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A study of certain types of surface waveguides Lewis, John Eugene 1968

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A STUDY OF CERTAIN TYPES OF SURFACE WAVEGUIDES  by  JOHN EUGENE LEWIS B.A.Sc, University of New Brunswick, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in the Department of Electrical Engineering  We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee  Head of Department Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA May, 1968  In p r e s e n t i n g  for  that  this  an a d v a n c e d  thesis  degree  thesis  make  I f u r t h e r agree  Columbia,  I  agree  i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and  that permission  for extensive  copying  of  this  f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my  Department  o r b y hits  representatives.  or p u b l i c a t i o n of t h i s  w i t h o u t my w r i t t e n  Department o f  f  thesis  ^ ' ^  f  ^  3/>  IH8  It  is understood  f o r f i n a n c i a l gain  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date  f u l f i l m e n t of the requirements  at the U n i v e r s i t y of B r i t i s h  the Library s h a l l  Study.  in p a r t i a l  £«*i"'e*,*e Columbia  shall  that  copying  n o t be a l l o w e d  ABSTRACT This work consists of two parts. The first part is a comprehensive study of surface-wave propagation along dielectric tube waveguides. It includes the derivation of the characteristic equations and expressions for group velocity and attenuation coefficient, the latter by a perturbation method. Mode designations are justified and the physical distinction between the HE-j-j and EH-JI  modes is further illustrated by showing three-dimensional plots of the  field configurations. Computed characteristics are given for a wide range of parameters, and are compared with those of standard rectangular waveguides. Finally, a method of shielding the tube from weather conditions is proposed and the resulting changes in characteristics are noted. The second part of this work is essentially a unified analysis of all slow-wave modes in eight cylindrical waveguides. Characteristic equations are derived and expressions are obtained for the group velocity and the attenuation coefficients by a perturbation method. Accurate propagation characteristics for the dominant  TMQ-J  mode are computed for four waveguides with no restric-  tions on their radial dimensions.  These guides are the Goubau line and a  coaxial cable with dielectric linings on the inner, outer, or both conductors.  ii  TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS LIST OF TABLES LIST OF SYMBOLS  v vii i x  ACKNOWLEDGEMENT  x i i  GENERAL INTRODUCTION  x i i i  I. THE DIELECTRIC TUBE WAVEGUIDE 1. INTRODUCTION 2. FIELD COMPONENTS 3. MODE SPECTRUM 3.1 Characteristic Equations 3.2 Radial Variation of Fields and Mode Designations 3.3 Cutoff Conditions 3.4 Field Configurations .. 4. PROPAGATION CHARACTERISTICS 4.1 Phase Velocity 4.2 Group Velocity 4.3 Attenuation Coefficient 5. POWER CONCENTRATION AND FIELD EXTENSION 6. POSSIBLE PRACTICAL SYSTEMS 6.1 A Series of "Standard" Dielectric Tube Waveguides 6.2 Comparison with Standard Rectangular Waveguides  9 16 22 24 28 28 32 38 42 47 47 50'  6.3 The Dielectric Tube in a Polyfoam Medium  52  APPENDIX A MODE CUTOFF CONDITIONS FOR THE DIELECTRIC TUBE  1 2  5  9  56  APPENDIX B DIELECTRIC TUBE FIELD COMPONENTS IN CARTESIAN COORDINATES ... 59 II. SCREENED SURFACE WAVEGUIDES 7. 8. 9. 10. 11.  