P . n nr ^or Thin Rods The appropriate expression for P^r when the rod is thin and an odd-multiple of a quarter wavelength is Eq. (84). The optimum value of 5,^ may be derived by setting the partial derivative of P with respect to equal to zero. Thus, cot(k & 2 q) = br - b i/(l+ b ? ) where J^ q is the optimum value of (109) • If the rod susceptance were capacitive, JZ^g lies in the range 0 ^ ^ (X /4) plus integral multiples of (X /2). § O If the rod susceptance were inductive, i.e., b r < 0 , &20 lies in the range 55 (X /4) < l 7n ^ (X /2) plus integral multiples of (X /2). 8 & 8 The optimum value of is similarly derived by setting the partial derivative of Eq. (84) with respect to b^ equal to zero. Hence, biQ = (br - cotK 2,0 )/[g^+ (br - cotKJL2)2 U (110 ) is the optimum value of b^. Optimal iris susceptance is inductive if cotKj^ > ^ is capacitive if cotKi^ < br * while it No iris is required if cotK.)!^ = ^r * Total conversion of available microwave power to heat in the rod occurs when = 1 or p = 0. Equating the real and imaginary parts of p in Eq. (83) to zero, simultaneously give, bio = ( l " g r)/gr and Eq. (109), with b^ replaced by b^Q . simultaneously (1U) Solving these two equations results in, cot(ic&20) = br + [gr(l - gr)T* (H2) Eq. (Ill) suggests that 100% conversion of available microwave power to heat is only possible when g S 1. Moreover, Eq. (112) indicates that there are two possible locations of the short-circuit that makes P nr =1. Note that Eq. (Ill) may be derived from the simultaneous solution of Eq. (109) and (110). II-4-b. Maximum P and E. --------- nr----- i ,eq The relationship between the occurrence of maximum and the position of the peak of the equivalent exciting standing wave may be 56 demonstrated by looking at specific examples. When the short-circuit is ideal, pg = -1; and Eq. (75) becomes, T e-jK(z+2&2) O' JKZ E. (*,*) - 3ln(l,x/a> ■■■ 1 >B<* ( l - p ' p ^ - J 2* (113) (l + P Q e ^ 2^ ) Eq. (113) may be written as, t. / E. (x,z)\ = Er eJ (eJ i,eqv + ,Mw e e “jKZ\ ) (114) where, E e*^ = sin(irx/a)/(I - p rp^e 3^K^1). (115—a) M (115-b) . -roe-32lcl2/(l + p0e-32'ci2) The location of the peak of E^ corresponds to the value of z that maximizes the magnitude of the part of Eq. (114) inside the brackets. Denoting this value of z by z^, 2 icz P where n is any integer. = $ + 2 mr The magnitude of E^ (116) at z = z^ is its maximum and it is, |E. ( x , z ) I = E (1 + M) ‘ i ,eq P (117) Note that the iris partly determines E while the short-circuit partly determines $, M and E. Eq. (114) and (115-a) suggest that the optimum iris size optimizes E and 0. Moreover, Eq. (115-b) and (116) indicates that the parameters determining z^ are T q , P q and Thus, $ = 77 + /Tq - 2 k &2 “ / l + poe ^ K^ (118) 57 If the short circuit were not ideal, (119) * = tjL + Zlo " 2lC^2 "/(1 ~ popse~32,ga2) and p s also partly determines M. The angled brackets in Eq. (118) and (119) denote the phase angle of the term inside. Substituting Eq. (118) in Eq. (116) gives, / ( I + pne -j2 K£2) - 2k Z 0 - 2 kz„ => (2n + 1 )* + (120) Eq. (120) indicates that the position of the peak is directly related to the position of the short-circuit by the term 2 k Z ^ and is dependent upon the transmission and reflection characteristics of the rod. The relationship between and the occurrence of maximum P ^ may be illustrated by considering the case when both 2,^ and Z ^ are oddmultiples of a quarter wavelength and when there is no iris, i.e., b^ = 0. Consequently, Eq. (90) becomes, Pmr - 4V [ (1 + V 2 + <*1 + *2>2'] (121) = 4Re{Z11}/|(1 + ZX1)2 | The values of culating x q and pq consistent with Eq. . From Appendix K, •to - 2 2 ^ /CZij 2 - Z 122 + 2 Z n PQ - (2 U 2 - (121) are determined by cal Z122 '12 - D /(Z U 2 v 11 (122) + i) - Z ..2 + 12 11 + 1) (123) Using Eq. (88 ) in (122) and (123), 2 (r2 + jx2) T0* Q l + r1) + j x j [(1 + r^ + 2r2) + jCxj + 2 x2)] (124) 58 1 + p. e"j2Ka2 = 1 - (125) 0 2 ^(1+R^) + j(x 1+ x 2)3 [r^R-j+rg+Z) + 2 r 2 - x 1 (x1+ 2 x2> + l] + j 2 [(x1+ x 2> ( r ^ l ) + r ^ ] (126) since 2k & 2 ” ir (2 n + 1 ) . From Eq. (124), J T q - arctan(x 2 /r2) = a r c t a n Q ^ / d + r . ^ ] a r c t a n f ^ + 2x2)/(l + ^ While from Eq. - + 2r2>J (127) (126), f \ 4- pn e 3 2 k Z 2 = arctanj^x^ + x 2>/(l + - 2 [](x^ + x 2 )(r1 + 1 ) + r 2 Xj3 (128) arctan< + 2 ) + 2 r 2 - ^ ( x ^ + 2 x2) + 1^ Using Eq. (127) and (128) in Eq. (120) and using l 0 and X /4 result ^ in, S 2 xZp *= 2mr + arctan (x2 /r2) - arctan[x^/(l + r-j_)3 ~ arctan[[(x^ + 2x 2)/(l + r^ + 2 r 2 )3 - a r c t a n Q x ^ + x 2 )/(l+R^)j 2 [(x1 + x 2^ ri + 1 ) + r 2 Xj3 (129) — arctan [^(Rj + r2 + 2) + 2r2 - x1 (x1 + 2x2) + l] When the rod impedances are all real, i.e., x^ = x 2 = 0, Pnr in Eq. (121) is maximum and the nearest value of z^ to the rod axis (z = 0) is zero. This means that maximum heat generation in the rod occurs when the peak of E. X 9 ©CJ (x,z) is on the axis of the rod. If the rod were thin, Z ^ - Z ^ — 0 and r^=*x^ — 0. Eq.(121) 59 becomes, 4r ---------- ir ---- T ~ (1 + r2 )2 + x 2 2 p nr <130> Pnr in Eq. (130) is maximized when x 2 = 0. When x 1 » x 2 = 0, Eq. says that the peak of E^ e^(x,z) is on the axis of the rod. (129) Therefore, the same condition for maximum heating is repeated when the rod im pedances are all real— z^ is zero. Consider the case for thin rods where the position of the shortcircuit is to be adjusted to maximize Pn r * het be an odd multiple of a quarter wavelength and assume no iris is used for simplicity; Eq. (84) then holds with b^ = 0. Hence, 4r, P 2 = nr (1 + r 2)2 + x 2^ + & 2^^X2 + X2 + cotfcA^] 2 (r22 + x22^2 (131) w h e r e the impedance and admittance representations of the equivalent shunt element of the rod are related by, gr = r 2 /(r 2 2 + x 22) br = -x 2 /(r 22 + x22) For a given value of r 2 and X 2 > P in. Eq. (132-a,b) (131) is maximized if, coti<£2q = - x 2 /(r 22 + x 22) (133) The position of the short-circuit &2 q for maximum P ^ Eq. should satisfy (133). Since there is no iris and Z ^2 or x^ 4s r^ — 0, Eq. (123) be comes, p n = - 1/(1 + 'Q 2 Z 10) ‘ “ '“12 (134) While Eq. (122) becomes, t0 = 2(r 2 +jx2)/[(l + 2r2) + j2x2J (135) where Z 12 = Z 11 r2 + *^x 2 * Figures (7), (8 ) and (9) show how P nr , (z /X ) and (Zof./X ) rep g zu g spectively vary with (x2 /r2) when J,2 is adjusted to the optimum value & 2 o» r2 *-S use<* 33 fcbe parameter in the graphs. The peak of P nr versus (x„/r„) occurs at (x_/r_) = 0 for any r„; z z z z z the peak is highest with a value of one when r 2 = 1. P nr The same value of occurs for a given (x0 /r_) in both r„ = 0.1 and 10, respectively, z z z For any r2> the peak of the equivalent exciting field is at the axis of the rod when x 2 is zero or infinite. (x2 /r2) occurs at x 2 = r 2 for any r2 - The peak of z^ versus With the same (x2 /r2) , z^ > z^ when r 2 < r2 '. The optimum short-circuit position is a quarter wavelength plus integral multiples of a half wavelength from the axis of the rod when x 2 is zero or infinite; and this is true for any ir2 . The peak of versus (x2 /r2) occurs at x 2 = r 2 for any r2 ; and the peak is higher the smaller the value of r2 . In Figures (7) to (9), x 2 > 0, i.e., the rod is inductive. When the rod is capacitive, the same value of Pnr results for a given [x2 [ and r„; also the same z Z Zp > 0. p results except for a change in sign, i.e., However, the value of becomes the supplement of the cor responding value of the inductive case, i.e., taking the range 0.25 < L. = (JL J* /X ) < 0.5 of the inductive case the corresponding value L § in the capacitive case having the same x2 and r 2 is L c = 0.5 - L^. 1— I------- 1--------- 1-------- 1-------- 1-------- 1-------- 1------- - b - 0 Fig. 7- 2 4 Normalized generated heat in the rod (P 6 x2/r2 ) as a function of rod resistance (rg) and reactance (x£). Ox H - 0.1= - 0.1 0 06 . - 6 x2 / r2 Fig. 8- Normalized distance of the peak of the exciting field from the rod (Zp/\g) as a function of rod resistance and reactance (x£). ^20A Fig. 9- g Normalized optimum distance of the short-circuit from the rod (i20A g ) as a function of rod resistance (r2) and reactance (x2). On U) Thus, the capacitive case has a minimum £^q at = r2 instead of a maximum as in the inductive case. The coordinates shown in Figure 4 indicate that a negative corresponds to a location of the peak of the equivalent exciting field, at the back-side of the rod; the front-side of the rod is facing the iris and the microwave source. Figures 7, 8 and 9 illustrate the effects of the trans mission and reflection characteristics of the rod on optimum P and nr £ 2 q and on the corresponding location of the equivalent exciting field. The variation in Pn r » &20 an<* Zp Pr^mar^ly due to the reactive part of the rod equivalent impedance. The highest heat generation occurs when x 2 = 0 giving - 0. The amplitude of the equivalent exciting field is 1 + M = 2 / (1-Pq ) since a 20 is a quarter wavelength. Using x 2 = 0 in Eq. (1 + 2r2)/(l + r2) . (134) gives, |E.^ eqi = When r 2 = 1, the amplitude is 1.5 while it is 1.091 when r„ = 0.1 and 1.91 when r 0 = 10. 2 2 Note that in this case, P 2 4(1 - (e. [) /r_. ’ i ,e q 1 2 requires a weaker r 2 = 0.1 and nr A smaller r„ corresponds to a larger <?; a larger a 2 exciting field to give the same Pn r - Thus, P^^ at r 2 83 1 are the same. When x 2 ^ 0, the peak of the equivalent close to the axis of the rod at optimum P r * exciting fieldis still Formally, the total heat generated per unit length of the rod is, P P = %f|'<Kx z) I nr a J1 f 1 dxdz (136) where the integration is over the cross-section of the rod and <j>(x,z) is the total electric field intensity that corresponds to the given E^ . Therefore, optimum P ^ corresponds to optimum ]<j>(x,z)|. 65 II-5. Thermal Stability and Temperature Control Under steady-state conditions, Eq. (136) may be generalized to, (137) P P = ( g(?)dV nr a j v where V is the volume of the rod and g(r) is the heating rate per unit volume at a given point, located by r, inside the rod. Thus, (138) g(r) = %|<Kr) \2 a(r) where in general a varies with r. While heat is generated at each point inside the rod, heat is lost by radiation and convection at the surface of the rod. If the rod is secured at each end by a holder, there is an additional heat loss by conduction to the holder. With thin rods this is a small portion of the total heat loss due to the very large surface area compared to the cross-sectional area. When a varies with temperature, in general it becomes non-uniform as the temperature reaches steady-state. In some cases, it goes back to a practically uniform distribution at steady-state as a consequence of a uniform steady-state temperature profile. This is experimentally observed in 6 -Al2C>3 , ZnO and NiO, Initially the heat losses at the surface tend to make the surface temperature lower than the axial temperature since the heat generated near the surface is dissipated first. The resulting temperature gradient makes it possible for the internally generated heat to be conducted to the surface for dissipation by radiation and convection. If the thermal conductivity of the rod is high, the required temperature gradient to conduct the same amount of heat is less. High values of o may actually lead to a smaller <|>(r) inside the rod than on the surface due to attenuation arising from the conversion of electromagnetic energy to heat. For instance, in thin rods with uniform a, <(>(r) is approximated by a jQ(k’r) distribution. very large, k' =» k/e^* When a is becomes a very large complex number and |jQ(k’r)| becomes smallest at the axis and largest at the surface— the reverse of the <J>(r) profile at low a. A practically uniform temperature profile is therefore possible, especially when the rod is thin, if both electric and thermal conduc tivities increase to large values at high temperatures. cause the inverted This is be- (r) profile tends to make g(r) smaller inside than at the surface while the high thermal conductivity tends to make the temperature gradients small. On the other hand, a combination of a rapidly increasing electric conductivity with temperature and a small thermal conductivity will lead to a very steep temperature gradient between the very hot axial region and the relatively cooler surface. It is possible that due to the rapid rise in electric conductivity inside, the amount of heat generated internally is more than what can be conducted to the surface even by the increasing temperature gradient. The internal temperature will then rise rapidly to the point of melting the core of the rod. When this happens, thermal instability has occurred. Theoretically, thermal instability corresponds to an indefinite rise in temperature to an infinite value. Experimentally, it manifests sometimes as melting of the core of the rod, like in B-Al^O^, ZnO and titania or sometimes as an explosion of the rod even before the surface temperature reaches incandescence, like in NiO. A quantitative analysis in Section II-5-a will show that thermal conductivity, rate of increase of electric conductivity with tempera ture, the available microwave power, and the diameter of the rod play significant roles in thermal instability. If thermal conductivity is high enough, it is possible that once the temperature gradient has increased to a value that makes the amount of heat conducted to the surface larger than the amount of generated heat, the internal temperatures eventually drop while those near the surface increase further, due to the heat supplied from the ■il internal regions and the increase in g(r) as temperature increases. This continues until the temperature gradient is reduced to the value equalizing the total amount of heat conducted to the surface and the total amount of heat generated. A stable temperature profile thus re mains at steady-state. It is shown in Section II-5-b that the important factors that affect the steady-.state temperature are the rod diameter, convection and radiation loss, and g(r). Achieving a more uniform temperature profile is a practical ob jective. In addition, a means of maintaining the steady-state temper atures at a desired value over a period of time is also practically desirable. Section II—5-c outlines how the steady-state temperature profile may be controlled to a certain extent by adjusting the avail able microwave power, the position of the short-circuit or by modify ing the thermal-loss mechanisms. II-5-a. Thermal Instability The transient temperature profile in the rod may be calculated by using the heat equation [36]subject to certain boundary conditions; mC ||(r,t) - v[kVT(r,t>] + g£,t) (139) where, m = mass density of the rod Cp = heat capacity of the rod material k = thermal conductivity of the rod T(r,t) = temperature profile g(r,t) = generated heat per unit volume profile In general, m, Cp and k vary with temperature. The boundary condition at the surface of the rod is expressed by, -k(r,T)VT.n - h(r,T)(T-T ) + e(r,T)oT4 - R (140) where, h = convective heat transfer coefficient e «* emissivity of the surface T & = ambient temperature n = outward normal unit vector a - Stefan-Boltzmann constant The first two terms on the right-hand side represent convection and radiation loss, respectively, while the last term, R^, represents the radiative heat transferred to the rod from the walls of the cavity due to the non-zero temperature of the walls. An appropriate boundary condition at the ends of the rod, e.g., T = T q , will also have to be specified. The heat loss due to conduction toward the ends of the rod is usually a negligible part of the total due to the relatively much larger surface area compared to the cross-sectional area. 69 In general, Eq. (137), (139) and (140) will have to be solved simultaneously in conjunction with other boundary conditions on temperature. When o varies with temperature, the above equations will have to be solved simultaneously with the wave equation for the elec tric field and its associated boundary conditions. -X. In this case, the U problem of determining T(r,t) and <f>(r) are coupled. The whole problem then becomes extremely difficult to solve except in a few simple cases, e.g.,C38.40J . An additional impediment to the solution of the coupled problem is the absence of data on how m, C , k and e * of the material of inp r terest vary with temperature especially in the incandescent tempera ture range. For the purpose of uncovering the roles that the pertinent fac tors play in bringing about thermal instability, it may suffice to look at a simple case. Consider a uniformly excited rod, i.e., $(r) is constant relative to r; assume a linear variation of cj(T) with temperature; thus, g(r) = 80 + A T ^ ) (141) where A is a constant of proportionality and g^ is a constant. Note that the assumption of a constant <j>(r) is closely approximated in rods with small cross-sections. Finally, assume that k, m, and C P are con- stant. It is not necessary to calculate the temperature profile through out the rod; it only suffices to examine the region where the maximum temperatures occur. The hottest zone along the axial direction, i.e., z-direction, is normally at the middle and temperature tapers off 70 toward both ends. The peak along the z-direction therefore corresponds &X to ~— = 0. oz -*-> (*0 is uniform, the temperature in the hottest zone Since is expected to vary with the radius only. test zone, Eq. Consequently, in the hot (139) and (141) reduce to, , „.m T?Z r tfr g. \ V T k Va/3t (142) where a = k/mC^ is the thermal diffusivity in the rod. The boundary conditions become, f £ + Yl = r(T), r = R at t > 0 (143) *-v all r at t = 0 (144) where y = (h/k). As derived in Appendix L, the result is, T(r,t) = 0 (r,t) exp£(aA/k)t] - (Sq M ) (145) where, 0(r,t) = £ 1(0 -B)exp(-aB. 2 t) + B exp (-aAt/k)lc.J (6 r) j_qL 0 2 J 3 u 3 — Jfe (r,T) exp (aAt/k) - (g./A)]] u y t=0 00 1 c J (3 r)(aB 2) exp(-a 8 2 t) j= Q J U J J J exp [a (8 j 2 - A/k)x]^ dr (146) F(T) = f o - M T 4 (147) B = (f /y - g ()/A)8j 2 /(3j 2 - A/k) (148) The constants c^ and 8^ are determined by, (149) 3=0 71 YJ q CSj R) - where, = o (150) R (B.) / J (B.r)r dr J 0 •* c = N 3 N(B.) J (151) R ? / Jn (B.r)r dr i=0 ^ (152) Note that Eq. (145) and (146) take into account convection and radia tion loss at the surface. Eq. (146) may be evaluated numerically by the "method of successive approximations" £ 8 ]• The more problematical type of instability that occurs experiment ally is the sudden explosion of the rod even though its surface tempera ture is still below the incandescent range, i.e., T > 800°C. In this case, radiation loss may be neglected, i.e., in Eq. (146) the value of M may be set to zero. Thus, 00 T(r,t) = . ? c.J (B.r){(9 -B)expQ-a(B. - A/k)t] + B} j=0 j ^ j ^ J I (g0 /A) (153) Eq. (153) suggests that T(r,t) -*■ «, when, A/k > Bj2 (154) This means that instability occurs whenever, (AR2 /k) > Z2 (155) where Z is the smallest zero of Eq. (150). Since A °< |0(r)|2=< P P 3 nr , Eq. (155) indicates that thermal in- stability may be prevented by using smaller values of ^aPnr and/or smaller radius and/or increasing the thermal conductivity of the rod. 72 Smaller values of may be achieved by reducing the output power of the microwave generator and/or detuning the cavity to reduce P nr . Increased thermal conductivity may be achieved by reducing the degree of initial porosity of the rod as in sintering applications. Finally, rods made of materials whose electric conductivity de creases with temperature can never become unstable since in this case A > 0 and Eq. (155) is never satisfied, although in practice there is an upper limit to the temperature levels, e.g., melting point. II-5-b. Stable Temperature Initially, the temperature of the rod increases even when elec tric conductivity decreases with temperature because the total amount of heat generated inside is greater than the total amount of heat losses. Convection loss and particularly radiation loss are very low at the start because the surface temperature is near the ambient temp erature. The elevated temperature levels finally stabilize when the total amount of heat loss is equal to the total amount of heat gener ated. When the temperature of the rod is uniform, the total amount of heat generated with a given g(T) is proportional to the volume of the rod while the total amount of heat losses is proportional to the sur face area of the rod. Since for a given length the volume varies as the square of the radius while the surface area varies linearly with the radius, a given g(T) will result in a lower stable temperature with smaller diameter rods. 73 When the rod is thin and exhibits a practically uniform tempera ture profile, a first approximation to calculating the stable tempera ture is to treat the heat generation and heat dissipation problems as uncoupled. The amount of heat generated may then be calculated inde pendently from the calculation for the amount of heat losses. The value of temperature, assumed uniform throughout the rod, that makes the calculated generated heat equal to the calculated total loss is the calculated stable temperature. Knowing how a and vary with temperature, the formulas shown in Sections II-l and II-4 may be used to calculate P nr as a function of temperature; the total amount of heat generated is then found by multiplying Pnr with the available microwave power Pfl. A very good approximation to the total heat loss, especially in thin rods, is the sum of radiation and convection losses. This is be cause the heat conducted to the surface is much larger than those con ducted to both ends of the rod in view of the much larger surface area compared to the cross-sectional area in addition to the much shorter conduction path to the surface than toward the ends. A first approximation to the amount of convection loss assumes a uniform ambient temperature throughout the length of the rod. When the rod is vertical, the convection loss per unit surface area is em pirically given by [25 ]» convection loss ®* 1.42 AT^^/I/*-^ watts/m^ (156) where, AT =» temperature difference between the surface and the ambient temperature L = length of the rod in meters 74 A first approximation to the radiation loss assumes uniform sur face temperature and that the material radiates like a gray body with a certain amount of emissivity. The radiation loss per unit surface area is then [42], radiation loss = ecT 4 s watts/m 2 (157) where, e = emissivity c Stefan-Boltzmann's constant, 5.67 x 10 ® watts/m -°K^ Tg = surface temperature in °K The rod also receives thermal radiation from the walls of the cavity. Using the simplest model of radiative exchange between two surfaces , the amount of radiant heat received per unit rod surface area is expressed by Eq. (157) where e and T s are replaced by the emissivity and surface temperature of the applicator walls, respect ively. The total heat loss per unit rod surface area is therefore, heat loss a 1.42 A T ^ ^ / L ^ ^ + c ( e T ^ - e T ^ ) s s w w watts/m^ (158) where the subscript s refers to the surface of the rod while w refers to the inside surface of the cavity walls. Eq. (158) may then be used to calculate the total heat losses as a function of temperature by multiplying it with the total surface area. The temperature at which the calculated generated heat equals the calculated total heat losses is the estimated stable temperature as sumed uniform throughout the rod. A more detailed calculation for the stable temperature profile in volves the solution of the heat equation as in Eq. (139) and the 75 accompanying boundary conditions as in Eq. (140). ■A* The solution will be in the form T(r,t) which gives more informa tion than the simplistic calculation above. It indicates the condi tions for the onset of thermal instability in addition to giving the steady-state temperature profile, which is the remaining expression after evaluating the limit as t -*■ 00. Since the problem is still treated as decoupled from the determi nation of <{>(r,t), the expression of to be used in g(r) may be approximated by using the equivalent exciting field approach and the -A. Galerkin method discussed in Section II-2. The magnitude of (f>(r) may be related to the available microwave power by Eq. (85) and to the tuning of the cavity by the equations in Section II-3-c. knowledge on how a and e In addition, vary with temperature Is required. —V To illustrate, consider the case when g(r) Is as expressed in Eq. (141); let the rod be axially translated through the cavity at a constant velocity V q towards the +z direction. Since proximated as constant, the temperature is only radially axially; hence, Eq. (139) becomes, and r'2j + i H + JCT . ( l a w l r ^ + r 8r 3? V a /3z |(j)(r) | is ap expectedto vary r-A-a1 t + ( . „ X 3T _k. KSo ■ a 8T . . } subject to the boundary conditions in Eq. (143) and (144) with the con dition in Eq. (144) extended to all z. In addition, appropriate boundary conditions at the tips of the rod must be specified. Note that gg and A are zero outside the region 0 £ z i L. T(r,z,t) may then be determined by solving the above system of equations. 76 Finally, a more exact analysis will have to solve the coupled problem wherein <{>(r,t) and T(r,t) are simultaneously determined. II-5-c. Temperature Profile Control Increasing P will generally result in increased temperatures _v. and steeper temperature gradients due to higher g(r). a If the rod is vertical and exposed to the air, the upward flow of heated air by convection makes the upper portion of the rod hotter than the lower end. If the rod is very long and actually extend out side the cavity, the ends will have temperatures near ambient values while having an extremely hot zone in the portion exposed to microwave excitation. These are examples of resulting temperature profiles forced by thermal boundary conditions. Modifying the thermal boundary conditions will generally result in a change in temperature profile. Translating the rod down at the right constant rate will counter act the upward heat transfer by convection moving the steady-state hot zone to the middle of the cavity. The same result may be achieved with the rod stationary by eliminating convection entirely through evacuation of the cavity. Turning the applicator 90° to make the rod horizontal may also minimize the skewness of the axial temperature profile. These are some ways of controlling the temperature profile by modifying the convection loss mechanism. Increasing the thermal emissivity of the cavity walls and therm ally insulating the cavity from the outside will reduce the net radia tion loss. Minimizing the amount of radiation and convection loss will re sult in minimum radial temperature gradients and smaller P SL requirement to maintain the same temperatures. If the rod axis were displaced from x = a/2, a non-symmetrical temperature profile along x will result due to the simrx distribution of the exciting electric field. Adjusting the short-circuit farther and farther away from the rod translates the peak of field in the same direction resulting in a the exciting gradually higher surface temperature at the back-side of the rod than at the front-side, i.e., the side facing the iris. Eq. (119) and (120) indicate the direct re lationship between the distances of the peak of the exciting field z^ and the short circuit from the rod axis. Hence a more radially uniform temperature profile may be achieved by putting the rod at the middle of the waveguide and adjust ing the short-circuit position to give the same temperature at the front and back sides of the rod. Adjusting the iris size to optimum will result in higher tempera tures at the same P . a Automatic control of temperature levels at a constant value may be implemented by sensing the surface temperature and/or Pnr and auto matically changing P by negative feedback of the sensed variables. The details of the scheme are discussed in Section III-1-f. 78 II-6 . Materials Characterization The values of o and e r of the rod material heated by the rec- J tangular cavity applicator may be determined by measuring the voltage standing wave ratio and the distance of a standing-wave minimum from the iris plane in the waveguide feeding the cavity. The variation of a and with temperature is similarly deter mined by maintaining the temperature constant and repeating the same measurements on the standing wave outside the cavity at each desired temperature. Incidentally, the same data on the standing wave and the derived values of u and e are used to calculate P , P , P . and P utilr nr n ni ns izing the equations in Sections II-l, II-3 and II-4. Better accuracy in measuring the standing wave ratio and the position of a standing wave minimum is obtained by using a single probe running along a slotted waveguide. The measurements are only done after the iris and short-circuit are adjusted for maximum P^ to maximize the sensitivity of the cavity. A four-probe system, de scribed in Section III-l-g,may be used to quickly determine the oc currence of maximum P , i.e., the magnitude of the reflection coefn ficient of the cavity plotted by the system is minimum when maximum P n occurs, The external standing wave data is actually used to determine the measured equivalent admittance of the cavity. It can be shown from transmission line impedance equations that the normalized measured equivalent admittance is, 79 V = xm + 1b = VSWR - 3 tanK:z0 m -----------------1 - j (VSWR) tamc^-j (160) where Z q is the distance of a standing-wave minimum from the reference plane (iris) and VSWR is the standing wave ratio. Note that attenuation in the waveguide is not taken into account in E q . (160). To minimize the errors in measuring VSWR and z q due to attenuation and the presence of higher order modes near the iris, it is advisable to use the position of the third nearest minimum to the iris and the maximum next to it on the microwave source side. The equivalent admittance at the reference plane is expressed by the variational model of the rod, the actual admittance of the iris and short-circuit, and the cavity dimensions Z ^ and Z ^ - The equivalent circuit of the cavity is used to derive the calculated equivalent admit tance of the cavity. The value of a and that makes the equivalent admittance derived from the equivalent circuit of the cavity equal to y^ are the measured values of electric conductivity and dielectric constant of the rod material. The equivalent circuit of the cavity can only be completely quanti fied after measuring the diameter of the rod, the admittance of the iris and short-circuit, the distances Z ^ and Z ^, the width of the wave guide "a" and the frequency and wavelength of the microwave field in the waveguide. Using the ideal representations of the iris and short- circuit instead of the measured ones will lead to serious errors in de termining a and e^. 