INTRODUCTION FORMULATION OF THE PROBLEM FIELD COMPONENTS CHARACTERISTIC EQUATIONS GROUP VELOCITY AND ATTENUATION COEFFICIENT 11.1 Group Velocity 11.2 Attenuation Coefficient 12. CHARACTERISTICS OF PARTICULAR STRUCTURES 12.1 Goubau's Surface-wave Transmission Line 12.1.1 Design charts iii  61 62 65 6 8  72 75 75 80 83 83 83  Page It.1.2 Accuracy of the design charts 12.1.3 Comparison with experimental results 12.2 Dielectric-lined Coaxial Cables APPENDIX C ATENUATION COEFFICIENTS OF DIELECTRIC-LINED COAXIAL CABLES BY THE PERTURBATION METHOD  89 91 93  r  98  APPENDIX D THE EXACT SOLUTION OF QUASI-TM SURFACE-WAVE MODES IN DIELECTRIC-LINED COAXIAL CABLES  102  APPENDIX E THE Q OF RESONANT WAVEGUIDES CONCLUSIONS  106  \  107  BIBLIOGRAPHY  ':  iv  109  LIST OF ILLUSTRATIONS Figure  Page  2.1  The Dielectric Tube Waveguide  3.1.a  Mode Spectrum of the Polyethylene Rod  3.1.b  Mode Spectrum of the Polyethylene Tube,  3.1. C  Mode Spectrum of the Polyethylene Ijbe,  3.2. a  Radial Dependence of Fields, HE^ Mode, P = 0 . 1  17  3.2.  b  Radial Dependence of Fields, HE^ Mode, P = 0 . 0  17  3.3.  a  Radial Dependence of Fields,  Mode, P = 0 . 5  18  3.3.  b  Radial Dependence of Fields, HE-^ Mode, P = 0 . 5  18  3.4. a  Radial Dependence of Fields, EH-^ Mode, P = 0 . 1  19  3.4. b  Radial Dependence of Fields, EH^ Mode, P = 0 . 0  19  Radial Dependence of Fields,  TMQ£  Mode, P = 0 . 5  20  3.5.b  Radial Dependence of Fields,  E H ^  Mode, p = 0 . 5  20  3.6  Cutoff Conditions,  3.7  Cylindrical and Cartesian Co-ordinate Systems for the Dielectric Tube Waveguide  3.8.a  Electric Field Configuration for the Polyethylene Tube, Mode, P = 0 . 9 .  3.5.  a  3.8. b 3.9. a 3.9.b  TEQ^  5 13 P  =0.5  13  p=0.95  14  23  , E-| , EH^ and HE^ Modes Q  HE-,-, !!...  26  Magnetic Field Configuration for the Polyethylene Tube, HE-,-, Mode, P = 0 . 9 !!...  26  Electric Field Configuration for the Polyethylene Tube, EH-,-, Mode, P = 0 . 9  27  Magnetic Field Configuration for the Polyethylene Tube, EH-,-, Mode, P = 0 . 9  4.1.a  24  !!...  Normalized Phase Velocity of Thick Tubes, HE^ and  TEQ-J  27  Modes .  29  4.1.  b  Normalized Phase Velocity of Thin Tubes, HE-J-J and TEg-j Modes ..  29  4.2.  a  Normalized Phase Velocity of Thick Tubes, TMQ-J Mode  30  4.2.  b  Normalized Phase Velocity of Thin Tubes, TMQ-J Mode  30  4.3. a  Normalized Group Velocity of Thick Tubes, HE-j-j and TE^-j Modes . 3 6  4.3.b  Normalized Group Velocity of Thin Tubes, HE -j -j and v  TEQ-J  Modes ..  36  Figure 4.4.a 4.4.  b  4.5. a  Page Normalized Group Velocity of Thick Tubes, TMQ^ Mode  37  Normalized Group Velocity of Thin Tubes, TMQ-J Mode  37  Normalized Attenuation Coefficient of Thick Tubes, HE,-, and T E Modes ............!!.. Q 1  4.5. b  Normalized Attenuation Coefficient of Thin Tubes, HE-,, and TE  Q 1  Modes  .  !!....  39  Normalized Attenuation Coefficient of Thick Tubes, TMQ-J Mode  40  4.6.b  Normalized Attenuation Coefficient of Thin Tubes, TMQ-J Mode ...  40  5.1.a  Power Concentration Characteristics of Thick Tubes, HE-J-J Mode .  43 43  4.6.  a  39  5.1.  b  Power Concentration Characteristics of Thin Tubes,  5.2.  a  Field Extension Characteristics of Thick Tubes, HE-J-J Mode  44  5.2.  b  Field Extension Characteristics of Thin Tubes,  44  5.3. a 5.3. b 5.4. a  HE-J-J  HE-J-J  Mode ..  Mode  Fraction of Power Carried in Medium 1 of Thick Tubes, HE,-, Mode !! Fractions of Power Carried in Media 2 and 3 of Thick Tubes, HE-J-J Mode Fraction of Power Carried in Medium 1 of Thin Tubes, HE,, Mode .... !!  45 45  46  5.4.b  Fractions of Power Carried in Media 2 and 3 of Thin Tubes, HE,, Mode ....!! 4 6  6.1  Comparison of Rectangular Waveguide and Dielectric Tube Attenuation Characteristics 49  6.2  Normalized Phase Velocity of Thick Shielded Tubes, HE-j-j Mode ..  53  6.3  Normalized Group Velocity of Thick Shielded Tubes, HE-j-j Mode ..  53  6.4  Normalized Attenuation Coefficient of Th-ick Shielded Tubes, HE -j -j Mode  6.5  54  Division of Attenuation Between Media 2 and 3 of Thick Shielded Tubes, HE Mode 54 n  8.1  The Dual Surface Waveguide  8.2  Types of Surface Waveguides  8.3  Steps for Obtaining Solutions for Cases 2 - 8 from Case 1  vi  65 .  66 67  Figure  Page  12.1.a  Surface-wave Transmission Line Design Chart  12.1.b  Surface-wave Transmission Line Design Chart for the Millimeterwove Region 86  12.2  Surface-wave Transmission Line Dielectric Attenuation Characteristics  12.3  Attenuation Characteristics of the Goubau Line  90  12.4  Attenuation Characteristics of Dielectric-Lined Coaxial Cables  96  vii  85  88  LIST OF TABLES Table  Page  3.1  Parameters for Radial Field-dependence Plots  16  6.1  Normalized Characteristics of Polyethylene Tubes in Free Space, HE Mode, p = 0 . 9  4  8  n  :  6.2  Rectangular Waveguide and Dielectric Tube Characteristics  6.3  Normalized Characteristics of Polyethylene Tubes Surrounded by an Infinite Polyfoam Medium, HE^ Mode, p = 0 . 9  9.1  51  55  Functions Describing E . and  70  11.1  Limits of Integration for the Eight Waveguides  77  11.2  Evaluated Integrals for Media 2 and 4  78  11.3  Evaluated Integrals for Medium 3  79  12.1 12.2  Measured Characteristics of the Goubau Line 92 Comparison of Propagation Coefficients Obtained from Perturbation Theory and from Exact Theory for Coaxial Cables with One Dielectric Lining 94  12.3  Comparison of Propagation Coefficients Obtained from Perturbation Theory and from Exact Theory for a Coaxial Cable with Two Dielectric Linings 97  viii  LIST OF' SYMBOLS constants functions of Bessel functions J'V i a  constant field extension ratio (see definition of r^) fraction of power transmitted within r=rp functions of modified Bessel functions <"ri  c  i o o» k  Z  < ri o » / < i o E  k  c  Z  / s  8  '  longitudinal, radial and azimuthal components of electri field, respectively, in medium i radial variation of E^., E^. and E ., respectively Q  frequency constant functions of Bessel functions wave number of medium i  Hankel functions of the first and second kinds longitudinal, radial and azimuthal components of magneti field, respectively, in medium i radial variation of H ., H . and H ., respectively Q  modified Bessel function of the first kind Bessel function of the first kind phase coefficient of free space  modified Bessel function of the second kind length of a resonant waveguide mode subscripts total power carried ard power carried in medium i , respectively h,r  V j  total power loss per unit length and power loss per unit length in medium i , respectively end-plate losses and waveguide losses of a resonant waveguide quality factor of a resonant waveguide radial co-ordinate radius of an interface between two different homogeneous media radius at which the longitudinal field components become C times their values at the tube outer boundary  f  see definition of Cp integrals of functions of Bessel functions  integrals of functions of modified Bessel functions time loss tangent of medium i speed of light in free space, group velocity and phase velocity total energy storage per unit length and energy storage per unit length in medium i , respectively total energy stored by a resonant waveguide cartesian co-ordinates Bessel function of the second kind  accurate and approximate waveguide impedance impedance of free space accurate and approximate attenuation