80 Depending upon the diameter of the rod relative to the freespace wavelength of the microwave field, the simplest accurate ap proximation to the variational model is used. II-6 -a. Thin-Rod Case The rod is considered thin when Z ^ — Z ^ and the equivalent circuit reduces to a simple shunt element with an admittance jb • + The equivalent circuit of the cavity becomes the one shown in • Figure 6 . The equivalent admittance of the cavity yc is obtained by using Eq. (94) to (96) in Eq. (98) resulting in, [1 + ( S j+ g ^ g i - Cb ] + j [ ( b r + b 1 ) g i + (g r + g 1 >bi ] y c ----------------------- ( g ^ ) + a ( b ^ ) ---------------- (161) where g and b are functions of a and e . r r r The measured value of a and correspond to y = y^; this means that the real parts of y^ and ym are equal and their imaginary parts are also equal. Solving these two equations simultaneously for g^ and br gives, gr = b (g g .)2 + (b - b .)2 &m °i m x " S1 (162) (b - b.) = -b 1 ----------- m 2 1--------- j Cgm - g ± >2 + (bm - b ± )2 (163) Note that an ideal short-circuit makes g^ = 0 and b^ = cotK^* Since Eq. (98) is only valid when is an odd multiple of a quar ter wavelength, a similar restriction applies to Eq. (162) and (163). 81 z i2* t*ie siraPlest approximation to the variational When model for g^ and may be derived from Marcuvitz's approximation [28] to the 1 x 1 approximation, j Z 12 ^ (a/2Xg)csc2 (irx0/a) [2 /a2 (er* - 1 ) - SQ - (er*- 3)/4(er* - 4)] (164) where Z ^ 2 1 = gr + jbr 2 (165) £ sin2 (mrxQ/a)^n 2 - (2a/X)2J ** - n ^ (166) However, Eq. (164) is only accurate when |er*| < 16 and R/a < 0.05. Eq. (164) may be simplified farther if a is very much less than one; hence, (a/X ) * a 2, (er* - 1) <167> giving, gr o' (a2 Xg/a)(o/beq) (168) br ^ (169) (o2Xg/a)(er - 1) Note that in its simplest form is proportional to a and br is pro portional to er while both as the square of the rod diameter. vary For larger values of a better accuracy can be obtained by using Eq. (164) instead. From Eq. (164) and (165), (8a2X /a) , g as— =— 8- 5- (8 + 2a )(a/cite0) r M + N (170) 82 C)er - C - where B = (4Sg + 1)®^ M - C - Be r (171) B[e r2 + (a/meg )2] ) C = 8 + (4Sg + 3)a2 (172-a,b) N 53 Ba/ayeg (172-c,d) Note that gr is approximately the same as in Eq. (168) except for a weak dependence on e^. However, b^ is a strong function of both c and e^. that even while er > 0 , It is possible can be negative if the values of o and/or are large enough. This emphasizes the severe limitation of the useful ness of Eq. (169) because it cannot take into account the possibility of the susceptance becoming inductive at large values of cr and . For instance, if the rod were made of perfect conductor its admittance is purely inductive £ 29 J. Of course, the validity of Eq. (170) and (171) is restricted by R/a < 0.05 and Jer*| < 16. A much better approximation to use is the l x l the variational model expressed by Eq. (10). approximation of The only restrictions are R/a < 0.05 and e^* can be any value except those near resonance. It may be pointed out again that the imaginary number j in Eq. (10), as well as in all the other variational model equations in Section II-l and Appendices A-F, should be replaced by negative j when used in equivalent circuit impedance calculations because the time dependence in the transmission line equations is exp (+jtut) while it is exp(-jwt) in the derivation of the variational model. Hence, - j Z ^ Is just the right-hand side of Eq. (10) divided by 83 In using Eq. Eq. (10) for finding the values of a and e (162) and (163), a computer is necessary to first that satisfy determine the value of B that simultaneously satisfies the two equations. An algo rithm for doing this is described in the next section. Once the root B q is found, cr and er are calculated from, er - j (c / o j b q ) = (B0 /°0 2 " (173) where Bq is in general a complex number. II- 6 -b. The General Case For better accuracy, the T-equivalent circuit of the rod should be used with the values of “ z i2 ^11 + ^12 eva^uate<^ to t*ie desired degree of accuracy. The equivalent admittance of the cavity then becomes, y + jtanicfc (174) yc = yi + i ' T where> yT = (zirzi2)^n +zi2) + Zn z3 * (175) 1 + j ( g s + j b s )ta ru c 4 2-l Z3 = (1 ? 6 ) gg + jCbg + tan< S,2 z^ is the equivalent impedance of the short-circuit at the axis of the rod. If is an odd multiple of a quarter wavelength, Eq. (174) simplifies to, yc = yi + yT ~1 The value of B that makes y a and r and e . r (177) c = y m determines the measured value of denoting this value by Sg> Eq. (173) is used to determine a 81+ II- 6 -c. Numerically Finding 8 ^ 3 g is found by looking for the roots of two functions, namely; F(B) = Re(yc - ym > = 0 (178) G(B) = Im{y (179) c - y } = 0 m On the complex 8 -plane, each of Eq. (178) and Eq. (179)’ should trace at least one curve representing the intersection of the surface that each represents with the complex 8 -plane. The intersection of these two curves define 8 g because it is the value of 8 that satisfies both equations simultaneously, which means that yc = y^. The numerical technique used to find the respective roots of Eq. (178) and (179) is the bisection method. Basically, the roots of a function of a real variable x, say f(x) = 0 , may be found by first specifying an interval of x where the roots are expected to be found. The interval is then subdivided into a number of smaller subintervals. The sign of f(x) is then examined at the ends of the first subinterval; if the signs are the same, a similar procedure is done on the next sub interval. This is repeated for all the other sub-intervals. Once the presence of a root is detected in a particular sub interval, zooming-in on the root is achieved by first evaluating f(xm ) at the middle of the sub-interval x . in Its sign is then compared to the sign of f(x^) at the beginning of the sub-interval x^; if the signs are opposite, f(x') is evaluated where x f = (x + x_)/2. m m m i If the signs of f(x’) and f(x-) are still opposite, the bisection process continues; if m l at a certain point the sign of f(x,f) and f(x,) are the same, f(y ) is m l m 85 evaluated, where y = (x" m m of the sub-interval is x„. 2. + x„)/2. The value of x at the higher end 2 The sign of f(y ) is m that of f(x^); if they are still the same, y' = m (y + x_)/2. If the sign of f(y') m 2 m is then compared with f(y^) is calculated, where still the same as that of f(x^), the bisection process continues on the upper range of the sub interval. If at a certain point, the sign of f(x^) and f(y^f) become opposite, the bisection process reverts to the lower range. The whole procedure is stopped when the value of the function is less than a certain specified small error. The complete details, Including the refinements incorporated in the actual algorithm used, are found in Appendix M. The bisection algorithm is applied to each of F(0) and G(0) . This is done by first specifying a span of |3 1 wherein the phase angle of B, say 0 , is constant, i.e., a radial line on the complex 0 -plane. The bisection algorithm then finds the magnitudes |3^| and | w h e r e F(p^) = 0 and - 0* T^e procedure is repeated at another value of 0 if |0^| is not equal to |0 2 1• The direction in which 0 is incre mented corresponds to a decreasing value of ]3^1 “ I I • tion procedure is stopped when [3 2 1 ” T^ e itera is less than a specified small error. The magnitude of the root Is then |3qI — |6 ^| ^ 1^21 an(* its phase angle is the last specified value of 0 . Note that the bisection method can only detect the presence of an odd number of roots in a given sub-interval so that if an even number of roots actually exists in a sub-interval, the bisection algorithm will not find even one of them. However, this shortcoming can be overcome by specifying a narrower interval. 86 Realistic values of e and a are confined to positive values. Thus, Eq. (173) suggests that only the range 0 > 0 > -rr/4 and 3ir/4 < 0 < ir of thecomplex 0-plane should be explored Ifnegative in looking for Bq. are included, the feasible range expands to 0 > 0 i -ir/2 and ir/2 < 0 < ir. However, note that a root in the range ir/2 - 0 - ir with 0 = 0 ^ and a root in the range 0 - 0 - -ir/2 with 0 = 0 ^ = 0 ^ - ir, correspond to the same phase angle of - jCa/weg) and is therefore expected to have the same magnitude if they correspond to the same a and e . r If whenever roots are found In the first and third quadrants of the complex 0-plane, they correspond to a < 0 , which means that the rod is producing instead of absorbing microwave energy. allowing Thus, even < 0 , realistic values of 0 q are at most only confined to the second and fourth quadrants. II-6-d. Numerical Evaluation of the Bessel Functions A central problem in the calculation of an^ ^ 1 + ^12 is the numerical evaluation of the Bessel functions. In evaluating the Bessel functions of the first kind with com plex argument 0 , the most direct way and the one least subject to accumulation error is to use their power series representation [33], [34]; J(B).l-s=stylg£. m =0 m! (n + m) ! am There is an upper limit to both the magnitude of 0 and the number of terms that can be used. I The actual limits depend upon the degree 87 of precision in which the computer is used. This is because there is an upper limit on the value of a real number that the computer can process; when this is exceeded, an "overflow" condition occurs. Never theless, realistic values of &g are well below those that can cause an "overflow", e.g., |Bq| < 8 . The same comments apply to the computation of the Bessel functions of the second kind, i.e., Yg(ot), Y^(a) and those of higher order. An alternative way for calculating the values of Yq (o) is to use the power series representations in evaluating Y ^ (a) and Y^(a) and then use the recurrence relation, Y . (a) = (2n/a)Y (a) - Y _ (a) n*ri n n-JL to compute those of order two and higher. (181) This procedure is less likely to cause an "overflow" condition in the computer. Unfortunately, a similar procedure using a similar recurrence re lation cannot be applied to the Bessel functions of the first kind be cause accumulation error builds up very rapidly [33} . Accumulation errors can only be made negligible if the values of two consecutive high orders are known and the recurrence relation used to calculate those of lower orders. expansion. Thus, it is more direct to use the power series Chapter III EXPERIMENTAL SYSTEM AND RESULTS A block diagram of the system used In conducting the various ex periments to be described Is shown In Figure 10. The circulator diverts the power reflected from the applicator to the calorimeter; in effect, the applicator is supplied by a matched microwave source because the calorimeter absorbs practically all of the incident microwave power and only a negligible amount is reflected back to the applicator. The interaction between the microwave genera tor and the applicator is also avoided making it possible for the generator to maintain constant power output regardless of the conditions in the applicator. As a consequence, the megnetron in the microwave generator is protected from possible damage that follows after an ex cessive feedback of microwave energy from the load. In the process, the calorimeter and the circulator also serve as a means of measuring the amount of reflected power from the applicator. However, the calorimeter is incapable of following rapid changes in the amount of incident microwave power. Thus, they are only useful in monitoring slowly varying, e.g., less than 0.2 Hertz, reflected power. The four-point probe on a slotted waveguide measures the instan taneous value of the reflection coefficient or equivalently, the in stantaneous value of the equivalent admittance of the applicator. In conjunction with the recorder it plots the instantaneous admittance on a Smith's chart. Under steady-state conditions, one of the four probes may be used as a running single probe for measuring the standing wave 88 feedback STRIP-CHART RECORDER PLOTTER CONTROLLER t CALORIMETER FOUR-PROBE SYSTEM PYROMETER MICROWAVE GENERATOR s.— CIRCULATOR Fig. 10 SLOTTED WAVEGUIDE APPLICATOR The experimental system. 00 VO ratio and position of a minimum of the standing wave outside the ap plicator. Measurement of the instantaneous surface temperature of the rod is made by an optical pyrometer that utilizes a semiconductor sensing element. Steady-state surface temperatures are measured by a pyro meter with higher resolution utilizing a calibrated hot filament. The temperature indicated when the brightness of the filament is the same as that of the rod surface is the temperature of the surface. The instrument is only useful for measuring steady-state temperatures because the adjustment for filament brightness is manual with the human eye as the sensing element. In both instruments, knowledge of the emissivity of the surface is required for accurate measurements. The pyrometer with a semiconductor sensing head is actually used as the controlled-variable sensing component of the closed-loop negative feedback control system that maintains the surface temperature con stant . The applicator is a rectangular cavity made from the same WE.430 brass waveguide that conveys the incident microwave energy. It is tuned by adjusting the size of the iris and the position of the short-circuit. Two types of experiments were performed: (1) steady-state, wherein the surface temperature of the rod is maintained constant throughout the experiment; and (2 ) transient, wherein the available microwave power (P ) is maintained constant throughout the experiment cl In the transient mode, "zero time" is marked by the instant P switched on. SL is 91 The steady-state type of experiments conducted were on: (1) opti mum heating conditions with and ZnO; (2) a and as a function of temperature in g-Al^O^ > ZnO and NiO; (3) the relation of the equiva lent exciting field to optimum heating conditions and the temperature difference between the front and back sides of the rod; and (4) temp erature profile control by short-circuit translation. The transient type of experiments conducted were on the relative response in time of and the surface temperature of B-Al^O^ and ZnO rods. Ill— 1. Components of the Experimental System Detailed descriptions of how the individual components function as part of the system as well as measured data on their performance are presented below. III-1-a. Rectangular Cavity Applicator A detailed diagram of the applicator is shown in Figure 11. Part of the incident microwave energy enters the cavity through the iris. An adjustable short-circuit at the end of the cavity helps to minimize the amount of net reflection of microwave power by the whole applicator structure when it is positioned properly. tance For convenience, the dis between the iris and the rod is fixed. Tuning is also partly achieved by choosing the right size and shape of the iris. The circular iris shown in Figure 11 is of the inductive type. An inductive iris is more practical than a capacitive one because it is less susceptible to arcing. The circular iris shown is fixed in size; therefore, it is less convenient to change than an adjustable inductive iris to be described in the next section. NON-RADIATING TUBE ADJUSTABLE SHORT O SAMPLERI O Fig. 11- Rectangular cavity applicator. 93 The rod to be heated and characterized is inserted into the cavity through the beyond-cut-off tube. The tube allows entry of the sample while preventing microwave energy from leaking out. ' However, it has been found that when the heated rod has a value of a > 0.1 mhos per meter and part of it is inside the whole length of the tube, leakage exceeds safe limits. This usually happens when the rod is stationary because the convective heat flow makes the upper portion of the rod red-hot even when it is not yet inside the cavity. Probably some amount of microwave heating eventually occurs inside the beyond cut-off tube in view of the red-hot rod and the tube acting as a co-axial line with the exposed portion of the rod as the pick-up probe. Translating the rod down at a correct speed keeps the hot zone inside the cavity; this helps to keep the leakage below the safe limits. The two peep-hole tubes on each side of the cavity are also beyond-cut-off tubes. The pyrometers make their measurements through these tubes. The beyond-cut-off tubes are designed to have a normal attenua tion of at least -120 dB. III-1-b. The Iris The circular iris is made of brass shim-stock inches thick. and is only 0.050 It was made thin to make its actual susceptance come close to that of an ideal infinitesimally thin iris which can be calcu lated from the model given in [30] . There is a lower limit greater than zero to the amount of 9^ realizable susceptance with a circular iris; it corresponds to the diameter equal to the height of the cavity. Figure 12 shows the measured values of the iris equivalent admit tance as a function of the diameter. A curve representing the admit tance of an ideal iris made of a perfect conductor and of zero thick ness is also shown for comparison. The equivalent shunt admittance of the iris y^ was measured with the cavity empty and the short-circuit adjusted" to a distance close to 1.25X g from the iris to minimize the contribution of the shorted waveguide to the total admittance at the iris plane. data were used in Eq. (160) to get y^; The single-probe measured y^ was determined after subtracting a small measured value of admittance, due to the shorted-waveguide at the iris plane, from y . Exactly the same pro cedure was utilized to measure this small contribution after the iris was removed. Incidentally, this data was used to determine the measured value of the admittance of the short-circuit at its front face; it is y The s = 1.3566 - j 3.1635. lossy nature of the brass plate manifests in the real part of y^ and in the reduced amount of inductive susceptance compared to the ideal. Figure 12 indicates that when the diameter is increased, the con ductance increases while the departure of the susceptance from the ideal widens. This is because a smaller iris has a larger plate area for intercepting the incident microwave field; therefore, the amount of losses is larger. When losses increase, the equivalent conductance increases and the magnitude of the reflected wave decreases; thus, the equivalent susceptance decreases in proportion to the amount of losses. •-tj LULU O O zz ■ gi < < £§ 8 - 09 DO - 0)0 O N 4- D O < Z 2 < cc o . 0 - + 0.29 Fig. 12- 0.33 0.37 0.41 IRIS DIAMETER/a 0.45 Normalized susceptance (-b^) and conductance (g^) vs. normalized iris diameter, "a" is the width of the cavity; curve is -b^ of ideal iris. 96 An adjustable Inductive iris was designed; and it proved very successful without any arcing even when the available power was at a maximum value of around 800 watts. drawing of the iris. Figure 13 shows a three-view It is basically composed of two sliding gates made of aluminum and chokes built into the housing to suppress arcing at the sliding edges of the gates. The housing is also made of alumi num to lighten its weight; in addition, aluminum has a larger conduc tivity than brass, making the losses smaller,, The measured values of the equivalent admittance of the iris as the distance between the two halves of the gate is varied is shown in Figure 14. The gates were displaced symmetrically from the center in these measurements. The corresponding values of the ideal iris with ideal chokes as calculated from the models in [301 are also shown as a curve for comparison. Exactly the same procedure used in measuring the admittance of the circular iris was used with the adjustable iris. The equivalent admittance was evaluated at the plane of the choke nearest the micro wave generator. As expected the inductive susceptance decreases as the size of the iris increases; the capacitive susceptance at large openings is due to the capacitive effect of the chokes. The inductive susceptance closely follows the ideal up to a maxi mum amount of around -6 . Moving the gate closer results in a drastic departure of the susceptance from the ideal towards a significantly large capacitive value when the iris is practically closed. The en tirely different characteristics at small iris sizes is probably due to resonance phenomena between the chokes and the iris. (a) choke ■'/’ / / .y and " housing /, / ‘i h h gate / 0 o _^1— M v % tri b-fl-H 0 0 HALF-FRONT VIEW SIDE VIEW Fig. 13a- Side view and split front view of variable iris, vo section (!) 'flanges choke flange /housing O housing I 1° !O o SECTIONAL SIDE VIEW ->• section HALF-FRONT VIEW Fig. 13h- Sectional side view and half-front view of variable iris housing and chokes. Mating waveguide flanges shown dashed. NORMALIZED SUSCEPTANCE 99 O) Fig. 14a- Normalized susceptance of variable iris vs. gate excursion into the waveguide (d). tance of ideal iris with chokes. Curve shows suscep 100 O O LlI M 2 4" J(cm) 4 0 (b ) Fig. l^b- Normalized conductance of variable iris vs. gate excursion into the waveguide (d). Line shows conduc tance of ideal iris with chokes. 101 Note that the breakdown in operating characteristic is also marked by the occurrence of very large losses in the iris. This is consistent with the occurrence of resonance. The effects of losses in the iris may be minimized by fabricating it from highly conductive metals like copper and silver. For economy without any significant reduction in performance, silver-plating the iris is more practical. When this is done, the maximum realizable in ductive susceptance is expected to be larger. III-l-c. Calorimeter The amount of microwave power absorbed by the calorimeter is deter mined by measuring the flow rate of the water running through it as well as the inlet and outlet temperatures. The absorbed power P ^ is then calculated from, P c » (T_ - T.)mC 0 i p (182) where T Q and T^ are the outlet and inlet temperatures, respectively; m and C are the mass flow rate and heat capacity of the water, respectP ively. Knowing P , the magnitude of the reflection coefficient and the c normalized power absorbed by the applicator may be estimated from, |pI - < v pa)!s (183) P (184) n - 1 - (P /P ) c a The values of P n calculated from the calorimeter data differed with that derived from the single-probe standing wave data in being 10 2 smaller by asmuch as 38% with ZnO and in being larger by at most 3% with B - A ^ O g in the cavity. The error in the values given by the calorimeter is due to the fact that the cavity has a very sharp frequency tuning characteristic with ZnO inside and a much broader one with B - A ^ O g inside. These are substantiated by data presented in Section III-2. The microwave generator was operating in the pulsed mode wherein the amplitude of the microwave output is varying by a small amount at 120 Hertz— corresponding to the full-wave rectified 60 Hertz supply voltage. The slight variation in the anode voltage of the magnetron re sults in a slight change in output frequency; as a consequence, the output frequency varies about the nominal frequency of 2.45 GHz by ±50 MHz With for each pulse cycle. ZnO inside the cavity, massive absorption by the cavity oc curs only during a very small fraction of the cycle due to its sharp tuning characteristic. With B-AlgOg inside, absorption occurs practi cally during the whole cycle. The very slow speed of response of the calorimeter cannot follow the sudden decrease in reflected power during portions of the cycle in the ZnO case, i.e., the heat capacity of the water and the slow re sponse of the outlet thermometer prevents the detection of the peri odic reductions in reflected power. power registered by In the B-AlgOg Hence, the average reflected the calorimeter is higher than the actual value. case, the difference is due to the small losses in the circulator; thus the amount registered in the calorimeter is a slightly smaller value than the actual reflected power. 103 The large error that occurs in the calorimeter reading when the applicator has a sharp tuning characteristic may be minimized if the microwave generator is producing a single microwave frequency. III-1-d. Microwave Generator The amount of available microwave power from the generator is varied by changing the anode current of the magnetron. values of P a The actual as a function of the anode current was measured by con- necting the calorimeter to the port of the circulator normally feeding the applicator. The other port was shorted. Figure 15 shows the calibration scheme. The single probe on a slotted waveguide measures the reflection coefficient p of the calori meter. The circulator was left in place to take into account the small transmission losses in the circulator; thus P^ is really the output power after the circulator. The magnetron anode current was maintained constant for each set of measurements. P c is calculated from Eq. (182) and P is calculated n a from, Pa - P c/(1 - Ip I2) (185) |p| = (VSWR - 1)/(VSWR + 1) (186) where, The resulting calibration curve is shown in Figure 15. III-1-e. Pyrometers The hot-filament pyrometer type was the Leeds and Northrup Model 8626-C; it is a triple-range instrument with the lowest ranging from DIGITAL VOLTMETER HP W A > SHORTING PLATE ' PROBE PGM-100 MICROWAVE -M V v- SLOTTED WAVEGUIDE GENERATOR CIRCULATOR (a) Fig. 15a- Microwave output power calibration scheme. CALORIMETER POWER (kW) OUTPUT 00 200 ANODE CURRENT(mA) Fig. Microwave output power vs. magnetron anode current of PGM 100 microwave generator. 