coefficient, respectively accurate and approximate attenuation coefficients of medium i , respectively attenuation coefficient of a TEM wave in medium 2 a /a 2  Q  phase coefficient a+jB percent phase-velocity reduction relative permittivity of medium i azimuthal co-ordinate free-space wavelength cutoff wavelength guide wavelength relative permeability of medium i V 2 r  conductivity of rectangular waveguide conductivity of medium i angular frequency  xi  ACKNOWLEDGEMENT The author is indebted to hi." research supervisor Dr. M.Z. Kharadly for his keen interest and assistance throughout the course of this project. Grateful acknowledgement is made to the Northern Electric Company for a graduate research fellowship for the academic year 1964-1965, and to the National Research Council of Canada for a Research Assistantship for the year 1965-1966, and for a Studentship during the period 1966-1968, and for support of the project under grant A3344. The author is also grateful to Mr. T. Bourk for his contributions to this work in deriving equation 12.20, and in writing several programs for Part I, and in particular, for his assistance in obtaining the three-dimensional field configurations shown in figures 3.8 and 3.9. The author also wishes to thank Miss L. Blaine for typing the manuscript and Messrs. B. Wilbee and P. O'Kelly for proofreading the final draft. Finally, I would like to thank my wife, Kathy, for her continued encouragement and patience throughout the years of study leading to the production of this thesis, for she alone knows its cost.  xii  GENERAL  INTRODUCTION  The increasing demand for high-capacity long-distance communications systems has recessitated the development of wide-bandwidth transmission systems, capable of handling large numbers of channels. This can only be achieved by operating at extremely high frequencies, and two promising types of systems are presently being investigated. The T E Q ^ overmoded circular waveguide is being developed to operate from 40 to 110 GHz.  Optical waveguides (beam, lens and  dielectric fibre) are also the subject of intensive study. These would be capable of handling extremely large numbers of channels.  In both cases, there  are still numerous difficulties to be overcome. The continuing development of very low-loss dielectric materials could make guides, such as the dielectric tube or rod, competitive as transmission media both at millimeter-wave and at optical frequencies. Part I of this work is undertaken to provide a comprehensive study of the dielectric tube waveguide, with a view to its possible application to low-loss transmission. One serious limitation of dielectric guides, being open structures, is that they are subject to radiation and interference. It would be useful if this difficulty could be overcome while still maintaining the desirable characteristics of the unshielded guides. For this reason, a study of screened surface waveguides is undertaken in Part II of this work, which examines several types of these structures.  xiii  PART I THE DIELECTRIC TUBE WAVEGUIDE  2 1. INTRODUCTION The first analysis of wave propagation along a dielectric tube was carried ouv by Zachoval^ in 1932.  The characteristic equation for TM modes was  obtained and solved graphically for several tube sizes and dielectric constants. 2  Two years later, the existence of these modes was verified by Liska , whose measurements of guide wavelength showed good agreement with Zachoval's theory. 3 It was not until 1949 that Astrahan  obtained the characteristic equations of  TE and hybrid modes. .Experimental values of guide wavelength for the TEQ-J  HE-J-J,  and TMQ-J modes for various sizes of polystyrene tubes agreed very well with  theory. Astrahan also gave data for TMQ-J and TEQ-J cutoff wavelengths, plots of the radial dependence of the field components and a field configuration of the 4 HE-J-J  mode in a transverse plane. Jakes obtained the attenuation coefficients  of TM and TE modes by expanding the characteristic equation in a truncated Taylor series about the lossless solution, thus separating the equation into its real and imaginary parts. The real part was solved for the lossless phase coefficient, while the imaginary part yielded the attenuation coefficient valid for small losses. Comparisons were made of the attenuation characteristics of these modes with those of the cutoff wavelength.  TEQ-J  mode in a circular waveguide having the same  The measured attenuation of  TMQ-J  and TEQ-J modes for poly5  styrene tubes agreed favourably with Jakes' theory. Coleman and Becker gave a theoretical attenuation characteristic which was obtained earlier by Beam^ in the vicinity of 100 GHz.  A technique for obtaining the attenuation coefficient  of any mode was outlined by Unger^ in 1954 using a method similar to Jakes', but the analysis was completed only for the  HE-J-J  mode on tubes with small  diameter-to-wavelength ratios. Unger also derived cutoff conditions for HE-j  m  and EH -j modes and gave a comparison of dispersion and guide radii between tubes and circular waveguides with the same attenuation. Some numerical results were 8 9 also given and these were extended in a later paper . Mallach made a rough  3 estimate of HE -j -j mode attenuation by measuring the radius at which the magnitude of the electric field fell to 1/e of its value at the tube surface. In 1954, apparently unaware of the work of Mallach and Unger, workers at the Bell Telephone Laboratories^ published the results of a study of the relative merits of the tube and the rod as long-distance wa/eguides. Calculations showed that the guide wavelength of the tube was more slowly varying with radius than that of the rod, thus requiring a less strict dimensional tolerance. However, this was considered insufficient to justify the additional problem of tube fabrication, and a decision was made in favour of solid-dielectric waveguides of rectangular cross-section which have the advantage of supporting modes of known polarization. The dielectric-tube antenna has been investigated by Jakes^ and 1112 Kiely  ' , who have calculated and measured radiation patterns of these  devices.  In the field of millimetre-wave generation, Pantell 13 and others 14'15  have used dielectric tube cavities resonating in TM and TE modes to couple electromagnetic waves and electron beams. Extensive data on cavity Q and field variation with radius for various sizes and dielectric constants of tubes have been given by these workers. From the above review, it can be seen that, to date, no comprehensive study of the possible usefulness of the dielectric tube as a low-loss waveguide has been made. The purpose of the present work is to provide such a study, particularly regarding the possible use of the tube as a low-loss waveguide at microwave and millimetre-wave frequencies. The field components for TM, TE and hybrid modes are given in Chapter 2 in terms of eight arbitrary constants. Relations among these constants are obtained and the lossless characteristic equations for all modes are derived in Chapter 3. Numerical solutions of these equations are obtained for various  4 sizes of polyethylene tubes. Computer programs are also written to