15t>- 106 1075°C to 1750°C. The manufacturer's manual states that it is accurate to within 10°F of the actual value. Of course, this is only true when the emissivity of the surface being measured is known at the wavelength of the red filter used in the instrument. This is because the filament temperature is calibrated against a black-body temperature. The semiconductor pyrometer type was the IRCON Model 230C; it is a five-range instrument with the lowest ranging from 900°C to 1300°C. The manufacturer's manual states that its accuracy is within 0.75% of temperature up to 2,000°C. It has an emittance knob which should be set to the value of the emissivity of the surface for accurate readings. The silicon photovoltaic detector that senses the thermal radiation from the surface being measured has a band-pass; radiation between the wave lengths 0.6yra and l.Oym are the only ones actually detected. In this regard, the Leeds and Northrup pyrometer is generally more accurate be cause in general the emissivity varies with wavelength. However, the usefulness of the IRCON pyrometer is in measuring in stantaneous changes in temperature making it a suitable component of the surface temperature feedback control system. The instrument pro vides an output voltage directly related to the amount of detected radiation. The speed of response of the instrument is limited to one second; this gives an upper limit to the temporal rate of change in temperature that can be accurately measured. In conducting the steady-state type of experiments both pyrometers are in operationg— the IRCON pyrometer as part of the surface tempera ture control system is focused on a single spot on the rod, and the Leeds and Northrup pyrometer measures t h e .temperature of various 107 section of the rod surface. Thus, the two pyrometers are looking at opposite sides of the rod through their respective peepholes. The Leeds and Northrup pyrometer cannot be used in transient type of ex periments because its manual-knulling procedure takes too much time. The setting of the emittance knob on the IRCON pyrometer is not * important when the instrument is part of the control system. This is because the objective of the system is to maintain the relative sur face temperature constant at a desired value. Since only the relative value and not the absolute value of temperature is important, the emittance knob is usually set to the lowest value of 0.2 to maximize the resolution and sensitivity of the instrument. The rod is always mounted at the middle of the waveguide in all the experiments. This means that the expected temperature profile is symmetrical about the axis along the x-direction, i.e., across the width cf the waveguide. For this reason, the temperature that the two pyrometers are measuring on opposite sides are practically the same. To determine the relationship between the readings of the two pyrometers, linear regression was made on 60 pairs of readings on g-ALjO-j, 80 pairs on ZnO, and 11 pairs on NiO. The relation is ex pressed by, T = mT + b n i (187) where T^ is the Leeds and Northrup reading and T^ is the IRCON reading and both are in degrees centigrade. The resulting values of "m" are 0.663 in B-A^Og, 0.764 in ZnO, and 0.588 in NiO; the corresponding 108 values of "b" are 369.872°C in B - A l ^ , in NiO. 154.585°C in ZnO, and 413.154°C An excellent fit on the data set by Eq. (187) is indicated by correlation coefficient values of 0.960 in B-A^O^, 0.955 in ZnO, and 0.977 in NiO. Using Eq. (187), Tn = T at 1098°C in B-A1 20 3 , 655°C in ZnO, and 1002°C in NiO; the IRCON reading is higher above these cross over points in all three cases. An estimate of the true surface temperature and emissivity may be made by assuming that the emissivity of these materials is constant over the spectral range of sensitivity of the IRCON detector and that the red filter in the Leeds and Northrup pyrometer has a characteristic wavelength of 0.65pm. Using the characteristic responses of each in strument and Eq. (187) will determine the true temperature and emis sivity. The IRCON temperature reading is proportional to the photon flux over the spectral range actually detected. The pertinent equations re lating the temperature reading T^, emissivity setting , the true temperature T and true emissivity e are derived in Appendix P. If a uniform spectral response is assumed for the detector, the result is, (188) where, P(X, T) - (189-a) exp(-C/TX)f (X, T) f(X, T) = (2T/C)(T/C + 1/X) + 1/X C = 14,388°K - pm 2 (189-b) 109 ** longer cut-off wavelength (pm) X = shorter cut-off wavelength (pm) If the detector quantum efficiency is taken into account, the re sulting equation has the same form as Eq. (188) and (189-a); the only difference is a slight correction in f(X, T) (Appendix P ) . The exponential term in P(X, T) is dominant for temperatures less than 3000°C and X less than one pm. The contribution of the terms in Eq. (188) involving the shorter cut-off wavelength X Xr is small at temperatures less than 2000°C. To determine the validity of Eq. (188), computed values of e and T were compared with the data shown in Table 3 of the IRCON manual. Using X^ = 0.6pm and X& = 1.0pm, the values of e agree to within 1% at 1900°F < T < 3130°F when e^ = 0.4; the values of T generally agree to within 1/10 of the corresponding the discrepancy in e. The lowest e^ data set of Table 3 was used because e^ is 0.2 in all the experiments described below. I Using the quantum efficiency data for [43 3 and X = 0.6pm and X Si = 1.05pm the shown in Figure 24 of values of e agree to within Si 0.3% in the 3000°F range and to within 1.5% in the 2000°F range. The accuracy is improved in the higher temperature range while it is slightly worse in the lower range. This is perhaps due to the use of 0.6pm instead of the shorter cut-off wavelength indicated in Figure 24 of [43] in addition to the use of a parabolic approximation to the curve given there. If the Si detector were assumed to respond in proportion to the total photon flux radiated over all wavelengths, Eq. (8.39) of [13] 110 gives (e/e±) - (T±/T) 3 (190) This assumption is a very approximate one because it does not fit the data shown in Table 3 of the IRCON manual. However, note that at very high T^ and T, f(X, T) dominates the exponential term of P(X, T ) ; so that in the limit as T and T^ approach infinity, Eq. (188) approaches Eq. (190). The response of monochromatic brightness pyrometers like the Leeds and Northrup pyrometer is governed by Eq. (8.43) of [13]; thus, (191) e = exp C(l/T - 1/T )/X n e where T^ is the temperature indicated on the scale. The true temperature T and true emissivity e may be calculated by simultaneously solving Eq. (188) and (191) or Eq. (190) and (191) given the readings T ^ , T^ and e^. The results for T as a function of T^ are shown in Figure 16 while the calculated (e, T) pairs are plotted in Figure 17. actually used to get the value of T^ given T^. Eq. (187) was Uniform spectral re sponse for the S. detector was used with values of X. = 0.6pm and X X Xr = U 1 .1pm, the latter being the band-gap cut-off wavelength of S^. The asymptotic relationship between Eq. (188) and (190) is exhi bited by the. ZnO case; the two give practically the same result in T above T^. = 1600°C while the discrepancy increases as decreases. The fJ-AlgOg results indicate a tendency to converge also at a much higher temperature. However, the discrepancies in the B - A ^ O ^ and NiO cases are very significant; in addition, the bottoming out in T derived from Eq. (188) at the lower ranges in both cases, is an indication of 2.0- B ---- A- ZnO B - A-ai2o C“ — r NiO LAW --PHOTO DIODE LAW Fig. 16- True temperature (T) vs. IRCON pyrometer temperature (T^). Ill 0.2> t war / / / 2 / LU / / / / / ^/ 0 T fC xlO 3 ) " H 1 .7 V 1.9 -PHOTO-DIODE LAW true temperature. 112 113 experimental error. This is because Eq. (188) has been shown to be more accurate than Eq. (190). Eq. (188) is very sensitive to experimental error while Eq. (190) is relatively not. To illustrate, consider the B - A ^ O ^ case at 1100°C where = 1099°C. = Eq. (188) and (191) give e = 0.654 x 10 ^ and T = 1721°C while Eq. (190) and (191) give e = 0.1342 and T = 1296°C. Let us denote the use of Eq. (188) as Case A while that of Eq. (190) as Case B. Suppose an error of +10°C was made in reading T for the same T . , XL J» rt Case A gives e = 0.4966 x 10 and T = 1794 C while Case B gives e = 0.1300 and T = 1312°C. The drop in e is 3% in Case B while it is 24% Case A; the increase in T is 16°C in Case B while it is 73°C in Case A. Now suppose an error of +10°C was made in reading Case A gives e = 0.8478 x 10 0.1380 and T = in for the same , and T = 1675 C while Case B gives e = 1292°C. The increase in e is 2.8% in Case B while it is 30% in Case A; the drop in T is 4°C in Case B while it is 45.5°C in Case A. Finally, suppose a +10% error was made in reading e.., i.e., from 0.2 to 0.22; Case A gives e = 0.8972 x 10 Case B gives e - 0.151 and T = 1282°C. and T The increase Case A while it is 12.9% in Case B; the drop in T = 1666 C while in e is 37% in is 55°C in Case A while it is 14°C in Case B. Hence the results derived from Eq. (188) are much more sensitive to errors in reading e^, sensitivity of Eq. and T^ than those derived from Eq. (190). (188) depends primarily upon the choice of X^ and Xe - Increasing X^ and decreasing Xe tends to bring T down. using X u The For instance, = 1.0 m instead of 1.1pm results in an increase of 37°C in T 114 for the same B - A ^ O ^ sample case above. X^ is second order. The effect of the choice in The resulting values of T when quantum efficiency is taken into account is generally higher;:this is probably due to a slightly smaller X^ (1.05pm) and a larger error that it entails at lower temperatures. Figure 17 shows that in both Case A and Case B the emissivity of ZnO peaks in the vicinity of 1700°C < T < 1800°C and tends to converge there. In all three materials, emissivity drops as temperature de creases. CaseA gives a value of e = 0.625 at T = 1601°C for NiO. For the same range in T^ the use of Eq. (188) gives a narrower range in T than Eq. (190). NiO exhibits the narrowest range followed by- 6 -Al 2 Q3;. this is apparently related to the slope in the linear re lationship between T^ and T^— NiO has the lowest slope while 8 -AI2 O 3 ; follows next. in Thus, a material that exhibits a slower rate of increase as T_^ increases will correspondingly have a slower rate of in crease in deduced T . As a final note to the question of experimental error, the lowest pre-set value of T^ that corresponds to the minimum detectable tempera ture T n by the Leeds and Northrup instrument are 1100°C in 8-A1-0, and £ j 1200°C in ZnO and NiO; the corresponding lowest values of T^ are 1087°C, 1075°C and 1100°C, respectively. In conclusion, the probable true values of T and e are between the values given by Eq. (188) and (190), especially at the lower temperature range. 115 III-1-f. The Temperature Control System The IRCON pyrometer gives an output voltage from zero to 10 milli volts; the upper limit corresponds to full-scale deflection of the meter on the face of the instrument. The surface temperature sensed by the pyrometer is therefore transformed by the Instrument into a DC voltage. This voltage is used to provide negative feedback that closes the control loop. The controlled variable is really the surface temperature of a spot that can be as small as 0.050 inch in diameter; this is because the optics of the instrument was designed to see a circular spot with a diameter that is 1/160 of the distance of the front lens from the sur face; and the minimum focusing distance is 8 inches. A detailed description on how the system works is given below. Measured operating characteristics are also presented. Appendix N lists the operating instructions for both the manual mode and automatic mode of operation. Ill-l-f-(l). Principles of Operation The control system is shown in Figure 18. The IRCON photo detector senses the surface temperature of the rod inside the cavity and gives out a voltage which drives the IRCON pyrometer to register the corresponding temperature on its meter. The pyrometer also pro duces a voltage output proportional to the deflection of its meter pointer. Full-scale deflection corresponds to 10 mV. The temperature control circuit amplifies the output voltage of the pyrometer 100 times before it is subtracted from an adjustable CIRCULATOR APPLICATOR sample # PGM-lOO MICROWAVE 1 GENERATOR JLL- DETECTOR control winding CALORIMETER U Rc TEMPERATURE CONTROLLER 0 © Power On Power On 4 Off Auto ( $ Mode Man o Temp Set Input from Meter ± Fig. 18- Control system interconnections. . 2 IRCON PYROMETER 117 reference voltage which can be varied between zero and 1.1 volts. traction results in negative feedback. Sub The difference voltage is then amplified- to produce an output current proportional to it. This output current is the DC control current flowing in the control winding of the saturable core reactor inside the Raytheon microwave generator. The anode current of the magnetron inside the Raytheon generator is proportional to the control current. Thus, the microwave power out put, which is proportional to the anode current, is controlled by vary ing the control current. Employing a negative feedback system results in making the surface temperature of the rod constant by the automatic variation of the micro wave power output. When the temperature tends to rise above the pre-set value, the increased negative feedback will result in a reduction of the microwave power output thereby lowering the heating rate of the rod, thus lowering its surface temperature back to the pre-set value. On the other hand, a sudden decrease in surface temperature results in a lower output voltage from the IRCON pyrometer and therefore a lower negative feedback. This in turn results in a higher microwave power output from the generator thereby increasing the temperature back to the pre-set value. The actual value of the stable surface temperature is pre-set by the actual value of the reference voltage inside the temperature control circuit. Figure 19 shows the schematic diagram of the temperature control circuit. The first stage amplifies the voltage output of the pyrometer to the same range as the reference voltage set by the IK potentiometer R. The second stage is a comparator circuit which basically subtracts 15 v cc Input (V^) from pyrometer control winding 100 K 12 K 2N651 100 K r =i k : 10K 10K 10K 10 OK Yr- reference voltage A- automatic B- manual R- manual adjust or temperature set Fig. 19- Voltage reference, difference amplifier and voltage-current converter. 118 119 the output of the first stage from the reference voltage. Therfore, the difference voltage is the output of the second stage. The third stage amplifies the difference voltage 3 times and maintains the am plified voltage across the 10-ohm resistor. The current through the 10 -ohm sensing resistor is practically the same as the control current through the control winding. Hence, a constant difference voltage re sults in a constant control current. The 330y F capacitor makes the third stage and the Darlington transistor circuit stable. In addition, it serves to divert the AC current generated by the control winding away from the control circuit. A Darlington pair is necessary because high voltage transistors have low current gains and the pA741 driving the base current of the trans istor can only give miniscule output currents. The diode across the emitter junctions of the Darlington pair protects the transistor from reverse voltage breakdown when the third stage has a negative output. Note that a reference setting of, say, 0.75 volts corresponds to 0.75 of full-scale deflection on the pyrometer. In effect, it sets the maximum surface temperature limit that the sample may attain. There are two modes of operation— manual and automatic. In the manual mode, the control loop is opened and the input from the pyro meter is grounded. The microwave power output is manually varied by manually adjusting the potentiometer R. A knob on the front panel connected to the shaft of R is provided for this purpose. A refer ence voltage output of 1 volt from R drives the magnetron to full out put power. In addition, a BNO terminal for monitoring the reference 1 20 voltage by a digital voltmeter is also provided. Grounding of the in put from the pyrometer is achieved by means of the double-throw single-pole MODE switch also mounted on the panel. The knob of the potentiometer R is marked TEMP SET on the front panel. III-l-f-(2). Operating Characteristics Figure 20 shows how the control current varies with the two con trol voltages, i.e., the reference voltage (V ) and the feedback volt age (V_^) . Figures 21 and 22 show how the corresponding magnetron anode cur rent and microwave power output varies with the control voltages. Note that the rounded dot data points correspond to the manual mode since V_^ = 0. matic mode with V The triangular data points correspond to the auto =■ 1 volt. They coincide with the dots. The relationship between the control voltages and the magnetron anode current is linear BetweenlO% and 90% maximum. with The same is true the microwave power output. Note that the peculiar characteristic of the saturable core re actor results in a small microwave power output, i.e., approximately 22 watts, even when the control current is zero. III-l—f-(3). System Interconnections Figure 18 shows how the temperature control circuit is inter connected with the rest of the components of the system. The actual front panel arrangement of the circuit is also shown. Figure 23a- shows the interconnections of the power supplies. 121 320t e < j= z240Ll I CC cr ZD O o ,6 0 1— o o 80" vcc= o-f 0 V| CmV) 8 50 volts H - 0A 4 Rg ■ 30 ohms — h- 0.8 Vr (V) 0 Fig. 20- Saturable-core reactor control current vs. refe rence voltage (Vr ) and input voltage (Vj_). V^=0 when using Vr and Vr“l volt when using Y^. 240" CURRENT C m A) 122 ANODE 160- vcc* 5o volts 0 Fig. 21- Rs= 30 ohms -- Magnetron anode current vs. reference voltage (Vr ) and input voltage (V^). V^=0 when using Vr and Yr =l volt when using V^. 123 t 0.6- OUTPUT POWER (kW) 0.8 0 .4 - 0 .2- 0 - O Vjtm V) Fig. 22- 0.8 0.4 8 4 V„ (V) O Microwave output power vs. reference voltage (Vr ) and input voltage (V^). V^=0 when using Vr and Vr = 1 volt when using V^; Vcc= $0 volts and Rg = 30 ohms. pilot light a 48-Volt DC ■o V,cc -h SUPPLY 1 S =D 110-Volt AC ■oV 3 -0+15 V 15-Volt DC SUPPLY -o - 1 5 v +lZI Fig. 23a-Power supply connections for temperature controller. 125 III-i-g. Four-Probe and Single-Probe Standing Wave Measurement Instantaneous changes in the input admittance of the applicator may be monitored by a multiple probe scheme. The four-probe system used in the experiments is based on a previous design [39]. The more accurate single-probe method was used under steady-state conditions. The fixed four-probe carriage in the previous design was replaced by a translatable carriage adopted to a Hewlett-Packard Model 809C probe carriage [ 1 ]; thus, one of the four probes may be utilized for single-probe measure ments as well. The inaccuracy of the four-probe system used in the experiments is primarily due to the presence of back-reflections from the source side. Moreover, non-linearities in the amplifiers of the electronic processing circuit contribute additional but relatively small errors. The VSWR measured by the four-probe scheme becomes less and less accurate as the amount of reflections from the source side increases. On the other hand, the VSWR measured by the single probe is independent of the magnitude of back reflections thereby giving the true VSWR of the applicator all the time. An analysis of the four-probe scheme presented in Appendix 0 shows that the net voltage V^ detected by the pair of probes giving the real part of the reflection coefficient p of the applicator is, Vr c* ___________ 4E0 1 2 !p I while the net voltage V^ part of p is, | p | cos__________ I PB | cos (0 + (192) - 2k I) detected by the pair giving the imaginary 126 2 y. i 4 Eo IP ---------- — 5-- ‘ r oc I sln 0 ' (193) 1 - 2 | a | | P ] cos (9 + 0 - 2 k L) ' B B where Eq p is the amplitude of the incident wave on the applicator while is the reflection coefficient at a reference plane on the microwave B generator side of the four-probes, located at a distance H from the load reference plane, e.g., iris plane of the applicator. The phase angles 9 and 0 B p = \p \ come from, p When =0, exp (j0 ) and p - | 'j exp (j0B) . the four-probe scheme, measures the true proportionate value of p ; however, regardless of the value of / p B , the measured 9 is the true 0. This is because arctan 0^1/ V^_) = 9 after dividing Eq. (193) by (192). Hence, the inaccuracy lies in a non-realistic measurement of the magnitude of p due primarily to a non-zero p and to a lesser degree from amplifier non-lineariries, i.e., amplification varies with the amount of detected voltage. The apparent shortcoming of the four-probe scheme may be put to use for measuring p knowing p . The square of the distance of (V^_ , V^) from (0 ,0) is, V2 r + V? x (4 [ 1 - 2 |y° | ^ (194) Eq)___________________ |p | cos (0 + 0^ - 2 k Z ) ] The maximum value of this distance occurs at, - 2 k JL = 2 n 9 + 0 B while the minimum value occurs at, (195) 0 + 0B — Eq. 127 (196) ZKl = n (195) and (196) actually define a line that is tilted from the axis by an angle, 0 = 2n - ( 0 - 2k I ) B (197) 0B is determined from Eq. (197) after inserting the measured values of 0 and SL . The ratio of the maximum distance to the minimum distance is, 1/2 l— ( V 2 + V? ) r x max ( 1 + 2 1? ( V2 + V 2 ) . r x mxn_ ( 1 - 2 1 ^ 1 The magnitude of p 1 ' 1V > (198) I ft I ) B is determined from Eq. (198) after plugging in the values of the distances measured from the plot of value of ant* t^ie known p . A convenient way of measuring p„ iJ is to use an adjustable short- circuit as the load. Translating the short-circuit is the same as varying 0. The maximum and minimum distances referred to in Eq. (198) are then measured along the "major axis" of the ellipse-like plot of (V , V.). r x. An experiment was conducted to verify the theoretical analysis. The load was an adjustable short-circuit and back reflections were exaggerated by putting a fixed iris at a suitable location on the back side of the four probes. The same Pa was applied with the different irises used. Typical results are shown in Figure 23b. The magnitude of the reflection coefficient of the 2-inch by 3 -inch rectangular iris is smaller than that of the 2 -inch diameter iris. Thus, the trace with the 2-inch circular iris emphasizes the elliptical + Rep a normal connections X longer input waveguide 128 © 2-inch dia. iris at source reference plane • 2x3 in? rectangular iris Pig. 23b - Effect of changing reflections from source side on the four-probe plot of reflection coefficient (/?) of the load, load is variable short-circuit. 129 nature of the plot of (V , V.) r i more. It is also indicated that with p , the centroid of the closed curve traced is displaced farther ’B away from the true center, i.e., center of the circular trace when p = 0 . larger The case with no iris departs slightly from being a true circle indica ting the presence of small reflections from the transition waveguide and circulator. Note that the three cases have different values of Eq arising from different transmission coefficients among the irises. To illustrate, let us calculate the value of Gg. of the 2-inch iris. O The angle of tilt of the "major axis" is 9 e* -45 ; since the length I is 4.1149 Xg, the value of calculated from Eq. (197) is + 128 . On the other hand, the phase angle of the reflection coefficient of the2-inch iris derived from the measured values of g^ and b^ presented inFigure 12 O is +141.5 . The 9.5% discrepancy is probably due to the additional re flections from the transition waveguide and circulator. Normally, p is the unknown quantity to be measured. In this case, the effect of Pg may be eliminated by the same scheme suggested in [17] to make V and independent from the available power level which is proportional to E^. The only additional modification needed is to put the pick-up probe for the dividing signal at the exact spot where it is proportional to 4E^/ [ 1 - 2 \p\ . | Cos (0 + - 2 «*)] . The complications associated with the four-probe scheme are absent in the single-probe scheme. When P g is not zero, the equivalent travel ling wave incident on the load Ej_. is, E± = Eq sin( 7rx/a) exp(j kz )/[ 1 - p p ^ exp(-j2«Jl ) ] (199) while the equivalent reflected wave is, 'V Er = E^ (z ** 0 ) exp(-j « 2 p (200 ) 130 = yo Eq sin( tx/a) exp(-j n z) / [ 1 - p p ^ exp(-j 2 /c 1 ) ] Therefore, I Er+ Ei 1 max VSWR — 1 + 1P 1 = I E + E. | . r x mxn and independent of (20 1 - I fi I ' since the constant in the denominator of Eq. (199) and (200 ) cancels out. The actual voltage detected by the probe is proportional to the square of | E + E_^| due to the square-law detection characteristic of the diode detector. Therefore, measured VSWR where V U l a X and Vm ,-„ U U .U respectively . = (V /V . )1/2 max' miny are the maximum and minimum detected voltages, (202) 131 III-2 Experimental Results of fi-kl^O , ZnO and NiO The optimum position of the short-circuit and iris size was ex perimentally determined by monitoring the normalized total amount of heat generated Pn the iris, respectively. while translating the short-circuit and varying The criterion for optimum conditions was mini mum output power requirement from the microwave generator to maintain the same temperature in the rod. The characterization procedure, Section II- 6 , was used to deduce the electric conductivity and permittivity of the rod, making it possible to deduce the corresponding amounts of heat generated in the rod, iris and short-circuit, respectively. Measurements were made under steady-state conditions wherein the temperature of the . rod was maintained by the automatic control system. experiments with The results of the /J-alumina and ZnO are discussed in Section II-2-a. The strong correlation between the location of the peak of the equivalent exciting field and the position of the short-circuit, shown in Section II-2, suggests that the temperature profile along the z- direction may be controlled by translation of the short-circuit. This was experimentally verified by an experiment using The results are discussed in Section II-2-b. /}-alumina rod. In addition, an analysis of the electric field distribution inside the rod using the experi mental data is also presented. Moreover, an analysis of the relation ship between the relative locations of the peak of the equivalent ex citing field and the short-circuit was made under optimum heating con ditions by using the data presented in Section II-2-a. The diagnostics precedure described in Section II-6 was used to 132 determine the variation of electric conductivity and permittivity with temperature of £ -alumina ZnO and NiO, respectively. The details of the experimental procedure and results are described in Section II2-c. Furthermore, an analysis of the respective effect of translating the short-circuit and changing the iris on the results of the characte rization procedure is also discussed. Finally, transient heating responses of ft -alumina and ZnO were experimentally investigated with constant output power from the micro wave generator. The results are presented in Section II-2-d. In par ticular, the variations of surface temperature and heat generated with time are explained. In addition, the roles of the thermal loss mecha nisms of radiation and convection in determining the stable temperature were experimentally verified. The first three types of experiments were conducted at steadystate. In all experiments, the samples were exposed to the air. length of the samples in these experiments .were 7.3 cm, which is slightly longer than the 5.46 cm height of the cavity. is hanged from an cut-off tubes. The rod The sample ar-alumina tube and extends equally into both beyond- It is held by an a-alumina pin passing through a hole in both the rod and the alumina tube. III-2-a. Optimum Heating Conditions 133 In these experiments, the temperature control system is in the automatic mode. Thus, each set of measurements were done with the surface temperature of the spot seen by the IRCON pyrometer maintained constant. The short-circuit is first set at a desired distance from the rod. The TEMP SET knob of the control system is then adjusted to achieve the desired temperature. After waiting for around 10 minutes to make sure that steady-state conditions have been reached, a final adjustment on the TEMP SET knob is made if necessary. Measurements on the standing wave outside the cavity are made after another wait of around 5 minutes. One of the probes in the four-probe assembly is used as a running single probe for the measurement of VSWR and position of a minimum. In addition, the inlet and outlet temperatures of the calorimeter and the magnetron anode current are recorded. Finally, the surface temperature at the middle of the rod-length is measured using the monochromatic pyrometer (Leeds and Northrup). This can only be done on the side of the rod oppo site to what is seen by the IRCON pyrometer; however, symmetry conside rations suggest that the two pyrometers are measuring practically the same temperature. This procedure is repeated for each setting of the short-circuit. The flow rate of the water passing thru the calorimeter is measured at the beginning, middle and end of the experimental run. A similar overall procedure was followed when the size of the iris (fixed circular type) was varied. In both experiments, the desired temperature can only be maintained in the vicinity of optimum short-circuit 134 position and op timum iris size due to a limited P , which is around 800 watts maximum. a Optimum conditions result in minimum to maintain the same temperature. The measured P r is calculated from the measured VSWR by using Eq. (201) and (78). Pn r , Png and Pn ^ are calculated from the appropriate equations in Section (II-3-c) and the measured admittances of the iris and short-circuit presented in Section (III-1-b). However, the measured a and e r are needed first before P nr and P ns can be calculated, Data on VSWR and position of the minimum are used in the characterization procedure discussed in Section (II-6 ) to determine the measured a and e r* The effect of losses in the iris and short-circuit on the amount of heat generated in the rod, as the short-circuit is translated, is shown in Figure 24, 25 and 26. For comparison, a curve showing the nor malized amount of heat generated in the rod when the iris and shortcircuit are lossless and ideal is included in each case; this represents the maximum P that the given rod can realize with the given cavity settings. It will be shown in Section (II-2-c) that in general, the values of a and er yielded by the characterization procedure vary with the po sition of the short-circuit and the size of the iris. It is suggested there that the value to use is the one under optimum P nx conditions. For the ZnO samples examined, the data nearest the optimum correspond to the 2-inch iris and no iris cases. The average values of these two cases were therefore used in deriving P nr a and and P ns . e r A shunt of 135 0.9 + A total # rod fl iris N short 0.7+ 0 ,5 + HEAT GENERATED ideal 0.3 M 0.1 0-86 0-88 0.90 lz/\ g Fig. 2+ - Normalized generated heat in the rod, iris and short-circuit vs. normalized short-circuit distance from rod (£2 /^g) * ;roc^ 1»200 C;O( = 0.0906; 2-in. iris; curve is for a lossless applicator.(^indicated) A total #rod □ iris 0.8- ideal 0.6" HEAT GENERATED Mshort 0 .4 - 0 .2- 0 .8 7 0 .8 9 0.91 X Fig. 25- ZnO under the same conditions as in Fig. 24 ex cept d=0.1078, the electric conductivity is 33 % higher at 0.722 mhos/m and the dielectric constant is 2 9 .4?6 lower at 494. 137 0 ~ o 0.8 ” LU £ cr L±J z w 06 < d total # rod H < M short LU ideal X 0.2 - 0-75 0.8 5 Fig. 26- Normalized generated heat in the rod and shortcircuit vs. normalized short-circuit distance from rod (j^/Ag)* A “Al2®3 a"k 1 1200*C; c(=0.11^-2; no iris; curve is for a lossless applicator, (^indicated) 138 equivalent circuit proved to be adequate for the the values of g r and br at optimum (i-A1^0 case; thus, were used, i.e., Note that the no iris case was near optimum for the 0.75.\g. /i-Al^O^ case. The data in Figures 24 to 26 all correspond to an IRCON pyrometer 0 reading of 1,200 C. The actual temperature are of course different in ZnO and /S-A^O^ due to their different thermal eraissivities. The ZnO sample in Figure 24 has a diameter of 0.353 cm and an of 0.0906 at 2.45 Ghz while the one in Figure 25 has an a The values of a and e r a of 0.1078. in the smaller rod were 0.5392 mhos/m and 700, ’ respectively, while they were 0.7218 mhos/m and 494, respectively, in the larger rod. The difference in a and e between these rods is related r to the fact that the density of the smaller rod is 6.14 grams/cc while it is 5.93 grams/cc for the larger rod. Figure 26 refers to a rod with a diameter of 0.445 cm and an a had a a of 8.0 mhos/m and an e r of 0.1142. The /5-A^O^ S - A ^ O ^ sample of 125. Figures 24 and 25 in relation to Figure 26 emphasize the more se vere degradation in the efficiency of energy conversion (pnr) occurs when the electric conductivity of the rod is small. 24 the maximum realizable that In'Figure 0.94 dropped to an actual value of around 0.5 while in Figure 25 it dropped from'an ideal value of 1.0 to an actual value of around 0.51. In both cases, a reduction in efficiency of around one-half occurred. On the other hand, the fi -Al^O^ case in Figure 26 shows a drop from an ideal value of 0.94 to an actual value of around 0.86. The order of magnitude increase in electric con ductivity plus the absence of a lossy iris helped to maintain a high efficiency with Looking at the M data of Figures 24 to 26, the normalized (^ ) amount of heat losses in the short circuit at maximum Pnr were around 0.20 with the smaller ZnO rod, 0.14 with the larger ZnO rod, and around 0.05 with the /i-A^O^ rod. Hence, with the same lossy short-circuit, a larger portion of the incident energy is converted to heat in the rod with a larger electric conductivity. However, a more accurate comparison must take into account the 2 o-R volume of the rod. Using as the basis of comparison and making ^ ffR of the smaller ZnO rod as the normalizing factor, the corresponding normalized values of ctR 2 are for the larger ZnO rod and therefore one for the smaller ZnO rod, 1.895 23.58 for the /J-AI2O 3 rod. At peak Pn r , the losses in the 2-inch iris were practically the same with both ZnO rods — a normalized (?n£) value of around 0.18 occurred. The losses in both iris and short-circuit were generally lower with the larger ZnO rod than in P the smaller one; the peak value of Pns and were 0.20 and 0.30, respectively, with the smaller rod while they were 0.14 and 0.25, respectively, with the larger one. The severity of the inefficiencies in the ZnO cases is further illustrated by the wider departure of actual maximum Pn from the ideal maximum PR which, incidentally, is equal to the ideal P^r . The reduction in P n is around 0.0 9 with the smaller rod while it is around 0.19 with the larger one. In comparison, the reduction in maximum Pn 0.01 with the fi - A ^ O g is only rod. Figures 27 and 28 show how the amount of heat generated varies as 0 .9 t 0 .5 - HEAT GENERATED 0 .7 - r-=-= 0 .9 5 4 5 0.3 o.i-2 Fig. 27- -4 Normalized generated heat in the rod*, iris □ , and short-circuit H vs. iris susceptance b£. ZnO at 1,200*0; o(= 0.0906; short-circuit distance from rod is 0.9l25Ag} A=total. Curves are for a lossless applicator. 141 0 .9 - 0 .7 Q LU *< cr 0 .9 1 2 5 LU 5 0 .5 — o I— -r=- < = 0 .9 5 4 5 LU X 0 .3 - 0 .1-2 -4 Fig. 28- Normalized generated heat in the r o d * , iris □ , and short-circuit m v s . iris susceptance hi. ZnO at 1,200*C; a = 0 .1078; short-circuit distance from rod is 0.9125Tig; A=total. Curves are for lossless applicator. 14-2 the size of the iris changed. In this experiment, the short-circuit position was fixed near the optimum value. Figure 27 refers to the smaller ZnO rod while Figure 28 refers to the larger one. Each of the data points in Figures 27 and 28 were taken with X = 0.9125 A . S For comparison, two curves showing the variation of ideal Pnr with 2 b^, when the iris and short-circuit are lossless and ideal, are also shown. One curve corresponds to 2 ~ (0*9125 + 0.0420) z A g ^ = 0.9125 A = 0.9545 A g and the other to g . In the actual short-circuit, ’ the minimum of the total electric field intensity occurs at 0.0420 X g behind the face of the short-circuit. Thus, the ideal Pnr corresponds to = 0.9545 A g curve that closely simulates the actual situation. The actual location of the minimum field was calculated from 180 O = -2nz + 149.7477 ° where 149.7477° is the phase angle of the measured min reflection coefficient of the adjustable short-circuit. The ideal P__ nr curve in both figures that correspond to X_ / = 0.9125A g clearly indicate that this value of jfc is near the optimum because the peak values are 0.927 for the smaller rod and 0.985 for the larger one. The peak is higher with the larger rod in view of a larger 2 <J R . The optimal normalized iris susceptance is -3.075 for the smaller rod and -2.325 for the larger one; this means that the rod with a larger 2 <rR requires a larger iris at optimum, i.e., around 2 inches in diameter for the larger rod against a diameter slightly larger than 1.75 inches for the smaller rod. The ideal Pnr curves that correspond to the variation of the actual Pnr X. 2 = 0.9545 X S delineate with iris size. They indicate the 143 approximate optimum iris susceptance that the actual situation requires; it has a normalized value of around -0.5 for the smaller rod and around -0.75 for the larger rod. The two ideal Pnr curves in both Figures 27 and 28 emphasize the degradation in efficiency due to losses in the iris and shortcircuit. The losses in the short-circuit follow the trend in actual P nr as the iris size is changed. The losses in the iris tend to increase as the diameter of the iris becomes smaller. However, the smaller rod case exhibited a peak in P . ni The two ideal P nr at a diameter of around 1.875 inches. curves in both figures indicate that the effect of moving the short-circuit farther away from the rod than the optimum distance is not only to reduce the maximum realizable P J nr but also to increase the optimum iris size. Hence, putting the short-circuit closer than the optimum distance will result in a smaller optimum iris size. Finally, the actual Pn A „ = 0.9545 X 2 Figure g is higher than the ideal Pnr with because of the losses in the iris and short-circuit. 29 shows the effect on the ideal P of a change in tem perature of the larger ZnO rod as the iris size is varied with an 0.954 5 X ideal P g nr of . Figure 30 shows the effect of a change in temperature on the of the smaller rod as the short-circuit position is varied with an iris diameter of 2 .inches. The effect of an increase in temperature from an IRCON reading of 1,200°C to 1,400°C in Figure 29 is to decrease the peak Pnr by 0.04 and the magnitude of the optimum iris susceptance by 0,25. The increase in temperature corresponded to a decrease in c from 0.722 mhos/m ■1A|4 0 .8Q LU H < cr LU 2- °-6 " CD 0 .4 - 0.2- ) A ,0 - 1200*C -2 B , # - 14001 IDEAL Fig. 29- Normalized generated heat in the rod vs. iris susceptance at different temperatures. ZnO with an (*=0 .1078; experimental data taken with/2=0 *9125Xg ; A and B curves for lossless applicator with 0 .95^5- (*indicated) 145 .0 t O 1200 C • 1400 C — IDEAL 1200 0.8 - C g 0.6<c LU X 0.88 0 .9 0 0.92 Fig. 30- Normalized generated, heat in the rod vs. norma lized short-circuit distance from rod (ig/Ag) a"t dif ferent temperatures. ZnO with an ot of 0.0906; 2-inch diameter iris. Curves are for a lossless applicator. (*indicated) 1A6 to 0.617 mhos/m while remained at 493. On the other hand, as shown in Figure 30, the same increase in temperature on the smaller rod resulted in a decrease in peak Pnr by 0.120 and a decrease in JL^q ^ crease in 0.0015. This corresponded to a de O' from 0.539 mhos/m to 0.517 mhos/m while increased from 700 to 702. The decrease in maximum P in Figure 29 and the general decrease in Pnr in Figure 30 at the higher temperature, are due to a smaller G. The experimental results shown in Figures 29 and 30 confirm the theoretical results of the effects of a change in temperature on the P nr of ZnO. The ideal curves in Figure 29 were computed at an JL„ £■ that gives approximately the same position for the exciting field as the experimental one. The experimental heat generated in the rod is lower than the ideal because of losses in the iris and short-circuit. Figure 31 shows typical distribution of the heat generated in the rod and the short-circuit as a function of the short-circuit po sition. For comparison, the values of P nr data points, respectively. are shown as circular No iris was used in both cases shown. The fi-AI2 O 2 case clearly shows that maximum occurs at a short-circuit position near the one that gives minimum Pns/(Pnr+ Pns) • The curve was calculated using the equivalent conductance of the rod at The 2 = 0.75 X and the measured admittance of the short-circuit. •''■g M represent the values of P / (P + P ) r ns nr ns derived from the T-equivalent circuit and the respective values of cr and the data of each of the short-circuit positions indicated. from Hence, 0.7 M measured constant 6 0 .4 " 0.2" <*=0.1142 (a ) <*=0.1078 0.9 0 ^ ____ 0.88 (b) + 0.92 ___ 31- Division of heat generated between sno?t-circuit(Pn s ) and rod(Ihr) vs. short circuit distance from rod(i2/^g)« No iris* (a)/J-Al203,l400*C; (b)Zn0,l300*C. 148 Eq. (105) and the measured P^ were used to derive the curve while Eq. (108) was used to derive the H data. In the ZnO case, the curve was calculated from Eq. (108) using the average value of 2-inch iris cases. data <T and er from the data for the no iris and A comparison of the P suggests that maximum Pnr nr and P ns /(P nr + P ) ns occurs at a short-circuit position close to where the theoretical and measured P / (P__ + p ) are ns nr ns equal. In both cases, the Pnr data points were calculated using the same data in calculating the respective curves. A situation where the equivalent shunt admittance representation for the rod is an excellent approximation to the T-equivalent circuit is illustrated by the /S-A^O^ case, i.e., the discrepancy between the curve and the measured values is minimal. III-2-b. 149 Equivalent Exciting Field The relationship between the occurrence of maximum heating in the rod and the location of the peak of the equivalent exciting field was examined by looking at typical data on heating as a function of shortcircuit position. Figures 32 and 33 show the distance of the peak of the equivalent exciting field relative to the rod axis (zp? as a function of theshortcircuit distance and pnr 5. For comparison, the corresponding values of P are shown. The data points on zp were calculated using Eq. (116) and (119) in conjunction with the T-equivalent circuit was calculated from the measured Pn n of the rod. P data and Eq. (108) since in both cases there is no iris. Inusing the T-equivalent circuit, the respec tive values of a positions and e deduced at each of the short-circuit shown were used. Figure 32 refers to a yS— rod 0.445 cm in diameter and the temperature was maintained at l,400°c as indicated on the IRCON pyro meter. The negative sign of z^ means that the peak of the equivalent exciting field is on the side of the rod facing the short-circuit; it is clearly shown that as the peak approached the axis of the rod, the amount of heat generated in the rod increased towards the maximum value. The trend suggests that the peak in P the value at l2 = 0.75 A g ted .by Figure 31a /(P P ns + p nr ) ns corresponds to was zp = -0.0176 A zp— 0, e.g., This is further substantia which indicates that the minimum value of occurs at around I 2 =0.70 A . g HEAT GENERATED 150 0.8 A total 0 rod Fig. 32- Normalized distance of the peak of the exciting field (Zp/\ ) from rod and normalized generated heat in rod and cavity vs. short-circuit distance from rod (i2/Xg). £-Al203 ; 0(= 0.1142; no iris; 1,4-00*0. 151 0 .9 o LU £ cr LU LU X 0 ~ ▲ total Fig. 33- •rod Same as Fig. 32 except: ZnO with 1,300 C. (^indicated) oi o f O.IO78 at Figure 33 refers to a ZnO rod 0.422 cm in diameter and the tempe rature was maintained at 1,300*C as indicated on the IRCON pyrometer. Again , an increasing value of Pnr is accompanied by a shrinking dis tance between the axis of the rod and the peak of the equivalent ex citing field. In addition, the sign of z is negative and the absoP lute value is larger than in the p - A ^ O ^ case. In both cases, the total amount of heat generated in the cavity and rod exhibit a peak in the range shown and the value of which this occurs is different from the value at which Pnr at is maximum. In /3-alumina, the negative sign of zp indicates that the rod was acting as an inductive post, as theoretically suggested in Section II4-b. An experiment was conducted to test the feasibility of controlling the temperature profile along the axis of translation of the shortcircuit by changing the position of the short-circuit. The rod was /8-A1 0 '.and was pre-sintered in a conventional oven at 900*C for an hour; however, it shrank from a diameter of 0.59 cm at the start of the experiment to a diameter of 0.54 cm at the end — more sintering oc curred during the experiment, which was conducted at 1,300 C as indi cated on the IRCON pyrometer. In contrast, all the data presented above were taken with rods pre-sintered by microwave heating to temperatures higher than temperatures at which the data were taken. Therefore,their dimensions remain the same throughout a typical experiment. The IRCON pyrometer was aimed on a spot midway between the front and back sides of the rod and the indicated temperature was maintained 153 o at 1,300 C throughout the experiment. The back side of the rod is the one facing the short-circuit. Initially, the short-circuit was at = 0.794 Ag when the desired temperature was achieved. The Leeds and Northrup pyrometer was then used to measure the temperature of the front and back sides at the same glan cing angle. In addition, the single-probe method was used to measure the standing wave ratio and the position of a minimum. The magnetron anode current was also read. The same procedure was repeated for each of the short-circuit settings made. At = 0.794 \ O , the back side was hotter by 9 C , i.e., as indi- z 8 cated on the Leeds and Northrup pyrometer the temperature on the back o o side was 1,119 C while it was 1,110 C on the front side. The short-circuit was then translated towards the rod in-steps of 0.5 cm. The closest setO ting was at £-2 = 0.625 which yielded 1,120 C on the front side and 1,094°C on.the back side— a difference of 26°C. The next set of measure = 0.828 \ , 0.5 cm farther away from the rod z g than the initial position. The short-circuit was then translated away ments was then made at from the rod in steps of 0.5 cm. The farthest setting was at ^ 2 = 0.895/lg O O which yielded 1,113 C on the front side and 1,161 C on the back side— a O difference of 48 C. A waiting period of around 10 minutes was observed after each short-circuit setting to insure that steady-state conditions have been reached. No iris was used during the experiment because earlier results (Section III-2-a) indicate that the optimum iris size is larger for lar ger rods; and that the optimum iris size for the relatively smaller 154$ -AJ^Og rod used earlier already corresponds to an almost no iris case. Figure 34a is a plot of the temperature difference between the front and back sides of the rod as a function of the short-circuit po sition. For comparison, the measured P„ and the deduced P are shown n nr in Figure 34b. culate P . nr the a and The 1 xl approximation and Eq. (104) were used to cal The equivalent circuit parameters were calculated using € r characterization data for each short-circuit setting, The corresponding positions of the peak of the equivalent exciting field are shown in Figure 35a. In addition, contours of z^ as a func tion of ff when <y and e are constant, are also shown. Figure 35b is 2 r a plot of the corresponding values of <7 and e ^ at each short-circuit setting. The results shown in Figure 34a demostrate the feasibility of ef fecting changes in the temperature profile by translating the shortcircuit. The corresponding variation in the position of the peak of the equivalent exciting field shown in Figure 35a suggests a direct re lationship between the temperature difference of the front and back sides of the rod and z . Following the constant (<T, £ ) contours on P Zp - S. 2 plane, the nearest peak of the equivalent exciting field is coincident with the rod axis when L 2 ^s t^ie r^nge 0.8 < l 2 / X ^ < 0.85. In this range of Jfc^, the back-side of the rod was hotter than the front side. Using the characterization data (Fig. 35b) for and 0.8277 \ j = 0.7264 \ in Eq. (77-b), respectively, the total electric field 8 intensity inside the rod, as a function of the radial distance, was 155 Fig. 34- (a) Rod front temperature minus tack temperature ■ and (b) heat generated in rod # and cavity A vs. short-circuit distance from rod (Jl^/Ag) • yS-AlgO^ at 1,300 * C ; ot=0.1385; no iris; curves at constant e*. (*C- indicated temperature) HEAT GENERATED i IV) O — $ o ------- h — -h o 0^ + P. o CD w oa FRONT-BACK TEMP.OC) _ IV) 15? -600 -*-200 Fig* 35- Normalized distance of peak of exciting field from rod M , electric conductivity# and dielectric constant ■ vs. normalized short-circuit distance from rod itg/Ag. at 1300*C; 0C=0.1385» no iris; curves- constant &*, 158 calculated. Figure 36 is a plot of the resulting normalized electric field intensity profile 2 ^ , on the x = a/2 plane, as a function of the dis tance from the rod axis. The profile when l0 is 0.7264 X shows a ba* g lance between the front and the back sides of the rod. This is consis tent with the observed zero temperature difference between the two sides shown in Figure 34a. Is 0.7264 X when Note that the electric field profile is primarily due to the J^(k'r) cosG term. 4> , On the other hand,' the contribution of the Jo (k’r) term is dominant when ^2 Is 0.8277 X § • 2 2 = 0.8277 X The plot for O suggests that the back surface of the rod should have a relatively higher temperature than the front surface in view of the higher heating rate resulting from the small hump of the ^ 2 profile near the back side. This is con sistent with the experimental result shown in Figure 34a. Note that the heating rate profile when is 0.8277 X is more evenly distri2 g buted azimuthally than the one when is 0.7264 X , as suggested by 2 g a comparison of file on the x the heating profile = a/2 plane. onthe The heating rate z = 0 plane with on the z= the same order of magnitude as on the x = a/2 plane when the pro 0plane is of t 2 is 0.8277 X while it is negligible on the z = 0 plane relative to the g profile on the x = a/2 plane when JL ±s 0.7264A • The more even • heating when compared to the is 0.8277 A S 2m , is consistent with the higher Pnr = 0.7264 A g case. Incidentally, the two short- circuit settings correspond to the two peaks in measured Pnr shown in Figure 34b. The increase in Pn with increasing i 2 > shown in Figure 34b, is 159 (a) < - 1.0 - 0.6 cr O 2 \ © = 0 Pig. 36- Normalized electric field intensity $ vs. norma lized radial distance. (a)i2 =0 -726^-^g; (b)>^2=0•8277?lg; R- rod radius? negative "r" on back side of rod; no iris; /3-Al20^ at 1300*C; oi~0. 1385• (^indicated) 160 is consistent with the result for /J-A^Og, shown in Figure 31a, which indicates an increasing trend of the fraction of Pn that is lost in the short-circuit for Z 2 greater than 0.7^. Moreover, the theoretical plots of Pnr versus shown in Figure 34b, suggest that the short-circuit position at which maximum Pnr occurs and the one which gives the more uniform temperature pro file are close to each other, e.g., within O.TX^. The theoretical * plots of P were calculated at constant e. . The two curves cor* respond to the value of e when I is 0.7264 \ and 0.8277 \ , r respectively. ^ o o In addition, the position of the short-circuit that gives a more uniform temperature profile is between the position that gives mtTm pnr and the one which makes zp =* 0. The characterization data shown in Figure 35b suggest that for O' increased as the degree of sintering increased. In addition, the drastic increase in O' and € r could be an indication of a major microstructural rearrangement. It may be pointed out that the maximum difference in temperature shown in Figure 34a is less than 4% of the average peratures of the front and back sides of the rod. between the tem Moreover, it is the surface temperature of a small spot, around 0.050-inch in dia meter, at 0 = 9 0 ’ system. that is maintained by the temperature control III-2 -c. 161 C h a r a c te r iz a tio n The variation, of the dielectric constant and electric conductivity of the rod with temperature is determined by operating the temperature control system in the automatic mode and repeating the characterization procedure (Section II-6) for each temperature setting. Measurements on * the standing wave are only made after a sufficiently long waiting time, e.g., 10 minutes, to ensure that steady-state conditions exist. Consistent with the assumption of the variational model (Section II-l) that the rod is homogeneous, only those rods that exhibit a uni form temperature profile can be characterized accurately. This is because a non-uniform temperature profile implies a non-uniform value of a and if these properties vary with temperature. The measured results do not account for non-uniformities in general, the measured cr and a and e r . Hence, in should be viewed as effective values. Figure 37 is a plot of electric conductivity surface temperature of ZnO. , a as a function of and NiO, respectively. Figure 38 is a similar plot for the dielectric constant e . r The results shown in Figure 37 and 38 for ZnO and were taken with the short-circuit position fixed throughout the experiment and no iris was used. The short-circuit was maintained at i for ZnO and 2 = 0.875 A g for 2 = 0.927 A 8-Alo0 Q . These conditions were near r l 5 the optimum short-circuit position and iris size. The results for NiO were taken using the variable iris. The iris and short-circuit were set at optimum for each temperature shown in Figure 37 and 38. The criterion for optimum settings was minimum g DIELECTRIC CONSTANT 30- O 9- 200 - 100 0.4 •-NiO i-/S-Al203 M-ZnO • - NiO ■-/S-Al.jOj H ” ZnO Fig. 38- Dielectric constant vs. true temperature, cl's are 0.096 for NiO, 0.1142 for and O.IO78 for ZnO. 162 Fig. 37- Electric conductivity vs. true temperature, ct's are 0.096 for NiO, 0.1142 for fi~AlzO^ and 0.1C78 for ZnO. 163 available power requirement from the microwave generator to maintain the same temperature. Moreover, the rods were pre-sintered by micro wave heating, cooled to room temperature and then characterized at the temperature shown in Figures 37 and 38. exposed to the atmosphere. slightly In addition, the rods were The upper half of the rods, were at a higher temperature due to convection since they were verti cally oriented. The diameters of the rods were such that a was 0.1078 for ZnO, 0.1142 for yd-Al^O^ and 0.0959S that the 1 x 1 for NiO. The values of a suggest approximation is sufficiently accurate. The constraints expressed by Eqs. (19) and (20) are satisfied in the three cases. There fore, it was not necessary to use the more accurate 2 x 2 approxima tion. The temperature used in Figures 37 and 38 are the deduced actual temperatures using the "T^ law" of Eq. (190). This is because the normally more accurate "photodiode law" ex pressed by Eq. (188) is much more sensitive to errors in reading the pyrometers, specially near the lower temperature limit of their sen sitivity. The electric conductivity of ZnO and NiO decreased while that of ;5-alumina increased as temperature increased in the range 1000 C to 1700*C. It dropped by 72% form 38mhos per ra, between 1300 C and 1700 C in NiO, and by 30% from 0.70mhos per m, between 1100 C and 1500*c in ZnO. It increased by 13% from 7.8mhos per m between 1200*C and 1700*c in -alumina. than in yfi-Al^O^ Note that O' is an order of magnitude higher in NiO and also in than in ZnO. 164 The corresponding dielectric constant remained the same at 500 in ZnO while it increased in NiO and decreased in jS-alumina as tempera ture increased. It increased by 500% from 35 in NiO and decreased by 13% from 130 in ft -alumina. The respective magnitudes and variations with temperature of <5 and of y6 -alumina (Figures 37 and 38) are consistent with the extrapola tion to the l,000aC- 2,000°C range of the results for flux-grown Nadoped yS-alumina crystals presented in [2 ]. Single-crystal ZnO is an n-type semiconductor with donors arising from either an oxygen vacancy or zinc interstitial [49,51]. The acti vation energy was found to be much lower than the 3.2 eV band-gap for ZnO, i.e., 0.1 eV in air [49] and the two types of donor states repor ted in [51] have typical activation energies of 0.04 eV and 0.20 eV, respectively. The results presented in [51] showed a decreasing DC con ductivity as temperature increased from 200°K to l,000aK due to a lar ger decrease in mobility compared to the increase in carrier concentra tion. This occured more often in the oxygen-deficient phase than in the oxygen-rich phase [51]. AC measurements on sintered ZnO varistors show a decrease in C with increasing temperature over a temperature range that shifts to a higher range as frequency increases [26]. Figure 10 of [26] is repro duced below. Extrapolating the trend of this shifting temperature range to 2.45 GHz (shown dotted on the reproduced figure), approximately the same order of magnitude in conductivity and temperature range, as in Figure 37, results. The observed variation of the loss tangent with frequency suggests that it is due to electronic states within the ZnO grain-intergranular material in , >V-109 Hz \ * ' \«* \ \ terface, i.e., surface states or \ electron traps with a preferred \ energy level around 0.36 eV. The 100 mechanism of conduction is thought to be of the "hopping" type. I05H2 In accordance with the theory of fre >- quency dependent hopping, the con ►— u Z O Z O o _I U J -J ductivity at the same temperature increases with frequency. £26 J These results suggest that —I the decreasing conductivity of sintered ZnO as temperature in 2 3 4 1000/T ( 5* G r *) sence of impurities in the rod sample. creases (Figure 37), is possibly due to non-stoichiometry and pre A presence of impurities in the form of intergranular material could result in a varistor-like variation of conductivity with temperature. The extrapolated room temperature value of 900 for £ r at one GHz, of the data in [26], is 80% higher than the temperature independent va lue of 500, above 1,000°C, shown in Figure 38. served The large values of ob £ r (greater than 900) in [26] is attributed to the effect on the capacitance method of measurement by the microstructure, where conduc ting ZnO grains are separated by very thin insulating intergranular ma terial. The €r of ZnO single crystals is reported to increase rapidly with increasing temperature, i.e., a value of 10 at 100 K doubled at 166 800°K [24], This is consistent with the result presented in [27], where in the e r of sintered ZnO varistor increased as temperature increased from -30°C to 109°C. The large value of 500 in deduced €.r , Figure 38, is consistent with these facts. Single crystal NiO is a p-type semiconductor arising from Ni vacan cies with conduction mechanism of the "hopping" type [12,15] having very low mobility [12]. The electric conductivity increases with oxygen par tial pressure and the dominant defects are at the crystal surface [12]. This finding is consistent with the report that the low-resistivity sur face layer masks the electrical behavior of the bulk [48]. These results suggest that the conductivity of sintered NiO could be dominated by phe nomena on the surface of the grains. Experiments with polycrystalline thin films of NiO, with intergra-. nular material minimized, revealed a reversible switching from a highresistance state to a low-resistance state when sufficiently high elec tric field intensity is applied. The "breakdown" field intensity rapid ly decreased as temperature increased. In the low-resistance state, the NiO polycrystalline film behaved as a degenerate semiconductor with a decreasing conductivity as temperature increases. It was suggested that conduction in the low-resistance state is in highly doped filamentary regions and field-ionized filamentary microplasmas. [15]. In view of the results in[12,15,48],it is not surprising if these filaments of con duction occur on the surface of the grains. The NiO rods heated in the cavity exhibited a very rapid transition to incandescence whereas in