calculate and plot the radial variation of the Held components. The verification of mode designations is established from these graphs, and cutoff conditions for n=0 and n=-l modes are given in terms of the behaviour of the longitudinal field components as cutoff is approached. Also shown in Chapter 3 are threedimensional plots of the field configurations for the first two hybrid modes,, obtained by numerical solution of the differential equations, with calculations and plotting performed by computer. In Chapter 4, the equations for group velocity and attenuation coefficient for all modes are derived by integrating the appropriate combinations of field components to obtain power flow, energy storage per unit length and power loss per unit length. Numerical results are given for the and  HE-J-J  TMQ-J,  TEQ-J  modes. The concentration of energy in the various regions and the  decay of fields from the tube surface are studied in Chapter 5. In Chapter 6, a comparison is made between the tube characteristics and those of rectangular waveguides.  Finally, a proposed practical system of embedding the tube in a  polyfoam medium is considered, and the resulting changes in characteristics are noted.  5 2. FIELD COMPONENTS ^ The tube configuration of interest is shown in Figure 2.1. It consists of two coaxial dielectric regicns of infinite length and relative and e ^ embedded in a vhird infinite dielectric of relative  permittivities permittivity e  where  v  e  r2 * r l l £  e  r3  FIGURE 2.1 The Dielectric Tube Waveguide In all cases, it will be assumed that the relative permeability of the i  region,  v.j > is unity; however, the relative permeabilities of the various media will be r  retained in the equations for completeness. Propagation is assumed in the zdirection, with t-e-z dependence of the form exp j(cot - ne - BZ) in the lossless case. Under these conditions, omitting the factor exp j(tot - ne - Bz), the field components are given by  E  - a, I ( r )  2l  n  hl  ,  rl  H  ny , k Z  i  1  I  - b, I (h,r)  2l  n  lo z2 2  E  1  r  n< 2 >  = a  J  h  r  R  = a A (h r)  IJ V 2 >  +  h  '  r  "  y  r ?  2  k  n  Z  n  2  n  h r  2  2  e2  :  H  a  - b  22  V2  " | 72  =  ^h f r^Z 2  =  J  VV>  a  2  =  +  2 n V  b  B  2  o  " J h7 2 2 >2 ) b  B  r  , r2 o - j ' r \ f 2 n(V) - i ^ 2 n(y) -o "2 h r e  •e2  V V>  k  a  A  b  2  B  a  "z3  K (h r)  3  n  3  ft  '  J H7 3 *3 n < V >  T3  p  n  K  r3  k  n  Z  n  2 h_ r  +  b  3 n< 3 ) K  h  r  < r< ~ h r *3 W  "93  2.2  " J ^3 T ^ > 3 ^ K  3  z3 3 n 3 >  H  = b  K  (h  n  r3  63  H  =  £  r  r3o k  3 j  a  o  3 W >  l  + j  3  b  3 n(V) K  er3_ ok a K (h,r) + - f - b K (h r) h, 3 Z0 "3 'V"3' 3 Q  3  n  h  n  3  r  where, from the wave equation hf1 = B - yrl , erl , o k 2 2 r2 r2 o  h  h  3  k  e  = 6  ' r3 r3o y  £  _  2  = y  B  2  k  The symbols appearing in equations 2.2 and 2.3 are defined in the list of symbols. Upon setting n=0 (no e-variation), equations 2.2 separate into two sets corresponding to the circularly symmetric modes designated The corresponding field components are given by  TMQ^  and  JF-Q M  TM MODL'S E -zl  a  TE MODES  l  0 < r< r  jt l W >  Hrl £  n o Z 3rl Q  z2 2 0 2  E  = a  A  (h  £  '62  'z2  r)  j  T2  a A (h r)2  Q  r2  2  r2 o _ Z B r2  2  2  -i £- b Bi(h r) 2  2.4.b  2  u k Z r2 o o  "82  a K (h r) Q  Q  M 0  0  3  o o H 6 rl J  b B (h r)  k  E  -z3  rl  -el  E  2.4.a  b  }  k  r  'el  n  z3  3  V2  b K (h r) 3  0  3  r <r< » 2  E  r3 ' h^ 3 0 3 =  J  a  £  H83  K  (h  r)  H  r3 o Z 3 r3  r3  k  =  J^ 3 V 3 > b  h  ^r3 o o k  r E  "83  0  r  2.4.c  Z  Hr3  For n^O, equations 2.2 describe inseparable combinations of TE and TM modes which are designated hybrid modes. In general, one or the other of the component parts of a hybrid mode is dominant. If the TE portion is dominant, the mode is designated HE ; if the TM component is dominant, i t is termed nm  EH^. The nature 'of TE or TM-dominance and the significance of the subscript m in the mode designation will be discussed in Section 3.2.  