both ZnO and were gradual. >5-alumina rods the transitions The optimum iris size at start-up for NiO was very small, 167 and the iris was almost fully opened to maintain high temperatures. These results suggest that the NiO grains were in the low-resistance state at high temperatures. Measurement of microwave conductivity, above 800°K, of NiO single crystals of 99.99% purity showed a typical semiconductor characteristic having intrinsically generated free carriers. The measurements were _3 made in an oxygen atmosphere of 10 torr. The activation energy above 1423°K of 3.67 eV corresponds to the intrinsic band-gap. activation energy below 1423°K was much lower at 0.15 eV. However, the The lower ac tivation energy was attributed to transitions between two holes and nickel vacancy. [32]. a Note that the measurements were made at low elec tric field intensities and very low oxygen partial pressure. Hence, "breakdown" did not occur and the surface defects were probably few. Nevertheless, it is possible, even under low field intensities, for a decreasing microwave conductivity to occur as temperature increases, if the increase in concentration of hole-Ni vacancy pair is less than the increase in the associated damping factor referred to in [32]. The electric field inside the cavity is at least 1,000 times more intense than what is normally used for characterization because of the heating requirements to maintain high temperatures. Hence high-field effects could be reflected in the deduced complex permittivity. It is highly probable that the decreasing conductivity of sintered NiO as tem perature increases, Figure 37, is due to "breakdown" phenomena [15]. It is also possible that the relatively high oxygen partial pressure, i.e., the rod was exposed to the air, contributed to the decreasing trend of conductivity by increased damping of the microwave absorption of a hole- 168 Ni vacancy pair. Theoretically, the characterization procedure should yield the same value of a and €r regardless of the position of the short-circuit or the size of the iris. This rests on the assumption that the rod remains homogeneous and the admittance of the short-circuit does not change. The effect of the position of the short-circuit on the deduced a and er was experimentally investigated by repeating the characteriza tion procedure at each predetermined setting of the short-circuit with no iris being used. The effect of iris size on the deduced a and er was similarly investigated by repeating the characterization procedure for each iris size with the short-circuit fixed at a given position. In both cases, the temperature of a spot on the rod's surface seen by the IRCON pyrometer was maintained throughout the experiment by operating the control system in the automatic mode. Figure 39 is a plot of deduced circuit distance for a yd-alumina rod. er as a function of short- Figure 39a is for a ZnO rod while Figure 39b is Their diameters were such that a ZnO and 0.1142 for yd-alumina. ficiently accurate. a and Again, the l x l was 0.1078 for approximation was suf The temperature of the ZnO rod was maintained at 1,300 °C while it was maintained at 1,400 °C for the yd-alumina rod, as indicated on the IRCON pyrometer. The electric conductivity of ZnO exhibited a peak near J?2 =0.9125Ag while the a of yd-alumina increased as increased. hand, the dielectric constant increased in both cases as The range of On the other Z 2 increased. Z 2 examined with ZnO was narrower than with yd -alumina due to a much sharper tuning characteristic with ZnO. The limited power 0.8T cr cr- • t 5I0 e r -i 0.7- 0.6-• © •■490 0.9 0.91 0.92 0.5 - a X t 600 ■ a -400 T cr t 80 b0.3- (a) i | 0.6t H -I h -800 -6 0 ■ a 0.75 0.8 0.85 X <r=# Pig. 39- Effect of short-circuit distance from rod on charac terisation results. Ho irisi(a)ZnO, 1300*C, 0i*0.10?8j (b) /S-Al^Oj , 1400*-C, ft=C. 1142. ( *lndlcc«d) 0 .4 - cr 1 (L) J-600 Pig. 40- Effect of iris size on characterization results, (a) cl*0.10781 lb) ft»0.0906j both, ZnO rods at 1,200*C. (*Indicated) H On NO 170 output of the microwave generator constrained the range of £ that rea lized the required temperature. The deduced conductivity dropped by 327. from a value of 0.775 mhos per m.over a change in of 0.0125Ag with ZnO while it dropped by 36% from 11 mhos per m over a change in JL2 of 0.125Ag with -alumina. The deduced dielectric constant dropped by 2% from 502 over a change in S.2 of 0.022Ag with ZnO while it dropped by 16% from a value of 65 over a change in X j 0.125Ag with y6-alumina. The relatively greater sensitivity of the deduced electric conduc tivity in ZnO to a change in £ 2 is due to *ts sharper tuning characte ristic compared to the yd-alumina case. However, in both cases, the de duced dielectric constant is less sensitive to a change in Si 2 than the deduced conductivity. Note that the data presented in Figures 31 to 33 and 39 were taken from the same respective experiments on the ZnO and yd -alumina samples. In addition, the measurements were made around the optimum short-circuit position and iris size. Comparing the experimental data in Figures 33 and 39a, the trend in the change of deduced ried. <? for ZnO is the same as that of P n as £ 2 is va A similar comparison of Figures 31 and 39b suggests that the trend in the change of deduced cr for y&-alumina is the same as that of the losses in the short-circuit as £ 2 is varied. Likewise, comparisons of Figures 31 and 39 suggest that in both the ZnO and yd-alumina cases, the trend in the change of deduced € c is the same as that of the losses in the short-circuit as X 2 is varied. Moreover, as shown in Section III-2-b, the temperature profile of 171 the rod depends upon the position of the short-circuit. These facts suggest that the dominant cause of the variation in de duced o- and e * as €.r with is varied. *-s t^ie changing non-uniform distribution of If <T and e: varies with temperature, the resul tant change in the heating profile as in the profile of 6 £ 2 is varied, results in a change and e r inside the rod. Hence, the effective <T and e r deduced by the characterization procedure depend upon jL^. The other possibility, though probably to a lesser degree, is the variation of the equivalent admittance of the short-circuit with Note that a constant short-circuit admittance of 1.3566-j3.1635 was used in the derivation of the results in Figures 37 to 39. This value was measured with i j & Q . 5Ag and at an incident power of 100 watta. In both ZnO and yS-alumina cases, the amount of heat generated in the short-cir cuit increased with in t*ie range of ^ 2 s*lown in Figure 39. the electric conductivity of metals decrease Since as temperature increases, a change in I 2 could result in a change in the admittance of the shortcircuit. The l x l approximation of z n ~ z i 2 *s 1:1/0 or^ers °f magnitude high er in the ZnO case than in the yS-alumina case of Figure 39. tion, z n 1SliZi 2 *-n /^“alumina case. In addi"? These facts and Eq.(77-b) sug gest that the field distribution in the ZnO and -alumina case is si milar to the one shown in Figures 36a and 36b, respectively. The decreasing trend of conductivity in ZnO as temperature increa ses (Figure 37) is consistent with the variation of <r with ure 39a. At ^ JL2 in Fig 0.9125A , the cavity settings are near optimum, i.e., 8 the setting that correspond to minimum available microwave power required to maintain the same temperature. 1?2 In addition, the efficiency of conver sion of microwave energy to heat in the rod is near maximum. The rela tively lower incident field intensity and the reduced reflections from the rod result in a lower total field intensity in the rod. Therefore, the peak temperatures in the two hot zones on the 0 = 0° plane are at their minimum. Thus the average conductivity of the rod is at its high est. At other values of the electric field intensity in the rod is relatively higher resulting in higher peak temperatures. The in crease in field intensity is larger for a wider departure of the shortcircuit setting from the optimum position. This is because the stand- * ing wave ratio in the cavity increases while the required incident field intensity also increases when the short-circuit is set farther from op timum. These temperature profile variations occur because the control system maintains the temperature of only a spot on the rod's surface. The results in Figure 39 suggest that the average temperature in the rod increases when the short-circuit is set farther from the optimum posi tion. Primarily the same phenomena exist with the / -alumina case. The deduced a increases as JL2 is increased because the conductivity of ji> -alumina increases with temperature (Figure 37). Note that the op timum I 2 f°r /-alumina is around 0.75Ag. The increase in deduced £r with increasing J?2 °f both ZnO and / - alumina and the similar trend in the heat generated in the short-circuit in both cases, suggest that the admittance of the short circuit varies with X 2 • This is because the dielectric constant of ZnO does not vary with temperature in this temperature range, discounting temperature pro 173 file effects. change in Moreover, the larger percentage change in € r per unit & 2 *-n t*le P -alumina case is probably due to additional con tribution of temperature profile effects, in view of the dependence of 6r on temperature in yfl-alumina (Figure 38). Figure 40 is a plot of deduced or and size. & r as a function of iris For convenience, the iris susceptance is actually the abscissa. Figure 40a refers to the same ZnO rod described above while Figure 40b is a similar plot for a smaller ZnO rod with an a of 0.0906. The tem perature was maintained at 1,200 °C in both cases, as indicated on the IRCON pyrometer. Note that these two samples were the same ones depic ted in Figures 27 and 28, respectively. The effective cases. e r was insensitive to the change in iris size in both However, the trend in the change of electric conductivity is si milar to the respective trends of the change in heat generated in the rod P _ shown in Figures 27 and 28. nr The drop in P as the diameter of the nr iris changed from 2 to 1.75 inches was 52% for the larger rod while it was 36% for the smaller rod. The corresponding drop in deduced cr was 53% and 46% for the larger and smaller rods, respectively. The primary cause of the reduction in deduced or as the iris size is decreased is the increase in electric field intensity in the rod. This increases the inhomogeneity of the rod due to a more non-uniform tempe rature profile. As a result, the average conductivity of the rod decrea ses . When an iris is present, the equivalent exciting field is expressed by Eq.(75). efficient The effect of the iris is represented by its reflection co embedded in a magnitude determining term of the equivalent exciting field. 17^ The effect of the iris is primarily to modify the magni tude of the field. After using Eq.(75) and Eq.(77-b), it is found that in all the cases shown in Figure 40, the corresponding field distribution |(E$t 2 is basically similar to the one in Figure 36a. The calculated field strength inside the rod increased as the diameter of the iris decreased. For the same iris size, the electric field intensity is higher in the smaller rod. Therefore the peak temperatures of the two hot zones are higher when the iris is smaller, resulting in a lower average conducti vity. This is because a of ZnO decreases as temperature increases in this temperature range (Figure 37), and the temperature as well as the cr near the z = 0 plane are maintained the same for any iris size by the control system. In addition to the difference in the degree of sintering, the strong er aligning field in the smaller rod could have contributed to a larger £ r and a smaller C, compared to the larger rod. The invariance of deduced er with iris size in both cases is a con sequence of the invariance of e r with temperature in ZnO (Figure 38). In view of the sensitivity of the value of deduced complex permit tivity to changes in the short-circuit position and iris size, the ques tion of which values to use arises. The position of the short-circuit and the iris size that give the most uniform temperature profile corres pond to the correct values of deduced sults for c and er . The experimental re j8-alumina and ZnO presented above suggest that this occurs when the amount of heat generated in the rod is in the vicinity of its maximum value. III-2-d. Transient Heating Response The relative occurrence in time of the total heat generated Pn and the surface temperature of the rod was investigated by operating the control system in the manual mode. The output of the IRCON pyro meter was fed to a recorder to monitor the instantaneous surface tem perature. the The instantaneous reflection coefficient was monitored by four-probe standing wave machine whose output was also fed to the same recorder. The output of the microwave generator was maintained throughout the experiment. A typical experiment starts by adjusting the TEMP SET knob to give a desired output power from the microwave generator. The generator is then turned off and the rod is inserted into the cavity. marked by the instant riment is stopped once Zero time is the microwave generator is turned on. The expe the steady-state temperature is reached. The effect of a change in the available power from the microwave generator was investigated by using a /J-alumina sample. On the other hand, the effect of a change in diameter was investigated by using ZnO samples. Figure 41 is a plot of heat generated PR versus time. It com pares two cases with different values of the available microwave power P . a The rod was S— alumina with an a r of 0.1142. and the short-circuit was fixed at a distance of No iris was used 0.875 A from the g rod axis. The cavity settings were close to the optimum settings. Eli mination of the iris increased the fraction of The lower P_ a P n generated in the rod. case reached steady-state in 55 seconds while the 176 # HEAT GENERATED (WATTS) 50+ . • • • * • • • • Pa = l7 2 W 10 - - _ ■ ■ ■ ■ ■ • I. "■ 1 " ■ ■ R, a = 112 W 70O ■ 30 Pig. 41- 0 4 0 SECONDS 80 Effect of available microwave power (Pa ) on tran sient generated heat. No iris; yS-A^O^ with d.=0.1l42. 177 other reached it in 60 seconds. Consistent with theoretical expecta tions, the normalized total heat generation P the same in both cases. n at steady-state was Thus, the increase in the actual amount of heat generated at steady-state was proportional to the increase in P . a However, the value of P^ lower in the larger were made. before reaching steady-state was slightly case, when comparisons at the same instants The difference was largest at zero seconds, i.e., approxi mately 4.5% less, and gradually diminished to zero at steady-state. Surface temperature was monitored at the same time as the heat generated. Figure 42 is a plot of the corresponding surface tempera tures versus time. The temperature of the lower Pa case increased O to a steady-state value of 1,212 C in 95 seconds while the higher P a o case reached a steady-state value of 1,563 C in 90 seconds. Both temperatures were the indicated values on the IRCON pyrometer. The heat generated in the rod is essentially P , i.e., from Figure 26, P ns is around 0.2 at Ji 2 = 0.875A . Hence, the increase in heat g generated, Figure 41, as temperature increased is due to the increase in electric conductivity of jS -alumina In the larger P (Figure 37) . case, the heat generated took 5 seconds longer, cl and the surface temperature took 5 seconds shorter, to reach steadystate, compared to the other case. in the rod at the higher P Si The larger amount of heat generated case required a slightly longer time to equilibrate and gave rise to a higher temperature. It is significant that in both cases the surface temperature reached steady-state later than the generated heat. later in the lower P a It was 40 seconds case while it was 30 seconds later in the other 1563- m• •••• • R = 172 W a. 212 ■ - b • I B ■ b R = II2 W i • ■ ■ ■ SURFACE TEMPERATURE (*C) 1?8 0 \/40 80 SECONDS Fig. h-2- Effect of available microwave power (P ) on transient surface temperature. with oC of 0.11 ^2. No iris; yS-Al20^ 179 case. This means that the temperature inside the rod is generally higher than the surface temperature. expectations cited in Section II-5. ple This is consistent with theoretical Furthermore, the yS-alumina sam used in this experiment is the same one used in the characteriza- tion experiment, i.e., Figures 37 and 38. The value of £ A of this r sample gives a typical electric field distribution similar to the one shown in Figure 36b. of This, and the fact that the electric conductivity yS-alumina increases with temperature, results in a higher tempe rature around the axial region. Figure 43 is a plot of PQ with different rod diameters. ® 's of 0.1078 and 0.0906. against time. It compares two cases The samples were ZnO with respective No iris was used and the short-circuit was fixed at a distance of 0.925 A from the rod axis. S were close to the optimum settings. The cavity settings The heat generated reached steady-state in 6 seconds in the smaller rod and 7 seconds in the larger rod. The total amount of heat generated at steady-state was smaller by 21% with the smaller rod than with the larger rod. the At zero seconds, the heat generated was 18% less with smaller rod than with the larger rod. Note that the same P Q. was maintained in both cases. In addition, the volume of the smaller rod was 30% less, and its radius was 17% less, than that of the larger rod-. Surface temperature was monitored at the same time as heat generated. Figure 44 is a plot of the corresponding surface temperatures versus time. The temperature of the smaller rod increased to a steady-state O value of 1,295 C in 19 seconds while it reached a steady-state value of 180 GENERATED 0.6+■■ HEAT 0 .8 t 0 .4 " c< = 0.108 ■ ■ ■ ■ ■ ■ cx= 0 .0 9 1 Pa= 2 6 5 W 0 .2+ H 0 1---------^ 10 20 SECONDS Fig. +3heat. Effect of rod diameter on transient generated No iris; ZnO rods; ordinate normalized to available microwave power. 181 12951250- Ba i o/ —0-091 ■ ■ ■ ■ ■ ■ ■■ ■ CC) ■ .* ^ '= 0 .1 0 8 TEMPERATURE ■ ■ ■ SURFACE ■ / / / / Pd . = 2 6 5 W H--- 1---- 1 ----1 0 20 I00 SECONDS Fig. bU— Effect of rod diameter on transient surface tem perature. No iris; ZnO rods. 182 O 1,250 C in 29 seconds in the larger rod. Both temperatures were read from the IRCON pyrometer. The normalized amount of heat generated at steady-state in the short-circuit was approximately the same at about 0.15 in both cases. This was determined from the data on optimum heating conditions (Sec tion III-2-a) taken when no iris was used. Thus, the average amount of heat generated per unit volume was slightly higher in the smaller rod compared to the larger rod. The decrease in generated heat, Figure 43, as temperature increased is due to the decrease in electric conductivity of ZnO (Figure 37). The smaller rod generated less heat because it has a smaller volume. The smaller difference in the amount of heat generated between the two rods at zero seconds compared to the difference at steady-state is due to the fact that a major portion of heat generation occurs internally and the effective volume in which this occurs grows, as steady-state is approached, at approximately the same rate. This also explains why steady-state surface temperature and generated heat was reached 10 seconds and one second faster, respectively, in the smaller rod. Figure 44 shows that, at the same instants, the surface temperature was higher on the smaller rod than on the larger rod. This is prima rily due to the lower, thermal losses from the smaller rod because of its smaller surface area, i.e., the surface area is 17% smaller. Second ly, the effective distance that the internally generated heat has to traverse while diffusing to the surface is shorter in the smaller rod. This also explains why the steady-state surface temperature reached steady-state 10 seconds faster on the smaller rod than on the larger rod. The phenomenon in yS-alumina wherein the surface temperature lags behind the generated heat also occurs in ZnO. The surface temperature of the smaller ZnO rod reached steady-state 13 seconds later than the generated heat while it was 22 seconds later in the larger ZnO rod. Hence, even in ZnO, where the electric conductivity decreases as tem perature increases, the internal temperatures are also generally higher than the surface temepratures. To verify the simple theory presented in Section II-5-b, was used to calculate the average surface temperature, given P the emissivity and diameter of the rod. Eq. (158) nr and The amount of heat generated in the rod was calculated by subtracting the amount of heat generated in the short-circuit from P P . A value of 0.2 n a used in the and 0.15 for P was ns -alumina and ZnO cases, respectively. Using the "T^ Law" to deduce the actual temperature and emissivity of the rod, Figures 16 and 17, the respective measured steady-state temperature and emissivity of the A O yS-alumina sample were 1,386 c O case and 1,665 C and 0.161 and 0.145 in the lower P in the higher P& a For the ZnO O sample, the corresponding steady-state values were 1,310 C and 0.195 o case. in the smaller diameter case and 1,269 C and 0.195 in the larger dia meter case. The calculated surface temperature was derived by equating the right-hand side of Eq. (158) to Pnr/S where'S is the product of the circumference of the rod and its length. O were 1,357 C in the lower P cl The results for o case and 1,526 c y8-alumina in the higher P cL case. The former is only 2.1% less, while the later is 8.4% less, than the I8if respective measured values. The results for ZnO were 1,430°C in the smaller diameter case and 1,471°C in the larger diameter case. The for mer is 9.2% higher, while the later is 15.9% higher than the respective measured values. The electric field profile inthe ji-alumina cases is that in Figure 36b while that of the ZnO cases is Figure 36a. similar to similar to that in These field distributions are consistent with the fact that the calculated average temperature is larger in the ZnO cases and smaller in the ft -alumina cases compared to their respective measured values. This is because the measured temperature is on a spot where practically no heat is generated in ZnO while a relatively significant amount is •A generated in fi-alumina. In addition, these field distributions are con sistent with the larger discrepancies between the measured and calculated temperatures in the ZnO cases relative to the because bution in ^-alumirra cases. there is more azimuthal symmetry in the electric This is field distri -alumina than in ZnO. The simple model discussed -in Section II-5-b is satisfactory for obtaining a first approximation to the stable temperature especially when the temperature profile is uniform. CHAPTER IV CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK The variational solution to the impedances of the rod, Section II-l, is accurate, regardless of the diameter of the rod or whether the rod is in a resonance. To maintain the same accuracy, the number of terms needed to calculate the impedances increases as the diameter of the rod increases. However, when the diameter is small, the solution converges quickly. A useful criterion for determining the required number of terms is the value of accurate. a . When a is less than 0.1, the l x l approximation is This is generally true when the rod is not at resonance. How ever, the occurrence of resonance may be avoided by a proper choice of a smaller rod diameter. When a is less than 0.2 and the electric con ductivity is zero, the l x l solution. approximation is the same as the convergent This is true for any e■ less than 200, including those around resonance. To avail of the accuracy of the variational solution, without using the very cumbersome higher-order approximations, it is recommen ded that the smallest possible rod diameter be used in characterization work. In addition, the diameter must be chosen to avoid resonance. A table showing the resonant values of the complex dielectric constant as a function of a will be helpful in determining the proper rod diameter. Aside from simplifying the calculations, a small rod has a more uniform 185 186 radial and azimuthal temperature profile than larger rods. varies with temperature, the rod becomes inhomogeneous as a When e* result of the non-uniform temperature profile. Hence, the deduced value of the complex dielectric constant is closer to the correct one, when the rod is smaller. However, there is a lower limit to the rod dia meter that can achieve the same temperature. The maximum output power of the microwave generator determines the maximum temperature 'that can be achieved, for a given rod diameter and cavity settings. For a given temperature, the required power out put from the microwave generator can be minimized by adjusting the short-circuit position and iris size to their optimal settings. Opera ting under optimum heating conditions maximizes the temperature range that can be achieved with a given microwave generator. The experimental results of fi -alumina and zinc oxide show that the deduced value of the complex dielectric constant at the same surface temperature varies with the position of the short-circuit and size of the iris. The analysis of the experimental results suggests that the temperature profile is more uniform when the amount of heat generated in the rod is maximum. The cavity settings that result in minimum available power require ment and those that maximizes the amount of heat generated in the rod are identical when there are no losses in the iris, short-circuit and cavity walls. The full advantanges of operating under optimum heating conditions, i.e., wider achievable temperature range and a deduced value of the 187 complex dielectric constant that is closer to the true value, may be realized by minimizing the losses in the cavity. Silver plating the iris and cavity walls and using a high-performance short-circuit, e.g., one that gives a VSWR greater than 300, are therefore recommended. A lossless applicator makes the task of predicting the optimum heating conditions easier. For instance, when the rod is thin and the electric conductivity is high, e.g., /3-alumina with an a of 0.1, the T-equivalent circuit may be approximated by the shunt equivalent circuit. In this case, Sec tion II-4-a, the optimum distance of the short-circuit from the rod axis lies between zero and a quarter wavelength when, the rod is capacitive while it lies between a quarter wavelength and a half wavelength when the rod is inductive. In addition, the optimum iris size is larger when the conductance of the rod increases. Finally, total absorption of available microwave power by the rod is realizable when the normalized conductance of the rod is less than or equal to one. The experimental results of zinc oxide and /}-alumina indicate that a good criterion for determining the relative iris size is 2 aR , i.e., the product of the electric conductivity and square of the radius of the rod. The experimental results show that even with large amount of losses in the iris and short-circuit, the optimum iris size is lar ger when a R^ increases. power increases as to This Is because the required amount of <rR^ increases; the iris size must be increased transmit a larger portion of the available microwave wards the rod. Note that an increase in oR 2 power to- correponds to an in crease in the rod conductance of the thin-rod case. 188 The normalized amount of heat generated in the rod P nr is also the efficiency of converting the available microwave energy to useful heat. When the losses in the iris and short-circuit are significant, the efficiency is small, e.g., the experimental results of ZnO show that the maximum efficiency was one half of the ideal value. The efficiency of energy conversion depends upon the settings of the iris and short-circuit. It also depends upon the complex dielectric constant of the rod and the losses in the applicator. Even with a loss less applicator, the efficiency will still be low if the short-circuit and iris were not adjusted to their optimal position and size, respec tively. On the other hand, with a lossy applicator, it is possible to have low efficiency even when all of the available microwave power is converted to heat, i.e., P “ 1. In either case, the temperature of the rod may be lower than the desired one. ZnO indicate that increasing ctR 2 The experimental results of ZnO and The experimental results of increases the maximum efficiency. (i -alumina show that the amount of losses in the iris and short-circuit varies with the short-circuit po sition. In particular, when no iris was used, the amount of heat losses in the short-circuit was minimum at maximum efficiency. The proper criterion then for optimal short-circuit position and iris size is not maximum P n but maximum P nr or maximum efficiency. When the losses in the applicator are negligible, it is easier to moni tor the incidence of maximum efficiency. P n and P nr are approximately equal and P r n by the four-probe standing wave machine. This is because, in this case, can be monitored easily In contrast, monitoring P 189 involves monitoring deducing the complex dielectric constant of the rod by using the characterization procedure and then calculating Pnr • The optimum heating characteristics when the rod is represented by its T-equivalent circuit is illustrated by the ZnO case. The experi mental results of ZnO suggest that when the rod has a small electric conductivity, e.g., less than 1 mho per m, and large dielectric cons tant, e.g., greater than 500, the T-equivalent circuit does not reduce to the shunt equivalent circuit even when a is small, e.g., a of 0.1. Secondly, the experimental results suggest that large dielectric cons tant makes the cavity tuning sharper, e.g., a given change in short-cir cuit position results in a larger change in ted theoretically by Eq. (84) P nr . This is also indica- for the thin rod case. Finally, the experimental results indicate that the optimum iris size increases when the short-circuit is farther away from the rod than the optimum tance. This is also indicated theoretically by Eq. rod case when the rod is inductive dis (110) for the thin and its conductance is negligible. The field that excites the rod is a standing wave. The total field inside the rod is related to the position of the peak of the exciting field relative to the rod. Since the amount of heating in the rod de pends upon the total field, the efficiency of energy conversion de pends upon the position of the peak of the exciting field. Theoretically, if the impedances of the T-equivalent circuit of the rod are real, the peak of the equivalent exciting field is exactly at the rod axis when maximum efficiency occurs. sistent with the experimental results of This result is con /$-alumina and zinc oxide 190 shown in Figures 32 and 33. The distance to the rod axis of the nearest peak of the equivalent exciting field was nearer in the case, at maximum Pn r - /5-alumina The rods have approximately the same diameters. Thus, the only significant difference between the two cases is the value of the complex dielectric constant of the rod. rod impedances Z + Z and Z Xm nary parts in - Z XX The real parts of the , were larger than the imagi- XX -alumina while it is the reverse in ZnO. Hence, the distance to the rod axis of the nearest peak of the equivalent exciting field is smaller in the ^-alumina case because the rod impedances are more real than in the ZnO case. shown in Figure 32 at The experimental results of /S-alumina = 0.75 A^ is also consistent with the theo retical results in Figure 7 to 9 which used a shunt equivalent circuit for the rod. Note that the valent circuit is accurate. yS-alumina case is one where a shunt equi At X & = 0.75 A , © mined x ^ /r2 was approximately 0.3 and r^ results for z , P p nr and optimum t 2 was 1.07. 2 is approximately 0.8 A g The theoretical in Figure 7 to 9 when r0 = 1 £■ are consistent with the experimental results. I the experimentally deter- The theoretical optimum while the experimental optimum I 2 is appro- ximately 0.75 A . Note that their difference is approximately equal g to the displacement of the actual standing wave minim um by 0.042 A beO hind the face of the short-circuit. Moreover, the values almost the same; and, the experimental of z^ are is lower than the theoreti cal by around 0.1 due to the losses in the actual cavity. In any case, when the rod impedances are complex, the distance to the rod axis of the nearest peak of the equivalent exciting field is non-zero, when maximum efficiency occurs. 191 The difference in temperature between the back side and front side of the rod is directly circuit. related to the position of the short- This is suggested by the experimental results shown in Figures 34a and 35a. The position of the short-circuit sets the distance of the peak of the exciting field from the rod; and, the position of the peak of the exciting field determines the profile of the total field in the rod. For instance, if the nearest peak of the exciting electric field is at the back side of the rod, face of the rod is higher than in the total field near the opposite region near the back sur the front surface, because of the decreasing amount of excitation away from the peak. As a consequence, the temperature of the front surface is less than that of the back surface. If the short-circuit is moved farther away from the rod, the peak of the exciting field also moves in the same direction.These results in a higher temperature difference bet ween the front and back surfaces of the rod. The electric field profile in the rod is determined not only by the position of the short-circuit but also by the impedances of the rod. The field profile is dominated by the ^ ( k ' r ) distribution - Z^2 is small relative to Z11 + ^12 * the cos0 J^(k'r) Z ^ 4- Z^ 2 * distribution when This is true when the 1 x 1 when while it is dominated by is much larger than approximation is valid. The experimental results shown in Figure 34 to 36 show that the distance of the peak of the exciting field from the rod is non-zero when the tem perature of the front and back surfaces of the rod are equal. This is because the electric field profile in the rod has more radial symmetry 192 on the x = a/2 plane at this distance than when the peak is directly at the axis of the rod. In addition, the experimental results suggest * that, when e is constant, this distance is closer to the rod hhan r * the distance that corresponds to maximum efficiency. In any case, the position of the short-circuit that gives maximum efficiency and the one that results in a more uniform temperature profile are close to one another, e.g. within 0.1 A . g However, it is recommended that further study be made on how the difference in short-circuit positions, betweer the one at maximum effi ciency and the one at a more uniform temperature profile, vary as a function of the rod diameter, and short-circuit. e , iris size and the losses in the iris r The case when both iris size and short-circuit positions are optimal is particularly interesting. Since the tempera ture profile is more uniform when the rod is smaller, it is more use ful to confine the study to cases when the simpler 1 x 1 is valid. approximation The objective is to determine how much is the difference between the characterization results taken at maximum efficiency and the ones taken when the temperature profile is most uniform. The electric field profile in a rod with higher electric conduc tivity has more azimuthal uniformity. is small enough that the l x l This is true when the diameter approximation is valid. These are sug gested by the experimental results shown in Figures 32 to 36. This is because, the higher conductivity cases, e.,g., 10 mhos per m, have Z a Z , while the lower conductivity cases, e.g., less than 2 mhos 11 12 per m, have a Z - Z^ of the same order of magnitude as Z ^ + Z ^ . 193 Therefore, the field profile in the high conductivity cases is domi nantly JQ (k’r) while it is dominantly cos© J-^k’r) for the other cases. The characterization results for ZnO suggest that the dielectric constant increases as the exciting field becomes stronger and the asso ciated electric conductivity decreases when the dielectric constant increases. & When varies with temperature, the characterization results at r the same surface temperature vary with the position of the short-circuit and the size and shape of the iris. The primary cause is the change in the electric field profile that results after changing the iris and the short-circuit position. suits This is because changing the field profile re- in a different temperature profile. Thus, the effective * changes as a consequence of a change in the non-uniform profile of * * The variation in deduced e. is particularly large, for a given change in cavity settings, when the rod has a sharp "tuning" characte* ristic. Nevertheless, the deduced e r that corresponds to the most uniform temperature profile is closest to the correct value. It is therefore recommended that a study be made to ascertain the *' amount of error incurred in using the "effective" value of £ , which * is the e of a homogeneous rod of the same diameter and having the r same reflection and transmission coefficients as the actual rod. This will involve finding the solution to the coupled problems of determining the electromagnetic field and temperature profiles at steady-state, cited in Section II-5. In this regard, the Schwinger variational so rt: lution may still be applicable with e being a function of space 19^ instead of being a constant. The solution will depend upon what tem* perature dependence, e.g., linear, is assumed for conductivity and heat capacity of the rod. e , the thermal r In sintering, the mass den sity and the dimensions of the rod varies with time. The more difficult transient solution will have to be found in this case. To provide some insights into the solution of the coupled problem, which is in', general non-linear, solutions to some de-coupled problems, * i.e., e. is not a function of temperature, may be studied first. The r * * * * following cases of e are suggested: (1) £ = e (r,0); (.2) £ = * r * r r r £ (r,0,z), for circular rods; (3) £ varying on the cross-section; r * r (4) £ varying in three dimensions, for non-circular rods; (5) homor * geneous rod of non-uniform cross-section; (6) £ varying on the crossr section and in three dimensions, respectively, for rods of non-uniform cross-section. In these cases, Schwinger’s variational solution for the equivalent impedances may be used. The variational solution for the homogeneous circular rod should also be extended to include the situation where the rod to be heated and characterized is enclosed by a tubular envelope, e.g.,quartz tube. This is the case when the material to be studied is a gas, e.g., plasma, or a liquid. The equivalent rod has two homogeneous regions with dif ferent dielectric constants. The effect of the envelope on the accu- -k racy of the deduced £ r will therefore be ascertained. The results will also apply to the case when the rod is actually a tube. * The small contributions to the error incurred in deducing to changes in the admittances of the iris and short-circuit £ r may be due 195 reduced by minimizing the amount of losses in these components, e.g., by silver-plating their surfaces. lizable This will also increase the rea susceptance, as suggested by the experimental results on the circular iris presented in Section III-1-b. * The effect on the accuracy of the deduced e of the perturbation r in the electromagnetic field near the ends of the rod, caused by the entry and exit holes, should also be studied. The length of the rod will be a factor to consider. The surface temperature of the rod reaches steady-state later than the generated heat. This is a consequence of internal heat generation and thermal losses at the surface, which results in higher internal temperatures than the surface. The generated heat reaches steady-state more quickly when the electric conductivity decreases as temperature increases, than when the electric conductivity increases with temperature. Radiation and convection losses from the cylindrical surface of the rod are the pricipal thermal losses that determine the stable tempera ture. The effects of increasing the available microwave power are higher temperatures and a reduction in the time lag, of the occurrence of steady-state in surface temperature, from the instant the generated heat reached steady-state. With the same amount of heat generated, a smaller rod achieves higher temperatures due to a smaller surface area, which requires a higher thermal loss per unit area. For the same reason, and because 196 of a shorter "thermal diffusion, length" to the surface, it reaches steady-state more quickly than larger rods. The experimental results of ZnO show that thermal instability cannot occur when the electric conductivity decreases- as temperature increases, because the usual temperature limiting process of thermal losses is aided by a self-limiting heat generation process. This is also suggested by the theorectical result in Section II-6. In addition, the theoretical results suggest the following "rule of thumb": thermal instability is more likely to occur when the thermal conductivity is low, the rod diameter is large or when the available microwave power is high. A more exact transient response analysis is necessary to study * the effects of certain variations, e.g., linear, of e , thermal conr ductivity and heat capacity with temperature on thermal stability. The transient solution to the coupled problem cited earlier is required then. As a first-approximation to the difficult coupled problem, the decoupled problem may be studied first. An interesting decoupled problem is expressed by Eq. (159), which considers the effect of translating the rod along its axis. Note that translation provides an additional means of "losing" the heat generated in the exposed region of the rod. make the heating process more stable. Thus, it is expected to Nevertheless, it will be interes ting to see the details of how translation affects thermal stability and temperature profile, specially looking at the solution of Eq. along the length of the rod, by (159). 197 The temperature control system senses the surface and maintains it at the desired value. temperature If the time lag of the surface temperature from the generated heat is long, thermal instability could occur,with the present control system unable to prevent it. The direct way of controlling the internal temperatures of the rod is to sense the amount of heat generated in the rod. probe standing wave machine. This can be done via the four- If the cavity has negligible losses, the four-probe standing wave machine is essentially sensing the amount of heat generated in the rod. The generated heat is more indicative of the average temperature than the temperature of a Hence, using the generated heat spot on the surface. in the rod as the controlled variable will improve the speed of response of the present temperature control system and will more likely prevent thermal instability from occuring. The outputs of the present four-probe SWR to the available microwave power. machine is proportional In addition, reflections from the source side introduce an error in the measurement of p , which be comes more severe as the amount of reflection increases (Section III-lg). It is therefore necessary to minimize the reflections from the source and remove the dependence on the available microwave power be fore the present four-probe system can be used as the sensing component of the improved temperature control system. [17] will remove the dependence on flections from the source. Using dividing circuits P and reduce the effect of rea This is the case if the dividing signal is picked up by a directional coupler in the vicinity of the four-probe. Reflections from the source can be minimized by using high-performance 198 circulator and by removing possible w a v e g u i d e , e.g., causes of discontinuity in the misaligned flanges.connecting the slotted waveguide to the circulator. An additional signal processing circuit is also needed to convert the two outputs of the four-probe system into a single output propor- 2 2 tional to P (1- -|p| ). This is because the quantity (1 - | p\ ), where cl p is the reflection coefficient measured by the four-probe system, is essentially the normalized heat generated in the rod if the losses in the cavity are negligible. The method of deducing the actual temperature and emissivity of a surface by using the measurements of a monochromatic pyrometer and a relatively broad-band pyrometer, whose wavelength-band includes the wavelength of sensitivity of the monochromatic pyrometer, was presented in Section III-1-e. It will be worthwhile to investigate further the accuracy of this method. The usefulness of the present characterization system can be greatly enhanced by using a microwave generator whose frequency can be varied and maintained precisely at the set value. This will add * the dimension of frequency-dependence of e. r into the results as well as make the task of "tuning" the cavity more flexible. 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"Elec trical Properties and Non-Stoichiometry in ZnO Single Crys tals," Physica Status Solidi (a), 66 , No. 2 (1981), pp. 635-648. Appendix A Derivation of the Marcuvitz Model for ^ ^ As suggested in [41], (er* ~ D k 2 j(zn + z12) Ka /<}>edxd2 - //(t>e (x,z)G'(x,y |x',z')<t>e (x',z') dx dz dx* dz ’ ( U j e dx d z ) 2 (4—1) where the integration is over the cross-section of the rod. Green’s theorem was used to transform the surface integral to a much simpler line integral around the circumference of the rod cross-section. Hence, Eq. (A-l) becomes _________ (er*-l)k2 J(2 U + Z 12) Ka (2 tt (e* - l)k2 0=0 3*e 3r - <f>e [ we ' o ^ / Rd0 0 f e e 3r -1 e=o (e* - l)k ft-r J 3r 3i|> \ 2 *e-if]) (A-2) Thus, 2 tt Rd0 9=0 e 3r e 3r (A-3) Z 11 + Z 12‘ e Sr 9=0 204 205 where 2tt w (x,z) = / e 0 ,=0 r Rd0' G* 3r* (A-4) 3r? (A-5) <J»e (r,0 ) = tJj 6 (x , z ) = sin -- cos cl (A-6) k z In evaluating the integrals in Eq.(A-3) and (A-4), the use of Schwinger's lemma and its corollary[41] makes it straightforward. Note that the constant 27tR (rod circumference) in Eq.(A-5) was chosen for convenience. Before proceeding, note that Schwinger's lemma states that "if u(x,z) =■ u(r,9) is any solution of the two-dimensional, sourcefree wave equation within a circular region of radius R, then for any r < R, Z tt / i. u(r, 0 )e d0 = J (kr)e m u(0) (A-7) jmD where the operator e is defined by " (A-8) Note that the coordinates are as shown in Figure A-l 206 x 4 I ’( r , 9 ) / 9 R x, »■ z Fig. A-l: Coordinates Used The corollary is i 1 2i , . .. jmQ r 8u(r,6) _ 8r 17 J “ e de jm J' (kr) e m jmD u(0) (A-9) As pointed out by Schwinger, Eq. (A-7) and (A-9) cannot be applied directly to 6 ’. However, if G' is expressed as G'Cx^lx'iZ') = r ,(x,z|x,,z') + G s (r,r') (A-10) where ((r,r*) = - j Yq = - 4 h (k|r - r'|) (A-ll) 1 Ym (kr)J(k r') e m “ in .00 when r > r' those equations can be applied to the I” (x,z|xf,z') part of the (A-12) 20 7 integrals in Eq. (A-3) and (A-4); the Gg part can be integrated directly. Note that G s is actually the infinite space Green’s function. Since 4>e (r,d) in Eq. (A-5) is not a function of 9, the typical integral will have as a result a term .0 eJ u(0) = Lim u(r,9) = Lim u(x,z) r-*0 X-MCq (A-13) z->0 As a consequence the problem'of evaluating- Lim Lim I” (x,z |x* ,z’) X-»-Xq x ’->Xq z->0 arises. z *-*0 This is because T ’ (x,z|x’,z’) — sin — sin icd & 3L sin k|z-zf j TITTX (A-14) + J *0 (k|r-r’|) and Lim Lim r >0 r ’-K) ± Y0 (k|r-r'|) - Lta [ ± «»(^)] - (A-15) It was therefore suggested£41] that the infinite series be re-expressed to cancel the limit of Eq. (A-15). First, 1 TTX ux0 sin — sin --- sin <a a a Lim T ’(x,z |x',z') 208 kz x ’-^o z ’->-0 •fK Z rnrx + (j/a) £ n =2 sin ■* * n T ■ e sin n (A-16) + f Y 0 <kr) The appropriate series is arrived at by noting that j Cd>^—<t»2^ (1) sin e sin <f>2 = Re - e (A-17) oo (2 ) I If we let (A-18) -S-------- ln(l - eJ*) = Orx)/a, (jig = 0rxQ)/a, 2 ” (ir/a)z an<* use Eq. (A-17) and (A-18), £ n=l — sin n<|>. sin nif^e n (-mr/a)z jn[<f>1-(Ji2+j(Tr/a)z] = |se^ I 1 — n e - I 1 , f ~ e n T j(<('1-H'o)-(Tr/a)z ■j Re< An[l - e Z “I r j (<(>.-<(0-(ir/a)zl J- Anil - e 1 ■ (A-19) J 209 The limit of the second term on the right-hand side of (A-19) approaches +» as x can cancel (A-15). Xg and z -*■ 0. This is therefore the term that Hence, j (2 itXq) /a Lim 1 - e = 1 - e = 2ttXq\ 1 - cos — — j ***C z-*0 2 ttXq - j sin — — (A— 20) j (<|)1-<j)2 )-(TT/a)z Lim 1 - e X-KXq z->0 = Lim [-J(x -X q ) + z] X-^Xq (A— 21) z-»-0 However, Re[ln(a + jb)] r 2 . . 2. jI tln(a + b ) Using Eq. (A-22) in (A-20) and (A-21), respectively, gives (A— 22) 210 irx = in 0 2 sin (A-24) Using Eq. (A-23) and (A-24) in (A-19), Lim I I x*xQ v n =2 I s i n ^ s i n ^ e - (nir/a)z n 3 z-*0 in in (2 sin r-K) V a / - I sin 2 ^ ir a (A-25) Hence Ik z J n Lim I” (x,z|x ,0) X-XQ r sin . mrx sin , Lim —1 ) tt “0 a x+Xq n =2 z-K) z-K) mrx, 0 a < i - (nn/ a) z mix. + Lim — T sin — — x+Xq " n =2 z-K) + Li ± ln(f) r->0 w 2 nirx0 ' J 2 31”: sin a-(mr/a)z nir 211 + i £ „ (l s i n ^ - ili. , 2 ”*0 . 1 .. n sin — — + — Lim £n 17 a ^ r-vO /ckr\ (— 5— ) 1 - — K 2 ' CO = ± I TT n =2~ sin 2 n7rx0 n p -( # 1TX, 1 4 2 "*0 sin --7r a fSSS.) sin — \ ir / a + 27^ (A-26) Evaluating w e (x,z), we obtain w (x,z) - f2lT J 0 '=0 Ao ,0t , 9[J <k'r’)] 3(I"-H3 ) ^nt de ; tr- + g_) 0 - J0 (kT') - s p - S 2tt ,2lT -k* J ^ B ) A 1 / 0 e’=o # 7 - cr* + G ) L 2tt Jo<6) A O 0I^O 8(1” + G ) 27" aP (A—27) where B = k*R = /e * k R = /e^* a Using SchwingerTs lemma and its corollary, (A-28) 212 2n f ,R, ju J_(a) Lim r* — e ’=o Z1T 2 tt f 9*=0 0 r’ = J (a) Lim 0 * ,»x 0 z'+O *'-«o z '-*-0 d 9 1 3T' U r - jU J ’<kr)[ 2tt 3r* Ir=R (A-29a) I” Lim T * = -kJj(a) x*-hx0 x Lim r* ’-X q z'+O z'-K) (A-29b) On the other hand, using Eq. (A-12) gives 2 ir ; 2 tt e '=o /^ 1 s 2tt u — 00 m m 0 '=0 " “ T Yo<kr>Jo(ot) 2 ir r2ir d6* 3Gs 2tt 3r * 9*= 0 (A-30a) ) 0-0 (A-30b) = YQ (kr)kJ1 (a) This is because '0 , n # 0 f2ir J 0 ’=0 i aQ' de* e “2? Z7r (A-31) il. n * 0 Using Eq. (A-28) , (A-29), and (A—30) in (A—27), w e (x,z) - | J L(g) JQ (a) [ Lim I" - -J- Y 0 (kr).] x ’-^-Xq z'-K) 213 - J0 (&) | J x(a) [-Lim r' + J Y 0 (kr)] x '-*xo z '->0 [ctJ^a) JQ (0) - 0^(0) JQ (a)][ Lim r 1 --i-Y^kr)] (A-32) *»-*0 z '-*0 Using Eq. (A-32), the numerator of (A-3) becomes A0W “Ji(c° V » - BJjCB) J0C.)1 /* ff j f x'->x. Lim r' - i r0 (kt) .z '-*0 Lim [-k'JjCk'r)] - JQ (k'r)- r ' “ J Yo (kr) z'-*Q 3r AO — Qo1 2ir X -*X, z ’-*0 Lim x*-«c 2 tt - V 8) / d 0 _ z ’-vO 2tr r' --5-Y0 (kr) dr (A-33) where Q q = aJj^cOJgCB) - BJ^CB) Jq (ci) ; Using Schwinger’s lemma and its corollary, 2tt de vlm x 0 JQ (a) Lim r’ Lim Y (A-34a) r’ x '**o z-»0 z '->0 J Lim • z *->0 T' d9 Lz’->0 2ir 3r = (ce) Lim V y z-*0 Lim (A-34b) r* *'-**0 z '->0 On the other hand, 2 tt ,q . 3Yn (kr) 2ir 3r - f V “> (A-35a, A-35b) Using Eq, (A-34) and (A-35) in (A-33), fa de4 3*e J I e aF" . - £ [oJ^a) J 0 (6) 3w \ **) eJ l(g) J-(a)] Lim Lim z-+0 + Tr ^ ( f O Y ^ a ) R Q 0 Lim z-*0 Lim z r->0 r* z *->-0 “ aJo (B) Y 1 (<X) I” + -J Y 0 (a) aJ 0 (g) Y^aflj (A-36) 215 The Integral in the denominator of Eq. (A-3) was shown in [41} to be, f 2ir J Rde 3<j> K a r - r-x. An 3^e,\ irrj m (A-37) Q o 3i" Using Eq. (A—36) and (A-37) in (A-3) gives r Z11 + Z12 Lim Lim x>x0 x ’->x0 = (-jica) esc z-K) z '-*0 1 BJ 1 (e)Y0 (a) 4 oJ 0 (B)J1 (o) - BJ 1 (3)J0 (a) aJ 0 (3)Y1 (a) (A-38) When ot « 1, V 00 J0(O and Y 2 ( a)» a (A-39a, A-39b) Y (a) Y 0 (a) - aJ 0 (3) Yj(a) - 8^ ( 6) JQ (a) Y.(a) - aJ 0 (B) J L(a) 3 - ^ 5- Yn (a) - 8Jj(3) JQ (a) ^ _ ajo (g) j i(o) § [Y (a) +.Y 2 (a)] J^a) where the last expression is derived from Eq. (A-39) Using Eq. (A-40) in (A-38) gives 2 irx0 Z 1 I + Z 12 = "jKa CSC ~a~ Lim Lim T' X-KXq x ’-^0 z-*0 z '->0 - 27 [ln (¥} 1 V e> 2it JQ (a) aJ0 (B)J1 (a) - BJ^CBJJqCcx) (A-41) Inserting Eq. (A-26) in (A-41) results in the Marcuvitz model for Z 11 + Z 1 2 : 2 niTXQ 09 z ^ + Z 12 = - j t c a C S C 2 **0 1 — IT r ) L r> n =2 , sin / n2- (ka/ir)2 217 1 V e) 2ir JQ (a) 2 UX0 ° “J T “ C8C 2 i sin n =>2 aJQ (3)J1 (a) - (3^(3) JQ <ct) 2 nirxb / n 2-(2a/X)2 T 2a + «.n % sin — 2- •n I i— ttR - ( f ) ^ q(«) 2 sin2 "*0 aJQ (6)J 1 (a) - SJjiOiJQte) (A-42) where k = (2ir)/X and k =. (2ir)/X. S Note that Eq. (A-42) is an approximation of Eq. (A-38) when a << 1 and that the final result is independent of the constant A^. Appendix B Derivation of the Marcuvltz Model for ^ - Z ^2 As suggested in [41], j(er* - l)k* <aCZlX - Z12) dx dz - J/(|>o(x»z) G'(x,z|x’,z') <j>0 (x’,z’) dx dz dx* dz' (B-l) where the integration is over the cross-section of the rod. As in the even case, Green’s theorem was used to transform the surface integral in Eq. (B-l) to a much simpler line integral around the circumference of the rod cross-section. r„ 2 ir R ds K 8 LW 0 Sr [ 0*0 = -(jxa) zn J 0=0 ^ *0 3r (B-2) 2 it “ zi2 - Thus, R d9 a^o 3wo * 0 .ar “ ^0 3r where 2rr f 0 ’=0 wQ (x,z) $o (r,0) = iJ /q (x ,z ) B1 = sin — R d0' G. 212-* SSL G *o j r JjCk'r) cos 0 sin (B—3) CB-4) (B-5) kz 218 219 Evaluating wQ (x,z), a[cos 0' JjCk’r 1)] f2ird9 w,Q (x,z) JB i u 0 ’=O (r* + g ) s 30” + g ) - cos 0* JjCk'r') --- ------- f> ^ JQ 1 A 8=0 - » . e ' « " + Gs> 3(r'+os) d6. 2, ' Jl< H e.j “ 3 9' & -(B-6 ) Using Schwlnger* s lemma and its corollary, 2tt r-ifl.Re r cos 9'. r' 2tt Qt J Q / e ’=o eJ 6 ’ Q-ni r sr* = Re< X-Xq [ % 2x A z ’-K) _ , . = J (a) .. Lim x '+Xq l ai” ^ jp- (B-7a) Z*-i-0 2tt Jf‘" cos G ar’ ---V e'=o 3r Re _ -de' ~5” Re Zir Iv<“>xLim f-*x, z ’-*-0 f 0 ,=o s39’ r. ar* il l 3z' + j 3x' 1 220 1 t v < a> x'-+x„ k 3r” ap- (B-7b) :*-H) From lq. (A-12), 2ir f e ’=o * d9' cos 9 * G 8 2ir 2ir / j 6 * = | J (e 2ir ( J9* . T 2 / Ve -j0 ' + e 9 ' =0 -j0 ’ +e 2 0 ' =0 ^ MI Js 2ir jm(9—9') / n ~ jmie-e-; ^ d0 I" T ) I ^m( kr)Jm 0 cr*)e 2tt m (B-8 a) J^(a) cos 9 Y^(kr) 3G 2ir J e»=o cos 0 ' 3r , ■^-KS s d9l 2ir = “ i f Ji ,(a) cos 0Yi (kr) (B-8b) Using Eq. (B-7) ind (B-8 ) in Eq. (B-6 ), wQ (x,z) B^ f . yce) ^ ( a ) T, i ar’ k a T •7- cos 9 Y. (kr) 4 1 x ’-X q .z ’-K) i ar* k az’ z'-K) 221 - -jr cos 0 Y^(kr) ^ [3 (3) J x(a) - 0^(3) *(ct) ] i_ 3r* Lim k 3z' x ’-vx0 z ’-K) - cos 0 Y^(kr) (B-9) Using the identity (B—10) In Eq. (B-9), wQ (x,z) B. j f [SJq(B) J x(a) - aJ,(3) J (a)] Jn i ar T . x o z' +O - -jr cos 0 Y^(kr) (B-ll) Using Eq. (B-ll), the numerator of (B-2) becomes 2tt J H, B, 0Wf aJ 1 (3)J0 (a)l 3r 2n / 0=0 d© 2 ttJ ■»j_ ill I .^iin 3z' k x'-wc0 z '+0 - ^ cos 0 Y 1 (kr) J 1 ,(3) cos 0 Lim 7~ 222 i ar' - -7- COS 0 Y t(kr) k 3z 4 7" "s— r x ^0 z ’-H) - cos 9 JjCB) 9r 2n B E-«i{ f Ji'<« f 1 srr “ ■ k"aP- COS 0 X 0=0 XQ z'-H) - 2tv / cos 0 _ J-i (B) d0 2tt cos 0 Y^(kr) 't * 1 ar* 1 Q v (kr) L^m k J p - - 4 cos 9 Y ] x -^b - z '->0 ' — I 3r 0 (B-12) where Q l = BJ0 (B)J1 (a) - <*^(3) ^ ( 0 ). Schwinger's lennna and its corollary give r2” ^ae cos J 0=0 1 3r' J-JJT 6 Lim X -**o z'-H) /n 2 = Jj(a) Lim Lim x-*x0 x'-*xn 0 z-K) 0 \0zV oz z'-H) (B-13a) 223 Lim f ill 3z' x'-»xr 2ir I =0 - z '->0 d9 1 cos 6 2ir 3r f OT '(a) Lim (i) Lim (B-13b) ***b x z-K) z'-K) On the other hand, 2 tt 2tt / cos20 Y ^ k r ) |2, = Y l (a) J 2tt J atYjCkr)] 2 cos 0 dQ gj 27 - j Y^a) cos20 | | a 2tt “ R Y 1 (0t) J 2 (B-14a) dQ COS 6 27 Using Eq. (B-13) and (B-14) in (B-12), 2ir J, / R 3<f>« ae( wos ^n\ r - *0i r \ ) - \-Tj C «i -<,^(3)^'(=.)] Lim Lim z-K) z?-+0 (i)* [6J 1 '(B)Y1 (a) - aJ 1 (8)Y 1 '(a)] I (B-15) 22^ Using Eq. (B-10) and a similar identity for Yn r(°0 in Eq. (B-15), 2tt / J R d0 =0 \ 3<frQ (wQ 3r 3wQ “ <l>0 3r Lim . (Vf Lim t xkJ ***b xt‘>xo z-H) z'-HD - cU^Y^a)] Evaluating the denominator of Eq. (B-2), 2ir / 0=0 f 3<j>0 /, !!o - A R de(^-5tr <jy m3ri \/ \ & 3 r ‘ ^0 2ir JQ B, f 1 Q J 3[cos 0 J.Ck’r)] tl>.------ — =------2ir I v0 3r Using Schwinger's lemma and its corollary, 32r* 3z3z' 225 B, f ^ ' ( B ) Re . y "3^o 3*0 ^3z + 3 3x j Jjte) Lim X+Xq J. z-K) - J l (B) Re z-H) = B, | T. (3) ^ ( 0 ) - 1**0 k i t CB-17) x o z-K) Using Eq. (B-10) and (B-5) in (B-17), 2 tt / 3^q 3rJ»0\ B T J R d\ + o i r - *0 i r j " r - Qi Lim — k K . TTX sin — cos tcz a ^ x0 z-K) B. irx0 IT «l I si” — CB-18> Substituting Eq. (B-16) and (B-18) in Eq. (B-2), ZU - Z12 4 k 2a 2 ’*< J — csc ~ Lim X-»X„ jZ+0 j_ gJ^^Y^a) - otJj^CBjYQCa) 8 “ BJQ(B)J1(a) - a J ^ e ^ C d ) Lim x'-^x. z r-K3 (if 32r» 3z3z 1 226 a CSC kR (4) Lim x W o z->0 z '-K) L- When a « Lim a2?* 8z3z * a2 pJ0 (e)Y1 (a) - a J 1 (P)Y (o) 8 8JQ (B)J 1 (a) - aJ 1 (8 )JQ (a) (B-19) 1, Eq. (B-19) can be simplified by observing that eJo (0)Y1 (a) - aJ 1 (3)Y(J(o) Y.(a) BJ0 (B)J i(°0 j ^ y Y (a) “ o J 1 (B)J0 (a) j o (a ) Y.Cot) 8J0 (6 )Jl(a) - ctJo(B)J0 (a) Y (a) ** BJ q C B ) ^ ! ^ ) (a /2) (2/a)Y.(a) - Y 2 (a) Jo(a> - — Y (a) " “J 1 C3)JQ (a) JQ (ot) « (a /2) " Y 2 <a ) » 1; J t(a) X. a/2 Y. (a) TS75T + «Ji<B)Jo(“)Y2 (ot) _ 2 Q 1 TT / “ „ ca 111 U 2 \ Ql ( ' ^ ? ) a 2 J 1 (B) ' ” 1 2 - aJ^B) -=j Tra / 4 \ 2 J,(e) J l ^ 'C755' ' ' Ql W 2) ' ’ ' J i(a) (B-20) 227 Therefore, when x 8 2 a <<1, BJQ (e)Y1 (a) - aJ^(B)Yg(a) a q: 2ir (B—21) 4-rr Jj(a)Q^ and is of order unity. On the other hand, the first term on the 2 right-hand side of Eq. (B-19) is of order a ; hence, this term can be dropped. Consequently, using Eq. (B-21) in (B-19) gives sin 7iXq/a Z 11 " Z 12 ICR As 1 1 . 2 2ir 4ir a J i(6) J 1 (a) _____________ *____________ BJ0 (8)Jn(a) - oJ 1 (S)Jn (a) «2.J1(e) :- y (01) aJx (B)J0 (a) - BJQ (3) J L (a) (B-22) which is the Marcuvitz model for - Z^- Note that the final result is independent of the constant B^, and that Eq. (B-22) is a good approximation of (B-19) when a « 2 land [BJq(3)Y^(cO - aJ^( 8)YQ(ct)] is not zero or of order a . The latter condition may arise at certain values of B for a given a. Appendix C Evaluating the Limit of / i \2 a2r , x*xQ L fm0 (£) x*^x z-K) z'-K) M ? - Marcuvitz[28] and Schwinger[41] did not explicitly evaluate the neglected term in Eq. (B-19). It is necessary to know the exact expression of the term for a more accurate value of Z ^ - Z ^ when the value of 8 is such that the retained term is actually less significant or zero. Differentiating Eq. (A-14) with respect to z* and evaluating the limit as .. Lim . x*^x0 z'-K) x ’->-Xq and z'-H3 gives 31" -r— r 3zf 1, irx = — sin — a a . ^0 sin------ cos a k z _ + 1 I a n =2 AK Z nirx_ j n s i n H 2[ * s i n ^e + y- cos 0 Y. (kr) 4 1 Differentiating Eq. (C-l) with respect to z and applying the condition x ->-Xq and z-K) gives 228 (C-l) 229 .2 I P *, Lim x->Xq Lim , x xg z-K) z'-K) -r ~ 3z3z -. i = Lim k-vXq a » _ T »nrv k n =2 n sin a sin nirxA D a j Kn Z e z-K) + Lim k2 -g- [Y (kr) - cos 20 Y£(kr)] ^ 0 z-K) r-»0 . “ nirx_ J n T j j v) tc s i . n nrx s . 