3. MODE SPECTRUM The field properties of the various modes are investigated in this section. The characteristic equations for all modes are derived, and solutions are given for several modes with n=0-and n-1.  The radial variations of the  fields a;re shown and the results used to give a consistent mode designation for the tube and the rod. Finally, three-dimensional field configurations are given for two of the lower-order hybrid modes. 3.1  Characteristic Equations By using equations 2.2 and equating the tangential field components  in media 1 and 2 at r = r-j, and those in media 2 and 3 at r = r^, eight homogeneous equations can be obtained in the eight unknov.ns a^, b.., i = 1,4. The most commonly-used method of obtaining the characteristic equation is to set the determinant of this system of equations equal to zero. However, by algebraic manipulation, it is possible to obtain two equations, one from the matched tangential fields at each boundary. These equations contain two unknowns, a^/a2 and b^/b^, which are found by applying Cramer's rule to the original eight equations.  The two equations are then solved simultaneously to  yield the phase characteristics of the tube. This approach has the following advantages over the alternative method of setting the determinant equal to zero: (i) The algebraic manipulation yields certain ratios of the arbitrary constants which are required in later parts of the work. (ii) The functions ^(^22^ ' ^n^22^ defined ky equations 2.2.b anc  appear in the characteristic equation explicitly. This proves useful in deriving cutoff conditions for the various modes in terms of zeroes of these functions (Section 3.3). Letting p.. - h.r., and matching tangential fields at r = r yields 9  h a  b,  >32> A (p )  K  3  n  K (p ) n  32  22  ' r2 n^22 r3 n 32 22 n^22 32 V?32 e  A  )  e  K  (p  }  +  1P  A  )  p  ]  hz n P22 , ^3 n P 2^ P22 n P22 32 n 32 B  (  }  B  (  }  K  (  ,3.1.a  3  P  K  (p  }  -J c.  2 323 and from matching fields at r = a  a  a  b  !2 _ VPll> l n P21 a  A  (  b,  b2  ]  e  A  (  A  )  >r2 n P21 \P l n^21> B  5  +  (  +  B  2  1  1  . 2 2 . \ ll 2l/_ +  p  ^rlVhl^ PnVPn^  J  -1  ngZ,  ^r^n^ll Pll^^ll)  )  (  IpjP^)  l " y ^ "  b  ./ r2 n P21 \P21 n P21  , we obtain  p  1  z0 k0 \p 2  3.1.b  1 +  n  p2  21  •J c.  l 2 l Equating the two expression for c from 3.1.a, and those for c-j from 3.1 .b.'yields a  a  b  3  the following two equations / r2 n P22 , r3 n P32 \ Ar2 n 22 , r3 n 32 \P22nP22^ P32 n P32^ l 22 n P22 P32 n P32 £  A  A  (  }  e  K  (  (  K  )  B  (  P  B  (p  )  (  y  )  K  (p  K  )  (  )  nB 1 J_ o\p p| /  3.2.a  +  k  22  2  and ng 1 1 3.2.b 2 , k 2 P /j ° n Equations 3.2 contain two ratios of arbitrary constants, a^/a and b^/b , which  ' r2 n<P2l) r l n l 1>\ K n 2 1 , MyPn>\_ 21 n 21 Pn P2i n 2i mvm n vn(Pn) ni £  A  £  T  ( p  A  P  A  (p  J  B  ( p  }  2  I  B  lp  T  Vp  21  2  2  have yet to be determined. Applying Cramer's rule to the original eight equations yields  CO  12 The ratio b^/b^ is obtained from equation 3.3 by interchanging the e .. and u .. Solution:, for hybrid modes are obtained by solving equations 3.2 simultaneously for 3 usinq the method of False Position, where a^/a and b^/b^. are