0 Lim in e -ja o n a a x-KCq n =2 z-K) + Lim ^xQ y - Jl.n(kr) + 2..cos-ge 41T (kr)2 (C-2) z-K3 rK) Note that both limits on the right-hand side of Eq. (C-2) are di vergent while the sum of the limits could be convergent. Eq. (A-17) and the approach in Appendix A of getting Lim Lim F' x+x0 x ’-x 0 z-K) z'-K) j suggest that an appropriate series of e n <j> can be constructed, and added to and subtracted from Eq. (C-2), determining the actual limit. Listed in Appendix A .6 of[10] is the Fourier series 230 = I jn* ne j* °e j* (1 - e ) (C-3) n=l Note that it is divergent as <f> 0< Let <f>l - J E(x - xQ) + jz] *2 = a [(X + V (C-4a) (C-4b) + jz] and using Eq. (A-17) and (C-3), 1 a a £ — “ a a n=l Sin HI* Sin --- 0 a 7T Re 2a -(mr/a)z a 2, n e n=l » n=l J*, J<f>i IT Re 2a j<f>2 n e- _ j +1 2 (1 - e- ') i<j>2 2 (1-e ) (C-5) - Note that Lim Lim <t>2 - 27rxO <j>j = 0 a ^ 0 z-K) z-K) Lim ( 1 - e x->x. z-K) (C-6 a,b) J*l) = Lim X^Xz-K) (-j^') Lim X + 3 tn z-K) [-j(x-xQ ) + z] (C-6 c) 231 Hence, p j +1 Lim Re [e x-*xQ (1 - e VI Lim Re x^ x0 [-j(x - xQ) + z ]2 z-K) z-+Q / s 2 x -(x - X q ) < =(f)2Lim X->Xj 2 + z [-(x - Xq)2 + z2 ]2 + [2 (x - xQ)z ]2 z-K) -[ (x-xQ)/r]2 + (z/r)2 [-(x-xQ)/r]2 + (z/r)2 2 + [2(x-Xg)z/r2]2 But note that from Figure (A-l) X “ Xq -----r = sin 9 and — = cos 0 r Using Eq. (C-8 ) in (C-7) gives (C-8) 232 S - Y Lin n r->-0 (C-9) r On the other hand, using Eq. (C-6b ) , ' i*2 Lim Re e j (2itjCq) /a .-2 (1 - e = Res j (2itXq) /al2 J [l - 2 z-*0 — ----- i - cos (C-10) 2irxo ---- Using Eq. (C-9) and (C-10) in (C-5), 1 Lim ® n/rx* — V — a n =2 a / / \ it j sin i!I5 s i n & .-<■»/*>*----- 2 sin a a a 1 “ w ~ z-K3 , Tj. 1 + Lim — a x >X q v nir ^ nirx j 0 ) — s i n ---- s i n ---*», .a a a n=l e -(mr/a)z z-K) ir , 2 ’"‘0 , it /, 2"*0 \-l * Ta aIn — + 74aT V' ‘ cos ~ /J it A 2 **0 - r I q- esc — a2 \8 a 1 2^ r+S0 — r ~ . 2 . 1 cos 26 sin — — J + rsr- Lim --- =— a / 2ir r-K) r2 Using Eq. (A-25) and (C-ll) in (C-2) gives cos 29 (C-ll) 233 nirx, Lim Lim x>x_ x'-’-x0 . 0 z-K) z’->z /'l'? 3 2I” (^> — r — I sin * n=2 fl\2 Lim sin u z-*0 nir a 3V s C4) “x->-x•„ k2 2 (nir/a) | i a n=2 6 -(mr/a)z [f - nirx sin _ 1 sin _ — k LT-^j 00 r I (ka) -(nir/a)z 0 nirx, 0 fIn . 2 sin — - 71 /n - n =2 IT _csc _ _ 1 2 ^0 IW,)2 - -k _ , 2 fX0 Sin (ka)‘ — -Iir f 1" ^[2 sin sln-r)a) + 0 L a z-K) it nirx„ ^n(kr) + — ~ r Lim 2-rrk2 rK) 2 sin2 — cos 26 r2 ( f) 2] . cos 20 2 Lim 2 2irk r-K) r 1 - Lim Jin — r-K) 23^ I sin2 (ka)‘ n™ 0 n=2 1 2 irx0 - 8 CSC . ( ka\ /2 n — /n /u / x 2 1 /" kc - (ka/ir) - ■=— V — 2n \ ir sin ■)•] 2 ” 0 — In \ 2iv 2ka I— (C-12) V ' Eq. (C-12) suggests that Lim Lim * * x0 x Z-vO a '->0 fi \2 a2r ’ (j) x0 is of order unity; hence the first term on the right-hand side of 2 Eq. (B-19) is of order o . Appendix D Second- and Third-order Approximations for and Z Schwinger[41] outlined the general procedure for calculating Z,,-Z,„ and Z,,+Z,0 given the terms D C , and D . 11 1Z 11 1Z va.% m mm . Hence, (e * - l ) k 2 -— = - j-J — -------i^a I‘- •B ■ mCm (D—1) where Zjj + Zj2 when m is even (D-2) Z = (Zj^ - * The intermediate quantity when m is odd is calculated from V D „ B. * C “ ml I m for all m (D-3) When only two terms of (J>e (r,0) and $Q(r,8) are used, respectively, a 2 x 2 approxmation results. For the even case, Eq. (D-3) becomes D00 B0 + D02B2 = C0 (D-4) D20 B0 giving . + ° 22B2 " C2 236 B2 " D z ^ 2B 0 0 '~ C 0D 20 D - D D 00 22 (D'5b) u02 20 Substituting Eq. (D-5) in (D-l), 1 _ (er*-l)k coD22 " C0C2^D02 + °20* + C2D00 - Z ll+ Z 12 Ka (er*-l.)k D 00D 22 D 02D 20 CQ - C 0C 2 (D02 + D 20)/D22 + C 2D 00 /D22 “3 ~ D 00 - <D 02D 20 'D 22 > ^ ( ^ } Using a similar procedure for the odd case, . u r* - m 2 _ z*11 n “ z 12 i? — — -i J xa + Dn + - (D 13D 31 /D 33) (D-7) When three terms of <j>e (r,0) and <(>q Cr ,0) are used, respectively, a 3 x 3 approximation results. For the even case, Eq. (D-3) becomes D00B0 + D02B2 + D04B4 “ C0 D 20B0 + D 22B2 + D 24B4 = C2 D40B0 + D42B2 + D44B4 C4 The determinant of E q . (D-8 ) , which is also the denominator in the expressions, is (D'8) 237 det " D0 0 (D22D44 D42D24) “ D20(D02D44 ” D42°04) + D40(D02D 24 “ D22D 045 (D-9) Hence, det x BQ = G 0 (D22D 44 - D 24D 42) “ C2 (D04D 44 " D 42D04* (D-lOa) + C4 (D02D24 " D 22D04) det x B 2 = -C 0 (D20D 44 - D 4 qD 24) + C 2 (DoqD 44 - D ^ D ^ ) (D-lOb) " C4 (1D00D 24 " D 20D04) det x B 4 = co^D 20D42 ” D 40D 22^ " C2^D00D 42 ” D40D02^ (D— 10c) + C4 (D00D22 “ D20D02) Using Eq. (D-9) and (D-10) in (D-l) , - . ~3 (e * - l)k2 r Z ll+ Z 12 ({c 0 - (D 22D44"D 24D 42) tC0C2 (D02D 44‘'D 42D04) " C0C4*D02D24"D 22D 04* - det * CC2B2 + C4B4)]}/(D00 - ® 22^ - D24D42rltD20<D0 2 V " °42D 04) “ D 40(D02D 24~D22D 04)1}) Following a similar procedure for the Odd case, (D-ll) where det x B 3 - -C 1 (D31D 55-D51D35) + C 3 (DU D55- C^O^D^- det x B 5 - C 1 (D31D 53-D51D33) - C3 (D 11D 53-D 5 1D 13) + C4(D11D33"D31D13) <D-13a,b) The generalized expressions for and Cm are ( 2 i r R ) ° Jrm (x,z)yjl(xtz) dx dz - (e^-D^J/Yjj^x^G'(x,z|x',z') Y 2irR C = m sin — cos a kz y sin kz m (x*,z') dx dz d x f d z r As (x,z) dx dz (D-14) when m is even (D-15) when m is odd (D-16) \ a sin y (x.z) dx dz m where the integration is over the cross-section of the rod, and 239 (j)(x,z) = y u m A -s— zttk Y (x,z) , A = constant. m m The final results for constants A . m calculations. and are independent of the The factor 2irR is added for convenience in the (D-17) Appendix E - Derivation of the Expressions for D . D . , and C ■— - ■ - ■— — *---------------iimr*-- rng ------- m of a Centered Rod As Indicated in Eq. (D-14) and (D-15), (2irR)^D mz = Jy (x,z)y (x,z)dx dz - (e* - l)k 2 in si, r f/v (x,z)G,(x,z/x*,z,)y (xT,z’)dxdz dx'dz’ 44 m /sin -a coskz /s i n sinicz y 4 (2ttR)C (E-l) z y (x,z)dx dz m m EVEN (E-2) m ODD (E-3) m J a m (x,z)dx dz where the integration is over the cross-section of the rod. Note that, D = D„ mil £m (E-4) because the Green’s function G'(x,z/x',zf) is symmetrical with respect to the primed and unprimed coordinates. Eq. (E-l), (E-2) and (E-3) may be transformed to line integrals around the circumference of the rod cross-section by using Green's theorem [41]. Hence, D«t " ( i r t ) - * } 24-1 where, 2ttt Z t wm (x,z) - r 9v (x\z') m 3r' £ ( f ^ ) G' (x,z/x' ,z’) A —/ I 0=0% *■ Ym (x’,z’) 3 G, (x^ { x-, -^z-) j TTX sin — a coskz (E-7) m EVEN (E-8 ) m ODD (E-9) t(x,z) = TTX sin — sinicz a For a rod located at the middle of the waveguide, (E-10) Y (r,6 ) = J (k'r)cos m0 m * m where, k' =» /e * * r k. E-l. Calculating D ------- =— a — mm First evaluate the first half of Eq. (E-7); 2ir , G'S [cos m0'J Ckr*5ft (f£>{ L- 2 TT/JOI .j0m 3r* » •— k £ w « -W » J . {cos m0'G'} -(A ) EW Wwhere, 0 = k*R and G ]- s (A-14), respectively. i ( f r ) cos + GJ (E—11) and r ' are defined in Eq. (A-ll), (A—12) and Using Schwinger's ldmma, 24-2 2 tt e’=o 2 tt de* 2tr / 4 1 .1.[ei"9'+e-J"9' cos 0’=0 2lT 2 -m Jm (°t)jmcos mD'tr’CoXl (E-12) where a => k R and the operator D' is defined by, cos D r = jk sin D * = 3z' JL 3 jk *3 x' (E-13) and r * (0 ) denotes, cos tnD* (rfCo)3 “ cos ® D ?L** ?(x,z/x* ,z ')3 (E-14) x?-"xc z’-^O On the other hand, 2 f it 0*4 , 2ir / ^ • f c e 3 ” 9 ' + <T > e,> 0’tsO A , 2u cos me'G 8 [- \ = " I Y (kr)J (kr’)ejn(0~0 © Jm (ot) Ym (kr)cOS me \V (E-15) Using Eq. (E-12) and (E-15) in (E-ll), dB* 2ir 0 Y„ g ’5 ^ I - 4 c w e) m j cos mD'r*(o) - D 7 - Y (kr)cos m9 q in Evaluating the second part of Eq. (E-7), (E-16) 2^3 271 s[r’+ G j . , </> cos me'j (k'r*) — — ;---- -5— > m a r Zir 3 r' 9*=0 2 it = Jm (8 ) $ cos m9 * m „ 0'=O -H*> ar1 k 3r 7 - j l V to)CJn-l(kt,) - J^dcr')] in(0-0 ’)1 d9* (E-17) J 2ir Using Schwinger's corollary to the lemma, 2ir ,fQ 1ST COS m9' I P -“ (t) Jm' <a>3mcos “D T ’fO) “ » (E-18) ^Jm-l(a) ” Jfflf1 (o)^ fflc°s mD'r 1 (0 ) On the other hand, 2ir a i , , „3G , s do ? ? - i r 2ir +~ <j) cos m 0 ' Q'=0 -f I7 Ckr) k f £ e in(0-0') d6 Ym (kr)cos 2ir (E-19) Using Eq. (E-18) and (E-19) in (E-17), d9 * 2 tt 3G1" 3r f Ik EWa) - W ^ V 8* 2R r 1jm cos mD'r’(0) - 7 -Y (kr)cos m0 L (E-20 tf m Using Eq. (E-20) and (E-16) in (E-7), w (x ,z) - [ A c s m D ’r ’(o) m { -k CJ»-1(8> Ym (kr)cos m0 - W * ! V “> -S - J m +1 244 Hende, w (x,z) = j^jm cos mD'r'Co) m Y^CkrJcos me] ^ (E-21) FSJ ,(B)J (a) - aJ (B)J - (a)] *- m—J. m m m—i where, the recurrence relation, nn-1 x m (E-22) m —1 was used. Using Eq. (E-21) and (E-10) in (E-5) gives, 2ir -<e* - l)k2D r mm 0 f r { v * ’z)£ £ cos 1<ejm(k’r>] cos m3J m (k r) I T “ (E-23) ] First evaluate the first part of Eq. (E-23), and let, V ~~ »V°> W £ Qm 0=0 w 277 m > — — = 8r! - aV i <<0V -Jj- fj 2R (g)J <f> (g) - j ® KL m_1 (E-24) B) cos me 0=0 2ir j [ jmcos tnD’T ?(0) - cos me] (E-25) Using Schwinger's lemma, 2ir / 9_ -r— cos m0{j cos mD'r’(O)} = J (a)j eLo 277 where r * (0 ,0 ) denotes, m cos mD cos mD'r'(0,0) (E-26) 2^5 cos mD cos m D T ' (0,0) = X**X V™ cos mD cos mD' (r '(x,z/x' ,z')] X (E-27) and, _ cos D = 1 3 — *-=— jk 3z . „ 1 3 sin D = — • — jk 3x (E-28) On the other hand, - , *| 2n n cos m 8 - 7 - Y (kr)cos m 8 = - 7- Y (a) J cos m 8 •=— L 4 m J 4 ® 9=0 27r 2ir j. 6 -x— 8=0 2rr - - J Y (a) o m (E-29) However, when m = 0, the right-hand side of Eq. (E-29) is just -1 V a) • Next evaluate the second part of Eq. (E-23), P I7 Y “g ~ = J ^ ) f> Qin 0=0 2* m 8 r m 9=0 cos itf^fj^cos tnDT’(O) - (E-30) y- Y (kr)cos msl A m J Using Schwinger's corollary to the lemma, 2 tt <|) 9=0 cos m9 [jm cos mD'r^O)] = ■ ^ ^ J^(ci) j2mcos mD cos mD*I [Jm_^(a) - Jm+ 1 (a)j j2mcos mD cos m D ’r'(0,0) ’(0 , 0 ) (E-31) On the other hand, So ** cos2”e(‘^)^rV fcr) ■ -fe t-Vi(a)- V i fr 21+6 - - K s ) [Vi<“> - W “>J < E - Again, when m » 0, the right-hand side of Eq. (E-32) is just 3 2 ) Y^(ct). Using Eq. (E-31) and (E-32) in (E-30) and using Eq. (E-29) and (E-26) in (E-25) and then using them in Eq. (23) gives, -(ej - D k ^ i = j2"cos m Doos in D'r'(0 ,0) • W 8)] - Jm'S> S £ W 6> ‘ W - - • W ‘>]) - £ ( V “>4i 6^ " V B> af "W “>J) (E'33) Using Eq. (E-22) and a similar recurrence relation for Y ^ a ) , -fa? - 1>k2n™, ^ J2”1™ 8 "D cos m D ’r '(0 ,0) - I CBYm (a)Jm - l (B) " aYm-l(a)Jm (fJ)3 <E~34) Thus, D“” ° ■ ^ f ^ 2 ( ^ 2“ “ s ”D c o 8 ”D 'r,<0’0) - I *Viw,.(#3) qJ > V “ >j„-:l(b> “ <E-35) However, when m ** 0, the factor 1/8 becomes 1/4. E“2 * Calculating Using D 111)1 and “ Ja (k'r)cos£0 in Eq. (E-5) gives, = ■ (e*~l)k‘ 3w j cos£0J^(k’r) — zzr) Tf (E-36) 2^7 The Y (kr)cos ra0 part of w in Eq. (E-36) has zero contribution bem m cause, 2ir J f cos m9 cos&9 J 0=0 2ir = 0 (E-37) Hence, D g will only be due to the cos mD'r'(o) part of w ; following mx id D procedure used to derive Eq. (E-35) results in, „= — (e*-l)ot ^ cos i v cos _— - a similar m D ’r ’(0,0) (E-38) On the other hand, using Eq. (E-10), h m m f S' ^ [oosm6 Jm (k-r>] - 3w» cosm9J (k'r) — r— } m 3r (E-39) and again the contribution of the Y^(kr)cos&9 part of w (x,z) is zero because of Eq. (E-37); therefore, ^ Am Qmj(m+a) o— (e*-l)a cos mD cos £D'r'(0,0) Note that in the calculation for Z11 - (E-40) and ® are odd » & and m are even. while in the calculation for From the identity [16], f “f <-l)k° 2 (°2 -22 )(n2 -42) k =0 [n2 - (2k- 2 )2] n> EVE[I (2k)! (E-41) cos n©. =< kr*0 /2 (-l)k fa2 -l) (n2 -32) (n2 -52) •••• [n2 -(2k-l)23 sin2fe6 n, ODD . (2k)! 248 it can be seen that the calculation for Z ^ + Z ^ only involves r'(0,0) and the x or x' derivatives of I” , while the calculation for Z ^ - Z ^ involves g2r f ( Q 0) ■— A — T~— 0Z <£Z d2r ' and derivatives of —— . with respect to x or x'. dZZZ Also, when a « 0 Til 1, the order of magnitude of, = 8J (a)J .(B) - aJ .(a)J (B) * a™ m m —1 m-I m (E-42) BY (a)J , (B) - aY (a)Jm (8 ) * [fl m m —1 m—l Laj (E-43) when the terms involving 8 are not small; Eq. (E-43) is primarily due to Y (a). m Moreover, ( jBt^ c o s mD cos aD'r’ (0,0) m, EVEN 1 ^ * (E-44) .(ka)-2 m, ODD since the leading term of cos mD cos AD'r'(0,0) dominates. E-3. Calculating Using Eq. (E-10) in (E-6 ). gives, c =— ” Ce*5 ) ? e ^ o ^ ^ f ^ 1-008 m0Jm(k,r)^ " COS meJm(k’r)f l } <E_45) Schwinger's lemma and its corollary can be directly applied to Eq. (E-45). Thus, 2ir -(sj-1 )k20in - ( ^ ) [ J n .l(B> - W * ! J (B) m p P t ^ 0=0 P : cos m0 -M- 9=0 2ir 3r 008 ’ 2k9 = 2R Jjj(<*) jmcos mlty (o) “ ^tn+1 (8 - J (B) ~ J ’(o)jm cos mDrJ»(0) m k. m - (rn Cjm-1<6) - V l < 6>] J» <“) - Jm«> a [V l C“> - W “>3 ) jm cos mDiKQ) “ RCBJm (ci)Jm - l C6) " aJm- l (a)Jm (e)] jmcos (E-46) Hence, C - Q l mcos mD^(O) - — 2------------ m (E-47) (e*-l)ka r Using Eq. (E-41) when m is even jm cos mD^(O) = jm cos mDQKx,z)J z->-0 .m Lim _r irx « j cos mDlsin — J »> 2cn ‘- a t coskzJ -* 2+0 .m Lim ( J WLkV * o z-T *-0 = j2n Lim tjr x^ x 0 m2 / 1 f 3% 21 3x 2 ^-t e ) 2 m 2 (m2 - 2 2) d S 4!(jk )4 + 3x4 - \ <-“ > z->z where, n = 0 ,1 , 2 ........... Lim ^ = Lim ^ 0 z-vO coskz sin — a TTXj x0 = sin — — * a ^ 0 z-»-0 Hence, for the even case (m = 2n) with b = (ka/ir) , (E-49) TTX jm co3 mlty ( 0 ) " (-I)™ sin --- 250 m , -2 . m 2 (m2 - 22) ,-4 1 " 2fb + ---- 4!----- b " •• (E-50) and, (-l)n+1Q m,even 2n (e*-l)ka 2y 2 m (m 2tt(sin ^ VC m2 -2 a J 1 " 2f b + „2 -2 ) ^-4 4! (E-51) b 2 „2 . , 2 .2 v | (-l)n+1Q -jl; Q„ , irXrtV n „ m (m -2 ) (m -4 ) •• • Im 2 -( 2&- 2 )2] t(e*-l)ka * iVi- "( ) ^“Q ][ (-1 ) 22, Vs i n a / (22.)! b For the odd case (m = 2n +1), jm cos mDifi(O) = Lim jm cos mD £sin — sirucz] z->-0 (if*x*xf I z-*0 2n+l _ . Lim x*-xf z^-0 - dz ) 35» - A -+ 2! (jk) [jcsin 8z9x 4!Mk'>: 41 (jk)‘> cosiczj (jk) (m2 -l) 4 3z3x b" 2 2! [*■ (m2 -l)(m 2 -32) t-4 ----- 41------- * .... •] - 1 ( m - 1) (m2 - l ) (ra2 - 3 2 ) , - 4 4 j-------------- b Using Eq. (E-52) in Eq. (E-47), -2 2T~b + •] (E-52) > •«xti4*3>— ( ) 2n+l T«T , 02 .n+l .. T C -— [fcjsln (e*-l)ka tn,odd 251 * [- 2! 1 ( ^ ,)-(ta2,z32)_ b- 4 .... 4! J .n+1. Q 2n+1 ^ k ^ sin (£*-l)ka TTX- £ (-1)*(m2 -l) (m2 -32) (m2 -52) »* » fm2 -(2&-l)2] 2% S,=0 (2£)! b E-4. a « (E-53) 1 Case From Eq. (E-42) and Eq. (E-51) and (E-53), when a « 1, (m-1 ) (E-54) (e£-l)k and using Eq. (E-42), (E-43) and (E-44) in Eq. (E-35) and (E-38), „m Dm m 'v* (E-55) (e*-l)a r a (uri-O (e*-l)a a (m+fi,-2 ) (E-56) (e*-l) E-5. Detailed Expressions fdr the 2x2 Aftproxlmation From Eq. (E-35) and Eq. (E-41), "00 Q 0 Lim Lim T ’(x,z/x’,z’) + *-l)a2 { (e*-l) r | Q 0 [gY0 (a)J1 (e) - aY1 (a)J0 (P)]} (E-57) 252 (e*-l)ct L r x^ xn ' z-*-0 U zf-> 0 U (E-58) ^ Q 1 [SY1 (a)J0 (3) - a Y ^ c O J ^ B ^ J D „ » — — ---- 5-i j4cos 2D cos 2D'r*(0,0) (e*-l)a I Z (E-59) | Q 2 [ B Y ^ c O J ^ B ) - aY 1 (a)J2 (B)]J - — °33 - c. :-l)a 1 ' r 0 I i 6cos 30 cos SD’r^o.O) I 3 (E-60) i Q 3 [BY3 <a )J2 <3) - oY 2 (a)J3 (B)]j Using the notation, gn+Jlj,, „ ,n 2. 9x 3 x a a 3z »» 3z3x (E-61) x ,nx& ax * = (E-62) r z , Z X , n X 5’ Eq. (E-59) and (E-60) may be made more detailed using Eq. (E-13) and (E-28) in Eq. (E-41), j cos 2D cos 2Dfr'(0,0) = Lim x+x z*0 Lim Lim [" I" - — ^ x*xft x'-»-xn L (jk) 2^ 0 ° z'-»-00 2 Lim cos 2D r * ____-— 2 r 'x x’-»-xn (jk) * zf-»-0 rJLr - T’ (jk) , + — ^ (jk) - r'(o.o) + (|)2 r^co.0) + ^ r ; . xI]a[(o,o> 1x'x'xxj (E-63) 253 j cos 3D cos 3D'r’(0,0) = -Lim Lim cos 3D — Jk r» .3 jk r z' , z - -r* 1 Lim (jk)4 r:.. 2 ’2XX (jk)4 z'x’x ’ Cjk) 0 x^ xo z*0 z'-*0 M -Lim Z_ r « _ z’ r',_. , + 2,zx'x z-K)° z*->0 16 z'zx'x'xxV (jk)6 .2 r (t) ri'z(o-w + < k ) o) rz'2xx(°’o ) + (l) (E-64) Note that a derivative of r' with respect to x' and x, respectively, are the same. Using Eq. (E-38) and (E-41), “* D02 " D20 0*2 .2 2 j cos 2D r ’(0 ,0) (e*-l)a r '(0 ,0) + -%• r ' (o),0)J (E-65) (e*-l)a: -*1*3 °13 " D 31 jA cos 3D cos D T ^ O , 0) (e*-l)a -QiQo p i t - ^ - 5- cos 3D [ -77- r ’ (0,0)] jk z' (e*-l)a ”^1^3 - - 2 Lim Lim rif_VJk7‘z ’z - <Jk)4 (e*-l)a x*xn x’-*-xn z+Q ( if (e*-l)a r . 1 z' zxx z zf-*0 +(t)2 (E-66 ) 25^ On the other hand, using Eq. (E-51) and (E-53), TTX, sin (E-67) (e*-l)toa (E-68 ) (e*-l)ka +0 (E-69) (e*-l)ka L- +Q< C3 - (E-70) (e*-l)ka r The detailed evaluation of the limit of the various derivatives of T ’ in Eq. (E-63) , (E-64), (E-65) and (E-66 ) as x and x' z T and z -*■ 0, are done in Appendix F. Xq and as Appendix F On the Limit of the Derivatives of T * and r*0 r f-K> the Series I e^ns The limit as r and r' + 0 of T' and T f, have been evaluated z z in Appendix A and C, respectively. It was found that both limits are related to the-series, U- I (F-l) n=l where, s = 1 -e j (x-x^) + jz[ (F-2) Note that E q . (F-l) and (F-2) may also be used with x ' and z ’. In particular, the limit of T' is related to Lim s->0 s / Ujds = Lim j°® s *0 = Lim s->0 s 00 00 / £ e Sjds = Lim £ e~*ns j°° 1 s-»-0 n=l n s js / — —— 7— jds = Lim {- &n(l — e j- 1 -e s-K) (F-3a) j While that of V '. is related t o , z'z 00 JJg CD Lim 4^- = Lim T — — *» Lim £ jne^ns s*0 1 93 s+ 0 n«l a r is,, is.-It )} (F-3b) (F-4a) 1 eJs ■ Lim "s«r Le <1-e > J ■s*0 Llm — -is 2 e*0 3 (l-e3S> (F-4b) It is apparent that the limit of the mth derivative of T ’ with respect to x, z, x T and/or z' as r and r* + 0 is related to the limit 255 as s 0 of the (m-l)th derivative of U with respect to s: 256 gCrrW+l).. “ 5 I’z ,zx,nxa " r+0 r '-*0 *■ “ j s-*-0 .. n+£+l as <F"5) Another fact which proves useful in the evaluation of the limits is, siw(>^ s i n ^ = Re^Ce If we let 4^ = <»> a } (F-6 ) x, <J>2 = ( “•) xg» 3115 $3 “ ( a ) z ’ then» n7rxrt J sin n=l - e sin — _— e a rnr a z r s igRe <*> £ e^ns Ln=l £ eJ n [ W 1-H>2 )+^ 3 ^ n=l CP-7) Note that the first series on the right-hand side of Eq. (F-7) is U. As examples.in the following cases of T' , , and r 1, , we have . XXX X z zxx 9 TJ q5|j to use — r; while for the case I” . 4 we have to use — =-! Hence, as3 2 zx as5 a3U r ^ 3 jns — T - - I jn eJ 8sJ n=»l ej2sT (F-8 ) ja3 s T l + 26ejs + 66ej2s + 26eJ3s + e34s] (p_g) js fl 4- 4ejs + jeJS 3 U V 3s3 n=l j 5 jns — 5 - I jneJ Cl - ejs)6 Note that using the second part of Eq. (F-8 ) and (F-9), re spectively, when s -*• 0 gives, 25? .. aJT l + 4eis + ej2sI .. 6 Lim ---- 1------- 7 — r •— - Lim -r 3+0 (1 - e-1 ) s*0 (-js) m^Tl + 26eJS + 66eJ2s 3*0 (F—10) 4- 26eJ3s + eJ4sl (1 - ejS)6 120 L i m ---s*0 (-js)6 T . (F-ll) In view of Eq. (F-7), (F-8) and (F-9), and using Eq. (F-2) in (F—10) gives, (F-12) Urn !*Re I n 3eJ" & - V j!3 {” /a) = Lim !sRe x*-x 1 x>xn [-j(x-xn)+zj z»-0U z+0 u However, the coordinate system used, as shown in Figure (A-l), gives, x-x ** sind (F-13a,b) z/r = cos0 Using Eq. (F-13) in (F-12) results in, Lim Site J x-J-xn 1 z-»-0 <*/“> = Lim r*0 3(a/»)Ve [-jsinQ +cos0] t o =s Lim 3(a/ir)^ (l/r^)Re(e+^^®) T+0 4/3COS40' to (a/7r)Y?cP-p -) = Lim r-*0 »-0 ^ r ' (F-14) Following a similar procedure on Eq. (F-ll) results in, Lim %Re J n5ejn[(x-X())+jZ] Cr/a) = Llm ( a M 6(6Ocos60\ x*-xn 1 r->0 r (F-15) 258 In the sprtial derivative of F ' (Eq. A - 14) we have to contend with the spatial derivative of YpCklr-r'l) and the consequent in finite limit as r and r ' 0. We will find that the limit as s + 0 of the corresponding derivative of U approaches exactly the same valuei Therefore, a cancellation procedure can be effected. Starting with Y^(kr), Lim Y ^ k r ) - L i m - (f)(4) <F-16> The corresponding limits of the higher-order derivatives of Yg(kr) may be precisely determined by using the recurrence relation, Yn+l (kr> - f? V kr) - Yn - l <kr) (F- 17> Hence, as examples, Lim Y 2 (kr) = Lim r^O r->0 - “ Y ^ k r ) - YQ (kr)J '[E - K c )] - £ (- t ) (&)* <F-18) Lim Y4 (kr) - Lim' [ ( £ ) Y3(kr) - YjCkr) ] - £ Lim Yg(kr) = ^ [ f j f [ ( A ) G D Y= H - ^ - ( (M) G D V kr)] " . Llm r^-0 v X e? ) 4 <f- 19> Y4 (kr> - »0)(76) (F.20) Tr(kr) since the limit as r + 0 of the highest order Yn <kr) dominates'the lower order terms. 2 59 To illustrate the exact procedure, let us calculate the limit as r and r ' -*■ 0 of I*', ,, T ' , , , r ’. and r ’, *— the other terms x'x” x'x'xx’ z'zxx z'zx^ needed to completely calculate the 2x2 approximation of and • F-l. Lim Lim T ', , x*x x'-»-xn x x z+0° z' + 0 ° For convenience, the expression of T '(x,z/x',z*) is reproduced below with <j> = (irx/a) and <j>' = (irx'/a); 03 I” (x,z/x' ,z') = -(l/ica)sin(f> sin4> * simcl’z-z'l + (j/a) J (1/k ) n=2 n sin n<|> sin n<t»' exp(jKn iz-z'|) + (k| r-r'|) (F-21) Since Y q is in terms of r and r ’, we have to use, 3 j r. t cose' - sine ' j p 3- + -p— 3 „, COSS JP- - 3 sinQ ' 3 jg-r /T, (F-22rt 3 J P ~— P ~ S T <F-2 2 W and express YgCklr-r'l) as co YQ (k| r-r' |) = I Yn (kr)Jn (kr’)exp[jn(e-e')] , r > r' (F-23) •OO Hence, using Eq. (F-22a) in Eq. (F-23) gives, 3Y 3x £ (k| r-r* |) = I Yn (5)eJn0(-jk/2)[jn_1 (S »)e“J(n”1)e' + + e - ^ n+1>e,Jn+1(5 *)] (F-24) 260 (k|r-r'|) » "I 1 Y (6) ejn0 <k/2)2 Gj (6 »)e“d (n-2)9 ' + as (F-25) 2e"jn0,J (S') + e"j(n+2)0’j 2 ( 6 ' ) J n n+z where the relation, dJ (6’) -T?— (F-26) - ( W 2 ) [ V l < S ’) - and 6 = kr and S ' = k r ’ were used. Differentiating I” relative to x ’ twice and using Eq. (F-25) gives, ^*x*x * " (ir/a)2 [f^ituclz-z’| - j ^ n 2fne;i'cn!z_z JJ - \ \ Yn (6)eJn0 J ^ n - 2 '^e_;i (n-2^0 ' + 2Jn (6,)e“jne' + Jn+2(5 ')e~j(n+2)0 '] (F-27a) (k/2)2 where, f n = (1/aic )sin(nirx/a)sin(nirx'/a) n (F-27b) Hence, Lim r', , ■ (i7/a)2 [fl0sinK:iz| - j \ n2f 0eJK“zl - (k/4)2 x ’-»-xn u n=2 -* z'+O0 jY2(5)ej2e + 2Y q (S) + Y_2(5)e”j20j where, fftQ =* (l/aKn)sin(mrx/a)sin(mrxQ/a) (F -28a) (F-28b) This is because, f 0, Lim J (kr') = < r ’+O m ll, m ^ 0 (F-29) m = 0 261 Therefore, CO Lim Lim I” , , = Lim (-j)(ir/a)2 £ n2f ne^KTiz x*-xn x ’->x_ x X x>x_. n=2 z->0 z'-»-0 z+0 Lim (k/4)2 [2 cos20Y (S) + 2Y (6)] r*0 (F-30) Note that the series and the limits of Y 2 (kr) and Y Q (kr) are divergent. However, if the series in Eq. (F-4a) is added to the above series, the sum of the two will be convergent; if the limit in Eq. (F-4b) is like wise subtracted from the limit of the Y ? (kr) term, the combined limit is zero while maintaining the equality in E q . (F-30). The same proce dure can be used to cancel the limit of the Yg(kr) term by using Eq. (F-3). The associated detailed expression using Eq. (F-3) is shown in Eq. CA-25), while that of Eq. (F-4) is shown in Eq. (C-ll). r;.at,(°,0> “ Lim x-*xQ z-*0 (ir/a)j - £ ie fnQ[ng |n=*2 Y k f n Tn + b2/2n]g > n-2 n n0 Therefore, n2/.Gi2-b2)?S + (%n)b2g j - + Lim [cos20/2irr2 - nJ r*0 (k2/47r)J.n(o)] (F-31) 2 2 where Eq. (F-18) , icq = j(ir/a)(n -b ) » Sn “ exp(-rorz/a) , 5 = kr, b = (ka/ir) and Lim Lim z-vo z ’->0 •. x*x0 X ^ X I*', , =» T ' , ,(0,0) were used. K X using E q . (A-25) and (C-ll), X X Finally, (k2/4ir)Enn(2h1) - 2h2 - Lim 8,n(irr/a) + Lim &n(kr)i] 1 1 r*0 t^-0 (it/a2) i I h 2 [n - ^ / ( n ^ b 2)5* + b2/2n] - l/8h2 + h2 U=2 n 1 1 (b/2)2 |s,n(2bh;L) - 2h2j| - (F-32) where h = sin(riirx./a) . n u Note that, rx*x’(0>0) = ri c (0’0) CF"33) because the same procedure can be used on the unprimed coordinates while using, Y0 (kjr-r*|) - J Yn (kr’)Jn (kr)ejn(0 ,_0), r' > r (F-34) —CO and taking the limit as r + 0 first. F-2. Lim Lim T ', , ^ . xx'xx ^ x0 x x0 z+0° z ’-M)U Starting with Eq. (F-27), we can proceed to getting I**, . by X X XX differentiating it twice. First, using the unprimed counterpart of Eq. (F-22a), [Yn (6)ejne] “ ” (k/2)2 [ej(n”2)6Yn-2 (5) + 2ejn6yn (6> + ej(n+2)6Y^ (fi)J (F_35) 263 Therefore, r' ’x ’xx = “ (7r/a)4 jfjSinKlz-z'l - j ^ n 4fneJ<nlz-z ^|+ X k I [y (kr)ejn6l •f"e"J (n_2)e 'j (5») + 2e_jn0'J (6 ^o3X2 L n J L »-2 n e-j (n+2)0 * (g ’)J + (F-36) and, L£” x'-^xz'-»-0 " - (' /a)4[ £l(finK!! " J ? "4£nOejl<nZ] L n=2 J + iS^ i- [6Y0 (5) + 8c o s 20Y2 (S) + 2cos40Y4 (6)] (F-37) Hence, using Eq. (F-18) and (F-19), GO r',x. ^ (0,0) ® Lim j (it/a)4 £ n4f . e ^ ^ + Lim x->x_ n=2 r*0 z->0 ] 16cos20 96cos40 i ir(kr)^ ir(kr)4 j “ (6/ir)£n kr L (F-38) The limit expressed by Eq. (F-14) exactly matches that due to the cos40Y2 (kr) term. Also note that the divergent series in Eq. (F-38) can be made convergent by subtracting the divergent series of Eq. (F-14) from it. Hence, as before, cancellations can be effected while main taining the equality of Eq. (F-38) and getting the finite limit in the process. It may be emphasized at this point that the limit as r •+ 0 of the highest ordered Bessel function in the equation Yn (kr), dominates those of lower order. Thus it suffices to introduce the (n-l)th s-derivative 26b of U Into the equation to cancel the divergent limit of Lim cos n6Y (kr). r*0 n However, this does not necessarily make the combined infinite series convergent. A convergent series may be constructed bysubtracting the appropriate series forms fromthe expansion of the original series com bined with that introduced by the (n-l)th s-derivative of U. To main tain the equality, the corresponding limits of these additional series must be added to the equation. that the divergent parts of their Note respective limits will be dominated by the limit of the Y^Ckr) term because they necessarily correspond to Ym (kr) terms of order m > n. Hence only the finite parts of their respective limits will be retained in the final expression. To illustrate, use Eq. (F-14) in Eq. (F-38); hence, ri'X'XX(0'0) ■ h/a)3| J 2Kn£n0[nV'c”z/<n2-b2)’5 - ,.3Sn] + z -j-0° 7 n \ f «g 1 u=2 n n0 a J - Lim r*0 (3/tt) cos40 (F-39) ~ ? r The combined series in Eq. (F-39) is not convergent as x -> xQ and z ->• 0. Using the binomial theorem on the square root term, n4/(n^-b2)^ - n2 = n^J^l - (b/n)2/2 + 3(b/n)4/8 - ••• + •••J - n 2 - -(n/2)b2 + 3b4 /8n - •••• + ..... (F-40) It may suffice to stop at the term with n ^ because it corresponds to Yg(kr). Eq. (F-40) clearly shows that the series in Eq. (F-39) diverges a s x + Xj and z •> 0 because the corresponding infinite sum of the two 265 terms shown In Eq. (F-40) is divergent. Subtracting the respective series associated with the two terms in Eq. (F-40) from Eq. (F-39) and adding their corresponding limits as expressed in Eq. (A-25) and (C-ll) result in, n. X X X X I <^n0 (0,0) = Lim ( W a ) 4 , — x*x_ I n=2 o I n=2 2+0 3b4 /8 n )J + . g (n3 . nb2/2 _ “n n n nu l *- f 2f n f n 0 Sn ( n 3 - nb2 / 2 + 3b4 / 8 n ) j - Lim 3cos40 /t r r 4 (F -4 1 ) r+0 A f t e r s im p l i f y i n g , r \ , x'x'xx (0 ,0 ) - <»3/a A) I I h3 [n4/ ( l>2-b2)!i - n3 + nb2/2 - 3b4 /8nl [ n=>2 a/16)|h"2(l + 3 c o t2 - £ ■ ) + 16 h2] - (b 2 /2)(l/8h2 - h2 ) + (3 b 4 / 1 6 ) j i n t f h j ) - 2h2J > (F -4 2 ) where, Lim W . ) 3 I n \ f g„ = ft)Llm Re f I n V 8- \ x*x. n-1 n n° n *>xn Ln-1 n-1 t0 „0 2*0 2>0 (3/ir) Lim r->-0 r - (ir3/2a4)Re + & ejv (l + 4ejv + ej2v) (F-43) (1 - e ^ ) 4 V “ (2irXg/a) Re (’'/a> (F-44a) eJv (l 4- 4ejV ♦ V -j2v' l (1 - eJv >4 J = (l/8h?) (1 + 3cot2 ^ ) 1 as it follows from Eq. (F-6), (F-8) and (F-14). 2 <F-44b) 266 Note that it was not getting Eq. (F-32) because necessary to use theaboveprocedure the series resulting in fromthecancellation of the limit of both the cos20Y2(kr) and Y^(kr) and Y^(kr) terms hap pens to be the correct one. 2 2 2 This can be easily verified by expanding the n / (n -b ) term using the binomial theorem. F-3. Lim r *, ***0 Lim , x xo t Z ZX X g>(T z*-»0 Since r'i , , = T f, , t , we may start with Eq. (F-27) and difZ ZX X X X z z ferentiate it with respect to z' and z, respectively. Using Eq. (F-22b) on the Bessel functions, we find that, (F-45) 8 7 [Yn Cfi)ein0] " <k /2)[®J<n“1)0Tfn„1 Cfi) - ejCn+1)0Yn+1(6)J (F-46) Applying Eq. (F-45) and (F-46) in (F-27), T* , , , = (v/a) fic^f sinic| z-z'| x x'z z 1 £ (nx )^f e^K n ^z z ^ - (1/4)(k/2)4 n n e-J<I'-l>°,J i_1 <{ ') - e-J<n+1)e'jn+1« ' ) (F-47) Taking the limit as x' -*• X q and z' ->■ 0, Tk'*'z'z = ("/B)2 *k 2 ” 2 1 f1QsinKZ - j £ (tucn ) fn0exP ^ Kn Z)J “ n—2 z -K) (l/4)(k/2)4 ^-2cos40Y4 (6) + 2YQ (6)J (F-48) Hence, 26? from Eq. (F-19), 09 ri'*'z'z(0>0) ■ Llm «><"/«? I <n*n)2f 0e ^ nZ x*xn n=2 z->0 . ^ [(7) ^ - +(?) ( 7 ) * » H w-495 Using Eq. (F-43) and (F-44) in (F-49), while dropping the limit due to the &n(kr) term, results in, r ' . ,_,_(0,0) = Lim X X z (ir/a) 3 ^ z ***0 z*0U r 3 r- v - r 2f 2 ,2.h jKnz £ ' n f n o Lu" ( n n-2 1 > e 3 1 ' n 8 n-J* (3 \ cos40 (F—50) The a d d i t i o n a l term s r e q u ir e d to make th e combined s e r ie s c o n - 2 2 v e r g e n t a r e d e te rm in e d by e xpanding th e (n - b ) H te rm u s in g th e b i n o m ia l theorem ; n 2 (n 2 -b2) ?S- n 3 - n 3 ( l - b2 /2 n 2 - b4 / 8 + ........... ) - n 3 = - nb2/2 - b4/8n + ................... (F-51) Hence using Eq. (A-17) and (C-ll) in Eq. (F-50), I r i ' * ' z ' z C° ’0) • L t o 1 n=2 * n £no[''2 ( n 2 - b2),Se3'< n Z - 2 „ < n 3 - n b 2 / 2 - Z-5-0 b4/8n)l + J i K f ng (n3-nb2/2 - b4/8n)1 n=2n n 0 n - J (F-52) r4 268 After simplifying, T\. , , (°,0) » (ir3/a4) ( I h2 \n2 Cn2-b2)h - n3 - nb2/2 - b4 /8n] x x z z In=2 L J (1/16)[(1 + 3cot2 | )/h2 + 16h2J - (b2/2)(l/8h2 - h2) + (b/2)4 [an(2h1) - 2h2] J (F-53) Note that, I” , , , (0,0) - I” f (0,0) x'x'z'z * xxz'z (F-54) since the above procedure can be repeated with primed and unprimed co ordinates interchanged. F-4. Lim xmc &»0° Lim Tx1,x , , ’xxz »z x'-*x0 ______ z ’-vO We could start with Eq. (F-36) and differentiate it relative to z ’ and z, respectively, or we may start with Eq. (F-47) and differenti ate it twice with respect to x; in both cases, we get, r ’f T , = -Gr/a)4 f K2f simcj’z-z'l - j J (n2K )2f e^K n ^2_z ^ t x n ix x'xxxzz L n=2a 4 jfL i <n“3>®Y (6) + e 3 (n-1)6Y (fi) n-3 n—x -*LCOL ej{n+l)0 Y^ ((S) _ ej (n+ 3)eYn+3(5)J. s-j(n-l)e *j n—l <n-3)e ’ (6 *) + , , } _ e-j(n+l)61j (g.) _ n+x e-j(n+3)0»Jn+3(g »)J (F-55) 269 Therefore, ri,x,Mez,z(0,0) " Llm (ir/a)5nsa2 2 (-a4)<n (n2 - b2>!5fn0e:lKnZ + X x xxz z ^ no z+or Llm (k^-2) r->0 [4Yq (6). + 2cos20Y2 (5) - 4cos40Y,(S) - 2 c o s 60Y6 (5)3 (F-56) Since the limit of the cos68Yg(kr) term is matched exactly by the one expressed in Eq. (F-15), apply Eq. (F-15) and (F-9) in (F-6) and (F-56); r V C 5 T', , , (0,0) * Lim Or/a)5 x x xxz z J / n £no L n «0 z-*0 n 5 V n k f rtg n nOen n=2 4.2 , 2 .% jKnzl - b > e + Lim (— ^ r-»-0 11 ' J - (F-57) r where, Lim fa/a)5 i x>x^ n=l a»0° - Lim (32) £22|5r»0 r - (i) fa/a)5Ra[e3v(l + u 26a3' ' + 6 6 e 3 2 v + 2 6 e 33v + eJ4v)(l-e3v)-6 | . (F-58) 3 The additional terms needed to make the series convergent are, n5 - n4 (n2 - b2)*5 = n5 - n5 [l - (b/n)2/2 - (b/n)4/8 - 3(b/n)6/48 - •••] ■ n3b2/2-+nb4/8 + 3b6/48n + ..... Hence, (F-59) 270 /2 " nb*/8 ‘ 3b<5'48»> < " ri'*'s*z'z(0-°> ■ (l,/a,5|J / n £no[8nCn5 ‘ 2*0° jKnz - n4,(n2 - _ b2 )’- a3 »l J - V n=2 e / 5 - n3v2 /« b /2 '60N cos66 - nb4/8 - 3b6/48n> I + Llm (^r) 6 r J r*0 v ' (F-60) Using Eq. (A-25), (C-ll), (F-43), (F-44a) and (F-58) in (F-60) and simplifying, T 1, , , x'x’xxz’z (0 ,0 ) = (it5 , 6. f r . 2f 5 4,2 .2.% 3.2/0 , 4 /a /a W £ In - n (n - b ) -nb/2-nb/8jn=,2 3b6 /48n] + h2 jjL - 2(b/2)2 + 2(b/2)4 + 4(b/2)6] (l/2h2)[h"4 + (13/16)h72 F + (F/2)2 + (b/2)4/2 + (b/4)2 1 Cl + 3cot2 f >] - 2(b/2)6 Jln<2h1) ) (F-61) where, Re ejv(l + 26eJv + 66ej2v + 26ej3v + e^4v) (l-e^)'6 = -(is) j^2h~6 + 13F/8h4 + F2h“2/8| F = cot2 (irxQ/a) - 1 (F-62) (F-63) Appendix G Shorted Waveguide Integral Equation G-l. Incident Field and Green’s Function Consider the rod in a shorted waveguide applicator* as shown in Figure 4, with the iris removed. Assuming an time depen dence* the incident wave of unity amplitude is ,, v . UX , +jKZ i|i(x,z) = sin — [e -jK(z+2b)-i - e J 'J . .V (G-l) The Green's function may be derived as follows [10,41], GU = £• n=l a sin S c ?L L- f„(z.z*) n (G-2) where, ( * — *2 +. ^O , fn^Z »Z '), = -S(z-z') — a 5" nl n V dZ V77 To make G q ” zero at z = -b, let -j k (z+b) f (G—3) = e +j k (z+b) - e (G-4a) and since the time dependence is e+Jut, —j k z f2n - e n . (G-4b) 271 272 Following Collin [10} , -jK W = fIn f2n* " f2n fln’ e = n (z+b) -j^Z + e +j k (z+b) n - e -j K jK J n (-jK )e n (z+b) e “jK z J n + j K (z+b) + j< J n e +j«nb (0-5) = +2jKn e Therefore fn = f2n / Z1 Cf1rr/W> C /a) 6 (z“z ’^ 3 dz (0-6) - fln / (0-6) (f2n/W)[-(2/a)5(z-z’)]dz z2 Let < z' and z2 > z*; when z >_ z ’, f = first integral n -jK (z+b) ® (+j/<n a ) e -jKn (z'+b)j (z'+b) jK e n n (0-7) - e When z < z *» f = second integral -jK b = (+j/Kna)e jK e n (z+b) -jK - e u (z+b) - j K Z* n (0-8) 2?3 Using Eq. (G-7) and (G-8) in (G— 2), , « A v I / a n=l . mrx . mrx * sin sin ---a jK (z'-z) a -j< (z+z'+2b) - e a Gn (x,z[x' ,z') = i Z>z' (G-9) oo . mrx . mrx' sin --- s i n --a a jK (z-z') n \ a n=l -jK^(z+z ’+2b) » z<z' - e Note that at z = -b and using Gq for z < z ', Gq = 0. G q (x ,z |x '»z ') satisfies the boundary conditions. Thus Also note that the first part on the right-hand side is exactly the Green’s func tion for an infinite waveguide with e+‘*0)t time dependence. G-2. Derivation of the Integral Equation The total electric field intensity $ ( x , z ) , the incident field ip (x,z) a and the Green's function G q (x sz (V2 + e * k2)(J> = 0 , r < R — (V2 + k2)<j> - 0, r > R (V2 + k2H = 0 r |x ' ,z*) satisfy (V2 + k 2) G Q - - <S(x-x')6(z-z’) (G-lOa) (G-ll) (G-12) Using Eq. (G-ll), (G-13) (V2 + er* k)<|> - (er* - l)k2 ip Let a = <p - tp; hence, /[Gq (V2 + er* k2)a - a(V2 + k2)GQ] dS* = f ( G Q V 2 a - a V Z G0) d S ’ + (er* - l)k2Ja GQ dS* (G-14) Using Eq. (G-10a), (G-12), and (G-13) on the left-hand side of Eq. (G-14), and using Green*s theorem on the right-hand side, gives -(er* - l)k2 /gq * dS + a(x,z) = ( £ (g 0 |2. - a - g ^ dV (G-15) + (er* - l)k2 [/<(. Gq dS* - fip Gq dS*] Rearranging, cj>(x,z) *» i|)(x,z) +^>[pQ j " a + ^Gr* “ 1^k2 /*** G (G-16) Comparing Eq. (44) and Eq. (G-16), Ei(x,z) = *(x,z) + C ^ G q || - a j d£» Note that in the infinite waveguide case the line integral in Eq. (G-17) is zero. (G-17) G-3. 275 Derivation of E^(x,z) The line integral in Eq. (G-17) is evaluated along x = 0, z =+°», x = a» and z = -b. Since GQ = (ji=i|) = a = 0 o n x = 0 , x = a, and z = —b , rk fr 9 \ 0 3n " “ (r iSL JQf (?0 3n “ ) 3G° ^ " x, 3n <j)(z = +°») = p sin — e"”‘I<z + ij>(x»z) 3n cl ) ^ +00 dx (G-18) (G-19) Therefore, aCx',00) = sin ~ ~ 3a e . N 3a P e ^KZ . ttx* 3n BJT-= (_JK) s i n T pe (G-20) -jicz' (G_21) For Gq (x ,z |x ',z') at z' = +«, „ , I, -W . TTX GQ (x,zjx’,z ) = (+j/a«) sin — . sin 7TX * r[e -jK(z-z') J -1K (z+Z *+2b) - e J 'J (G-22) !f° i f1°a „. f f . + (1/a) sin S 3n sin 12l [eJxC*-*') . ,-Jk (*+.'+»)] 3z' (G-23) Using Eq. (G-20) to (G-23), 276 x , J( G ' <«« Substituting Eq. (G-24) in Eq. (G-18) and (G-17) results in E^(x,z) = \p(x,z) = sin ~ £e^KZ - e 3K (z+2k)J (G-25) Appendix H Infinite Waveguide Equivalent to the Shorted Waveguide H— 1. Equivalent Sources From Eq. (58), the equivalent source at z = +® is (H-l) while the one at z = -<*> is e-jic(z+2b) (H-2) and they are simultaneously applied. H-2. The Integral Equation The source ^(x.z) produces a field (^(x.z) while the source <|/2 (x,z) produces a field <}>2 (x,z), in the rod. The total field in the rod is therefore (H-3) <j>(x,z) = ^(x.z) + <f>2(x,z) . The integral equation in Eq. (G-16) holds when the Green’s function of the infinite waveguide is used; it is therefore only necessary to evaluate the equivalent E^(x,z) or e^(x,z). In this case, iKx,z) “ ijij(x ,z ) + i|)2 (x,z) 277 (H-4) H-3. The Equivalent E^(x,z) Proceeding with the evaluation of the line integral, at z ' = +°° <Kx',«) a (Pq + ATQ)[sin ~ _ J e ”^KZ + ^ 1(x,z) (H-5) where -j 2icb a= 7 !' p0e e 1 + ( B ' 6 ) Therefore, »(x',®) =» [(pQ + A r ^ e -^102 t “ Hr - Ae~^ICZ j sin + ^ 3“ V - « « > [ -Cp0 + (H-7) t • From Eq. (A) and implementing the sign reversal in the argument of the exponential when the time dependence is e+ '*ut, at z 1 ■= +*», GCxjzlx’.z’) “ (j/aic) sin 3G 3G -— «= ”— r 3n 3Z * . 1 . ttx . ttx* j K ^z"z ^ “ + — sin — s i n e a a a Using Eq. (H-7) to (H-10), a x ,<30 a , jK(z-z’) sin ~X— e a (H-9) • (H-10) 279 On the other hand, at z* = <f>(xf ,-°°) = <t q + AQ q ) sin a(x',-«) = 3a 3a + ^ 2 (x,z) (H-12) IiKZ’ i*Z \ ttxI (Tq + ApQ)e - e J sin (H-13) , . J r e^KZ . 4 1 . Jkz’ 3^ = " 3P'= (_:1k) L(T0 + Ap0)e " e , G(x,z|x',z') = (j/ax) sin — 3G ^ _ sin 3G f. -i / \ j ttx ttx' T P - = (+ 1/a) S±n 1 “ S±n — 4 ttx’ J sltl“T j<(zf-z) e 3 k (z (H-14) (H-15) z) 6 (H“ 16) Using Eq. (H-13) to (H-16) x ,= a •' °(GH - “I?) *=’ x -° <H-17) From Eq. (G-16), Ei,eq(x’z) = * (x’z) +(^ ( G I t ‘ “ I f ) Using Eq. (H-l), (H-2), (H-4), E1>eqta,z) = sin dV (H" 18) (H-II), and (H_17) In(H-18), + (H-19) Appendix I Integral Forma of and Tq Jones [21] has shown the equivalence between the variational approach and the Galerkin method in calculating the scattered field from the rod. The following procedure Is in effect using the Galerkin method. When the source in an infinite waveguide applicator is at z = +<» and when the time dependence is e+^w t , the incident wave on the rod is j KZ Tjjjtejz) When the = sin source is at z e • = -oo, theincidentwave (1-1) ontherodis -j K Z lfi^Cx.z) = Eg sin ^ First consider the case when e . thesource is at (1-2) z = +■»;then - Eq. (44) becomes <f)1(xJz) - tJj1(x,z) + ( e r* - l)k2 As z /<(.1 G dS' . (1-3) -<», G(x»z|x,,z’) -> (j/ica) sin cl sin —— & . j k (z - z ') e . (1-4) This is because for a time dependence of e+^ut, the appropriate sign for is negative, i.e., k n 280 281 (1-5) <n = “j [(mr/a)2 - k2] ^ Consequently, after substituting Eq, (1-1) and (1-4) in (1-3), jKZ Lim ^.(x.z) = E. sin — 1 l a e jKZ + sin — e a , /<j), sin J 1 „ [(e * - l)k r a -j KZ ’ e dxf dz'] y = r|>^(x,z) [ 1 + ^ - (er* - l)k2 (j/ica) / ^ = (j/ica) sin T —j K Z * e dxf dz'J t q i( j1(x ,z ) (1-6) since the total field at z = -“ is the transmitted field. From Eq. (1-6), — J KZ * 1 rQ = 1 + ^r- (er* - l)k2 (j/xa) / ^ sin — - e dx' dz* (1-7) Therefore, i E ^ T q - 1) = (sr* - l)k (j/ica) / ^ “j KZ' sin ■— £- e dx' dz* (1-8) where the integration is over the cross-section of the rod. Considering the case when the source is at z = Eq. (44) becomes <j>2(x,z) = ^2(x,z) + (er* - l)k2jf <t>2 G dx' dz * (1-9) 28 2 Using Eq. (1-4) the total field at z = -» is J Lim <j>2 (x,z) = <J>2 (x,z) + sin ^ Z+— °° ° A [(er*-l)k (j/Ka) e , l$2 s^-n -jKz' e d x 1 dz'] +jKZ = ^ ( x . z ) + i|i2 (x,0) pQ e (1-10) since the total field at z = -» is the sum of the incident field +j<z t/ >2 (x ,z ) and the reflected field pgtj>2 (x,0) e . From Eq. (1-10), jKZ i^2 ( x , 0 ) p 0 e +j KZ = E2 Sin ~ e jKZ pQ = sin — e , J <J>2 sin — - [(er* - l)k (j/Ka) -jKZ' e dx* dz'] (1-11) Therefore, „ E2 Pq = (er*-l)k (j/Ka) J<j>2 sin , “ jKZ* e ' dx* d z ’ where the integration is over the cross-section of the rod. (1-12) Appendix J Heat Generated with Lossy Iris and Short Circuit The amount of power consumed in a circuit element with voltage V across it and conducting current I is P = Y Re{VI*} (J-l) where the asterisk denotes the complex conjugate. If V q is the amplitude of the equivalent voltage wave from the microwave generator, the total voltage referred to the plane of the iris is V = V Q (1 + p) (J-2) and if the equivalent admittance of the cavity evaluated at the plane of the iris is (g^ + j b ^ Y g , the current flowing is I - v(% + jbL )Y0 . <J-3) Using Eq. (J-2) and.(J-3) in (J-l), - T M 2 ®L Y 0 * < T V 0 2 V I 1 + p i2 « L Since the available power P a is • (J- 4) 284 we can express the normalized heat generated in the cavity as Pn * 11 + p 12gL (J-6) ' Y q Is the characteristic admittance of the waveguide. A more familiar equivalent to Eq. (J-6) can be derived by noting that yt - %. + jbL - H Using Eq. i • W-7) (J-7) in (J-3) and implementing Eq. (J-l) and (J-5), Pn - Re {(1 + p)(l - P)*} - 1 - |p|2 . (J-8) The amount of heat generated in the rod can be calculated by following a similar procedure. The amplitude of the voltage wave transmitted by the iris is V q |1 + PjJ* where P^ is the reflection coefficient the iris would have if it were in an infinite waveguide. This propagates toward the rod and undergoes an infinite number of multiple partial reflections between the rod and the iris. Using the rod axis plane as the reference plane, the total incident voltage there is Vr - <1 + Pi)V0 e j<Al/(l - p ,p1 e where p 1 is the equivalent reflection coefficient of the rod and the shorted section of the waveguide evaluated at the rod axis; then (J-9) 285 where pj. “ -y±/(2 + y±) (J-- 10 ) p' » (1 - y )/(l + y) (J-- 11 ) and y are the iris and the rod-short-circuit equivalent admittances, respectively. Following Eq. (J-l), the amount of heat generated in the rod is then PnrP a “ J Re tVr (l + p ’> f(1 + pt>v r <8r + J V Y 0 ]* } * (J - 12) Using Eq. (J-5) and (J-9) in (J-12), Pnr “ gr l (1 + p ')(1 + pi) / a ” p,pie where (g^ + jbr) is the rod equivalent shunt admittance. Eq. -13) (J 1}!2 Note that (J-12) and (J-13) are valid only when the rod is thin. -j2KJLj When is an odd multiple of a quarter-wavelength, e becomes -1 and -j2tc£ 1 - ■p P ’p.e *P ^ e = l1 + pp'( ,P i • (J-14) Using Eq. (J-10), (J-ll), and (J-14) results in -j2i<£1 (1 + p')(l + Pi)/(1 - p'pje L ) - [ 4/(1 + y) (2 + y±)]/[l - y ^ l - y)/ (1 + y) (2 + y±) ] » 2/(1 + y + yy^) (J-15) 286 Using Eq. (J-15) in (J-13), the normalized heat generated in the rod is Pnr = 4gr/|(l + y + yy±)|2 . (J-16) The amount of loss in the iris and in the short circuit can be calculated by first evaluating the total voltage across their respective equivalent admittances. At the iris plane, the total voltage is expressed by Eq. (J-2); follwoing Eq. (J-l), the total loss in the iris is PniPa * I Ke{V[V V * 1 - -3- I»I2b± • 0-17) . (J-l 8) Dividing Eq, (J-l7) by (J-4) gives -S i n = _£ The incident voltage at the rod axis plane calculate the losses in the short circuit. can be used to The total voltage across the equivalent admittance of the short circuit evaluated at the' rod axis, gj+Jbj, is x''V^, where x 1 is the equivalent transmission coefficient evaluated at the rod axis plane. Thus, 287 where P P . £L ns is the amount of loss in the short circuit normalized to From Eq. (J-5) and (J-19), -j2 KX._ P ns = g J r ’d 5 _ l)\ + Pl)/(1 - p'Pje * (J-20) Eq. (J-20) is useful when the rod is not thin; r f is then evaluated in terms of the impedances of the T-equivalent circuit of the rod. If is an odd multiple of a quarter-wavelength, Eq. (J-20) reduces to Png = gjJ-r'd + y)/(l + y + yy±) |2 • (J-21) When the rod is thin, r f = 1 + p r; Eq. (J-20) and (J-21) then reduce to P n s = gjl (1 + P ’)(l + P.j) (1 - p ' P j e Agj/J (1 + y + yy±)|2 » ^ 1)|2 - (2n + 1)(X /4) . (J-23) Note that in this case, Pns Pnr _ (J-24) sr after dividing Eq. (J-23) by (J-16). The total normalized heat generated in the cavity can be ex pressed as P = P . + P + P n ni nr ns . (J-25) 288 Note that when the iris is lossless (g^ = 0) and when the rod is thin. P nr - 8r V C«r + g l> (J- 26> derived by using Eq. (J-18) and (J-24) in (J-25). The normalized amount of heat generated in the rod when it has to be represented by its T-equivalent circuit can be evaluated by first calculating the total amount of power absorbed by the rod and short circuit combined and then subtracting Png of Eq. (J-20) from the total. The combined absorbed power is (Pnr + png)pa - J Re {(1 + p f)Vr [(l + p')Vr y Yq]*} i |V.|2 R e { (1 + p*)(l - P')*> \ |Vr |2 (1 - |p’|2) where Eq. (J-ll) was used to replace y by p '. (J-27) Using Eq. (J-5) and (J-9) in (J-27) results in 2k % pnr + PnS " (1 " Ip’I ^ K 1 + P±)/(l - p'Pje b\2 . (J-28) Subtracting Eq. (J-20) from (J-28), —J 2 k£ Pnr {(1 - |p'|2) - g l |T*|2 }|(l + P^) 12 /|(l“ P ’P;Le *)|2 . (J-29) 289 When is an odd multiple of a quarter-wavelength, Eq. (J-29) reduces to Pnr = {(1 - |p '|2) - t '|2>|Cl + y)|2/ [(1 + y + yy i>|2 where r' can be evaluated from the given (J-30) and diameter of the rod. Note that dividing Eq. (J-20) by (J-28) gives Pns ^ Pns + pnr g j b ’ l2 1 (J-31) 1i- l.l2 p' Eq. (J-31) is useful in calculating P value of Jl, if (P + P )> 1 nr ns t ', ns and P nr regardless of the and p' are known. Appendix K Expressions for p^ , p', t ' and p^ The equivalent circuit of the rod sitting in an infinite wave guide is shown in Figure 3. The normalized equivalent impedance z is, ze - <zll - ZX2> + (1+ Z11 * ZX2)ZI2/(1+ Zll> The reflection coefficient ^ is, p Q = (ze - l)/(ze + 1 ) (K-2) Inserting Eq. (K-l) in (K-2), "O - (Z1X2 - Z122 - - Z122+ 2Z11 + “ The transmission coefficient across ZQ to the incident is the ratio of the total voltage voltage V Q . The total voltage at the input side is V q (1 + Pp); thus the total voltage V1 * V 1 + t,0>Z12n h l - <K-3) across ZQ is, Z122+ Zll> <K"4) Inserting Eq. (K-3) in (K-4) gives, T0 ■ (V V - 2Z12/(ZL - ZI2 + 2ZXX+U <K‘5) The same equivalent circuit applies to the derivation of p ' and t ' except for the terminating impedance being Z^/y^ instead of Z^ where is as expressed in Eq. (95), i.e., is the normalized equivalent admittance of the short-circuit at the rod-axis plane. Thus, where zL = (1/y^. Using Eq. (K-6 ) , 291 p ’ = (z* - l)/(z' + 1 ) e e Z 1(Z11 ~ 1) + (ZH 2 =* Z1^Z11 + ^ + ^Zn ~ Z12 ~ 2 11) 2 “ Z12 + Zll^ By a similar procedure used in deriving t ' = 2Z , , + D + z u + Z u 2 - Z122] « -8 ) The equivalent circuit that applies to the derivation of p ^ is the admittance of the iris y Zq. shunted by the characteristic impedance Therefore, the normalized equivalent admittance is, y" e = y. + 1 i (K-9) and, pi = (1 - y ” ) / ( l + y ” ) - - y ± / ( 2 + y± ) (K -1 0 ) Appendix L Transient Temperature Profile of the Hottest Zone The first step in the solution of Eq. (142) to (144) is define a new dependent variable to reduce the equations into a more manageable form [36j. Let T = W - (g0 /A) Then Eq. (142) to (144) become 32Wi4.*«1 , 2 r 3r 3W■ 4-.—A I IT J ss —1—3W■ 3r k a 9t 3W ■ g + y W - G(t) + g0 W = Tq + — , g0 t > 0, r = b t = 0 , all r where G(t) - F(T(t)). If we let W = 0 exp (aA/k)t then Eq. (L-2) to (L-4) become 293 -|^ + y0 = [G(t) + (yg0 /A)] exp[-(aA/k)t] = f(t), t > 0, r = b 80 0 - 0Q = TQ + ^ , t - 0, all r (L-7) . (L-8 ) The solution 0(r,t) is related to the solution 0(r,t,-r) by Duhamel's theorem, where 0 (r,t,t) is the solution of the auxiliary problem, 1 30 QN ~3r2 + 7 37 " « at 32§ , 1 30 (1^9) + y 0 =f(x), t > 0 , r = b 0 = 00 (L— 10) t = 0 , all r . (L— 11) When f(x) = 0, Eq. (L-9) to (L-ll) can be solved by separation of variables; let 0 ™ R(r)T(t); insertion in Eq. (L-9) gives 1 R d2R 1 dR _ 1 ^ 2 + rftd7 rtlZ ( 2 6i R - J0 (e±r), T = exp(-a$i2 t) . (L12) • (L-13a,b) Using Eq. (L-13) in (L-ll) with f(r) - 0, dJ_(S.r) U dr + YV^r) - 0, r - b * (L-14) To make use of Eq. (L-13) and (L-14) in the solution of Eq. (L-9) to (L-ll), let 0(r,t,t) = 9 q + [ f ( t ) / y - 0 q H 1 - I c i J 0 ( g j, r ) e x p ( - a 8 i t)> (L-15) where 00 2 1=0 (L-16) ciJo (3ir) = 1 Eq. (L-15) is the solution of Eq. (L-9) to (L-ll) because it satisfies these equations. The solution 9 (r,t) can then be derived from the solution 0 (r,t,t) by using Duhamel,s theorem: (L-17) Substituting (t - t) 9(r,t) =0_ + a U for t in Eq. (L-15) and integrating, / [f ( t ) / y - 0nl £ r=0 U i=0 1 U 1 Bj 2 i (L—18) where If radiation loss is taken into account, F (T) = fQ - MTA where and M are constants. (L-20) Using Eq. (L-l) in (L-20) and in serting the result in Eq. (L-3) and (L-7) results in f(t ) = {fQ - M[0exp(aAx/k) - (gQ/A)]^ + (YgQ/A) ) exp(-otAx/k) (L-21) where 0 = 0 (b,x). Inserting Eq. (L-21) in (L-18) and rearranging* 0 .z (fn/y - gn/A)/(B A/k)]exp(-ot& t) + e±2 (f0/Y - g0 /A)/(B±2 - A/k)exp(-oAt/k)}c1 J 0 (6 ir) — tl f [0exp(aA/k) - g„/A] e x p K - a g ^ t ) + 0 (3 ^ 00 I ciJ 0 (B±r)(a3±2) - A/k)x] dx (L-22) The solution to the original problem may then be expressed from Eq. (L-l) and (L-5): T(r,t) = 9(r,t) exp(aAt/k) - (8 Q/A ) (L-23) 296 where 0(r,t) is as in Eq. (L-22). Actual numerical calculation for 0(r,t) may use the "method of successive approximations" [8 ] to evaluate (L-22) at r = b; this gives 0(b,t). The value of 0(b,t) in conjunction with Eq. (L-22) and (L-23) determines the temperature profile. Appendix M Implemented Flow Chart for Bisection Technique G EijEiD Finds the roots of f(x)=0. Inputs* a, b f — -<+) £x <a < b) ax = (t>-a)/lO root = a a = a + 2£X > x-j = a X2 = Xi f AX yi = a H X II H X t AX x2 = Xl + A X )*f (x y = (x-l+x 2 )/2 o II a C D \r r stop.- ^ ...- Q decrease interval Cxx )*f C 297 stop J stop ^ 298 c yL = y yR = y y = (y+x1 )/2 m = m + 1 <JD y = (y+x2 )/2 m = m + 1 x = v - Ax xls yl 10n-l m - 100 yL < root < yR f (x ls> root = x C d ) m - 100 ?C1 S= X 1 S + A X / 1 0 ’ x v y i + i x / 1 ° n ~1 x 2 s= x 1 s +a x / 10 x 2 s= x l s + A x / l °n yR = y y = (y+xls)/2 m = m + 1 >r o lr* II <C L v, < root < r root =■ y y = (y+x2s)/2 m * m + 1 Ax/10n -S: Appendix N Operating Instructions for Temperature Controller What follows are the step-by-step procedures in activating and deactivating the system in either manual (MAN) mode or automatic (AUTO) mode. N-l. Manual Mode of Operation Activation Procedure STEP 1. Turn the microwave generator (RAYTHEON) power switch to ON. STEP 2. Turn the MODE switch of the TEMPERATURE SERVO to MAN. STEP 3. Turn the power switch of the TEMPERATURE SERVO to POWER ON. STEP 4. Turn the TEMP SET ADJUST (black knob) of the TEMPERATURE SERVO to its maximum counter-clockwise position. STEP 5. Turn the RAYTHEON RADIATE switch to ON when the READY light is ON. STEP 6 . Slowly turn the TEMP SET ADJUST of the TEMPERATURE SERVO clockwise to raise the magnetron anode current to the desired value. Deactivation Procedure STEP 1. Turn the TEMP SET ADJUST of the TEMPERATURE SERVO to its maximum counter-clockwise position. STEP 2. Turn the power switch of the TEMPERATURE SERVO to OFF. STEP 3. Turn the RADIATE switch and then the POWER switch of the RAYTHEON microwave generator OFF. 299 N-2. Automatic Mode of Operation Activation Procedure STEP 1. Set the IRCON pyrometer to the desired temperature RANGE and EMITTANCE. STEP 2. Turn the IRCON pyrometer power ON. STEP 3. Wait for one minute to allow the pyrometer to stabilize; when the GREEN light is on and the meter is reading zero, the pyrometer has stabilized. STEP 4. Connect the A(+) and B(-) output terminals of the pyro meter to the INPUT FROM METER terminal of the TEMPERATURE SERVO by a co-axial cable. STEP 5. Turn the MODE switch of the TEMPERATURE SERVO to AUTO. STEP 6 . Turn the RAYTHEON power switch to ON. STEP 7. Turn the TEMP SET ADJUST (black knob) of the TEMPERATURE SERVO to its maximum counter-clockwise position. STEP 8 . Turn the TEMPERATURE SERVO power to POWER ON. STEP 9. Turn the RADIATE switch of the RAYTHEON to ON when the READY light is on. STEP 10. Slowly turn TEMP SET ADJUST of the TEMPERATURE SERVO clockwise until the desired temperature is reached. Deactivation Procedure STEP 1. Turn the TEMP SET ADJUST of the TEMPERATURE SERVO to its maximum counter-clockwise position. STEP 2. Turn the MODE switch of the TEMPERATURE SERVO to MAN. STEP 3. Turn the power switch of the TEMPERATURE SERVO to OFF. 301 STEP 4. Turn the RADIATE switch and then the power switch of the RAYTHEON microwave generator OFF. STEP 5. Turn the IRCON pyrometer power OFF. N-3. Switching the Microwave Power Output Off and Oh While the System is in Operation In either mode of operation, manual or automatic, if it is necessary to turn the microwave power output OFF and ON while maintaining the set tings of the system, simply turn the RADIATE switch of the RAYTHEON microwave generator OFF and ON, respectively. However, this manner of operation may be done only when absolutely necessary to avoid undue stress on the system. TION procedures listed above should be followed. Otherwise, the DEACTIVA Appendix 0 Four-Probe Detection Signals The proper locations for the probes are shown in Figure 0-1. z LOAD REFERENCE PLANE SOURCE REFERENCE PLANE Fig. 0-1. Location of Four Probes The pair C and D monitor the real part of p while the pair A and B monitor the imaginary part of p. The probes of each pair are X /4 apart. g If E q is the amplitude of the wave initially incident on the load, the total electric field at probe C is E„ = E [exp(j<nX /2) + pexp(-jKnX /2)]S W w g (0 - 1) O while the total at probe D is Eq = E q £exp[j<(nXg/2 - *g/4)] + pexp[-jic(nXg/2 - Xg/4)]}S (0-2) where OO s = I m=0 (pPB)m exp(jm2icS,) (0-3) 303 is the sum of the infinite number of multiple reflections between the load and source. Since S is a geometric series, the sum is S = [1 - ppBexp(-j2icJ0] 1 • (0-4) Owing to the square-law detection characteristics of the diodes in the probes, the difference in detected voltage is Vc - VD = |Ec j2 - lEjjl2 = Eq 2 S2 ([(-1 - |p |cos 6 ) - 'j |P |sin 0 ]2 - [|p|sin 0 + j(l - |p|cos Q ) ] 2 ) = 4 E q 2 S 2 |p |c o s 0 • (0-5) Squaring Eq. (0-4) and inserting the simplified expression in Eq. (0-5) results in 2 . 4En V r = C D |p| cos 0 = • (0-6) 1 - 2 1pI|pB |cos(e + 0R - 2 k£) Note that p = |p|exp(j0) and pg = jpB |exp(j0B ). The voltage (Vc - V_.) is exactly proportional to the real part of p if p_ » 0. 1) O The total electric field intensity at probe B is EB = V e x P (j<L) + PexP(“j<L)]s (0-7) whereas the total at probe A is EA = EQ{exp[jic(L + 1^/4)] + pexp[-jic(L + Xg/4)]}S where (0-8) The difference between the detected voltages of probes B and A becomes VB - VA - 1 % ^ - IEa !2 = E 0 2 S2{[(1 + 1p|sin 0) - j|p|cos 6]2 - [jp|cos 0 + j(|p|sin 0 - l)]2 } ■ 4E 2 S2 |pI sin 0 P . (0-9) Squaring Eq. (0-4) and substituting in Eq. (0-9) gives V, <= V„ - V A = i B A 4E 2 |p|sin 0 --------: ---1 - 2 |p||pB |cos(0 + 0fl - 2kA) (0-4.0) The voltage V„-V. is exactly proportional to the imaginary part of O A Appendix P Photovoltaic Detector Response Characteristics If the spectral response of the detector is uniform over Its detection range X AJ <_ X <_ X , the resulting output voltage V n is ~ U U given by Eq. (8-37) of [13]: VQ oc e F(T) (P-I) where e is the emissivity of the target surface and F(T) is propor tional to the photon flux; F(T) - / U - 7 ----- — X^ -------X [exp(C/XT) - 1] (P-2) C is equal to 14,388 ym-°K and X is the wavelength of the detected photon. The integral in Eq. (P-2) can be evaluated by letting x = a” 1 and a = -C/T; for temperatures less than 4,000° C, the exponen tial is much greater than one when X is one micrometer or less. The silicon photovoltaic detector has Xu s 1 ym; therefore, F(T) s /Xu ea/XX~4 dX (P-3) is a good approximation, since the range of surface temperatures measured is less than 4,000° C. Je Hence, a/X ,-4 _ r ax 2 x“* dX - - /e3* ■& dx 305 306 2 ax „ x e . 2 r ax ------ 1- — Ix e a a^ , dx _ T ax 2_ xe 2 ax x e a I a u ax " _ e__ a 2 -I a/X a \X a/ - -L § P(X,T) (P-4) Using Eq. (P-4) in (P-3) and (P-l) gives VQ «e e[P(Xu ,T) - P(X£,T)]T/C (P-5) P(X,T) = exp(-C/TX)f(X,T) (P-6 ) f(X,T) = Z f (X" 1 + T/C) + X"2 (P-7) where The same output voltage will result if a given temperature has the right emissivity e^; hence, VQ - ei [P(Xu ,T±) - P(X^,T^)]T±/C (P-8 ) Dividing Eq. (P-5) by (P-6 ) gives f s - \ r p(xu»t> - \.& i J v V (P-9) In general, the spectral response is not uniform, i.e., the quantum efficiency of the detector varies with X. A plot of the quantum efficiency n of SI photodiodes is shown in Figure 24 of [43]; approximating the curve by a parabola, n sc 0.5[1 - b(X - 0.74) ] (F-10) where X is in ym and b = 10.41 ym -2 307 after equating ii to zero at X u = 1.05 ym Thus the new F(T) becomes £ 4.7 P(X,T) X^ (P-ll) X exp(C/TX) where P(X,T) is the same as before and the new f(X,T) is f(X,T) - -[2.215 + X“2 - CT/C + X-1) (3.279 - 2T/C)] . (P-12) The relationship between an assumed e^ and the reading T^ to the true e and T is still expressed by Eq. (P-9) with the modi fied f(X,T) as shown in Eq. (P-12). Eq. (P-12) departs significantly from the curve given in Figure 24 of [43] at the short-wavelength range; it gives X^ = 0.43 ym while the actual cut-off is at 0.26 ym. Nevertheless, Eq. (P-9) will still be a good approximation because P(XA,T) is very small compared to P(yu »T) for T < 4,000“ C. Appendix Q Electric Field Profile and the Impedances of the Rod in a Shorted Waveguide The following development relates the impedances of the T-equivalent circuit of the rod to the electric field intensity profile inside the rod when the l x l approximation The l x l to the variational solution is used. approximation to the total electric field intensity in side the rod may be expressed as, d(r,0) = (1/2 7r R) [aQJ0( k'r) where the unknown constants aQ + a^^ cosQJ1 ( k ’r)] and a.^ (Q-l) are determined by using the equivalent exciting field and the Galerkin procedure described in Section II-2. It can be shown, upon solving Eq. (52) with Schwinger's lemma and its corollary (Appendix A ) , that C0 “ ci (Doi + D10 )/2Du (Q-2) C1 " C0 ® o i + D 10 )/2D-qo (Q-3) D11 “ al = where Cq and field and D q q (D10 + D01 5 2 /4D00 are expressed in terms of the equivalent exciting and are as in Eq. (E-57) and (E-58) while, 309 Applying the general procedure for finding the pertinent limits of the derivatives of f' (Appendix F) to Eq. (C-l) yields, i\ © ar’ (o,o) (Q-5) dz 4ka The equivalent exciting field of the shorted waveguide applicator is expressed by Eq. (59). Substituting Eq. (59) for ^(x,z) in Eq. (E-47) gives, n Q0 co - - J1 = " t A \ ( 11 _- A ) X, 71 sin — — a nXQ . Qx (1 - A) (K/k) ( e* (Q-6 ) sin (Q-7) - 1 ) kor When the rod is at the middle of the waveguide sin ( Trx^/a) becomes unity. It can be shown after insertion of the detailed expressions of Dq q » D , CQ and C^in Eq. (Q-2) and (Q-3) and a comparison of the re sulting expressions with the l x l approximation for Z ^ - Z ^ and Z11 + Z12 » that art (j/27r) (1 + A) + (Zu - Z 12 ) (1 - A)/4 (Q-8 ) 2 7t R k aQ 0 (Z L v 11 (ka/27r )(Zn + Z 12 - Z 12 ) ) + (Z 11 - Z 12 )/16 .(1 -A)(Z1;l + Z 12 ) + (1 + A)/4 (Q-9) 2 jtR (Z11 + Z12 ) + (Z11 “ Z12 } /16 310 where Qg and are as defined in Eq. (E-24) and Che rod is assumed to be at the middle of the waveguide. Substituting Eq. (Q-8 ) and (Q-9) in Eq. (Q-l) give, (j/2?r ) l4 ( r , 9 ) - [(Z1;L + Z 12 ) + (.Z±1 - Z 12 )/l 6 ] [(1 + A) + (Z1;L - Z 12 )(1 - A). 4] j (k,r) ------------------------------- aQo l + j (Q-10) Q (ka) (ZH ~ Z12 > [(1 - A)(Z1;L + Z 12 )-(l + A)/4]cos9J 1 (k'r) > Ql as the total electric field intensity inside the rod located at the middle of the shorted waveguide applicator. Note that the amplitude of the incident electric field wave is assumed unity in Eq. (Q-10). Therefore the right-hand side of Eq. (Q-10) should be multiplied by the actual amplitude. VITA Name: JOSE C. ARANETA Date of Birth: March 16, 1947 Place of Birth: Cebu City, Philippines Educational Background Master of Science in Electrical Engineering University of the Philippines Diliman, Quezon City, Philippines Thesis: "Computer-Aided Design of Variable Frequency UHF Oscillators Using Bipolar Transistors" Graduated: April 1973 Master of Business Administration University of the Philippines Diliman, Quezon City, Philippines Graduated: April 1973 Bachelor of Science in Electrical Engineering University of the Philippines Diliman, Quezon City, Philippines Graduated cum laude: April 1968 Papers Presented "Theory and Practice of Microwave Sintering". J. C. Araneta and M. E. Brodwin. Northwestern University, Evanston, Illinois. 84th Annual Meeting, American Ceramic Society, Cincinnati Convention-Exposition Center, Cincinnati, Ohio, May 2-5, 1982. "Microwave Sintering of Thin Ceramic Rods in a Cavity Applicator". J. C. Araneta and M.E. Brodwin. Northwestern University, Evanston, Illinois. Fall Meeting, American Ceramic Society, Inc., Hyatt Regency, Cambridge, MA., September 12-15, 1982. 311

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