given by the 2  appropriate form of equation 3.3, and the three wavenumbers are then given by equations 2.3. Upon setting n=0, both left-hand sides of equations 3.2 separate into two factors, each of which equals zero. These are given by 'e-o A (p •rl "0 22 P n( 22 n  A  r3 0 32 P32 0 32  e  VF c  p  22  K  (p  K  (p  0  •3.4  r3 Q 32 P K (p  = 0  3.5  rl W l l ll 0 ll  = 0  3.6  = 0  3.7  1  i  %2 0 22 ,P2 0 22 B  (P  B  p  ( P  r2 V 2 1 ,21 0 21 p  A  (p  32  2  :  K  p  e  (p  p  Q  I  32  ( p  1  \l 0 21 21 0 21 p  B  (p  B  (p  M  T  o ii ( p  piTVhi  k  Solving equation 3.6 for a^/a and equation 3.7 for b^/b,, yields 2  P ) 2 " \ 0 21 /\Pn 0 PlT) n  a  Y  (p  )  -1 r2 0 P 2 1 \ / V l 0 l l .r2 0 P21 P J p^T/ IPnio^n) P2i o 2i  e  I  J  )  I  (p  )  £  Y  (  Y  21  )N  (p  3.8  )  0  -1 J  0 21 \A rl 0 n (p  )  J  I  , r2 o 2i Pll 21 0 2l" p  }  y  7  p  J  J  (p  )N  (p  ^rlVPn) »Pn o Pn I  (  VV2 0 P21 P2i o P2i , Y  +  )  (  Y  )N  (  3.9  }  The characteristic equation for TM modes is given by equation 3.4 with a^/a  2  defined by equation 3.8, while that for TE modes is given by equation 3.5 with b^/b defined by equation 3.9. 2  Solutions of equations 3.4 for TM modes, 3.5 for TE modes and 3.2 for n=l hybrid modes are shown in figures 3.1.a, 3.1.b and 3.1.cfor three sizes of polyethylene tubes in free space U =e =y -| y 2 M l > e =2.26, and =  rl  r3  r  =  r  =  r3  r2  p-r-j/r=0, 0.5, 0.95). The following main features of the mode spectrum may be 2  15 noted from these graphs: (i) For the dielectric rod (p=0), TEg and TMg modes hcve the m  same value of r /X at cutoff, as do E H ^ and HE-j M  2  m  modes (m>l).  (ii) The HE-J-J mode has no lower cutoff. (iii) If  the TEg and TMg modes no longer have the same value m  m  of r^/X at Cutoff. The same is true for H E ^ -j and E H - ^ modes. m+  (iv) As p+l, the phase characteristics of the Hg and HE-j modes m  m  become indistinguishable, as do those of the Eg and E H ^ modes, thus providing m  the physical distinction between H E and EH modes. (v) As p->-l, the n=0 and n=l modes appear in widely separated clusters, each cluster consisting of four modes ( H E ^ , TEg , TMg and EH^). m  m  (vi) The HE-| and Hg phase characteristics intersect at some value m  of r^/X.  m  In most cases, for values of r /X greater than that at the intersec2  tion, the differences in the two curves are too small to be seen graphically. However, the degeneracy of the HE^ and H Q modes for p=0.5 can be seen in 2  figure 3.1.b. The intersection of HE-j and TEg mode transmission characterism  m  tics will be observed in Chapter 4. Points (i), (ii) and (iii) are well-established facts. Point (iv) provides the distinction between HE and EH modes as well as the basis for their description. It is believed that (iv), (v) and (vi) have not been previously observed.  16 3.2 Radial Variation of Fields and Mode Designations ixcept for the mode designation for the dielectric rod given by Clarricoats* , which was based upon the sequence of solutions of the characteris1  tic equation, ither attempts to provide a consistent designation have met with limited success^ " . 6  19  In this section, the mode designation for the dielectric  tube is obtained by examining the sequence of solutions as well as plots of the radial dependence of the field components for various values of p and rg/x. It is shown that this designation agrees with that of Clarricoats for the special case of the rod. The functions describing these radial variations are given by E  zi  = E ./(cosne cossz) zl  E~ = E^cosne sinez)  H •= H -/(sin ne cosez) H zi  zl  ri  ri  E~ = E /(sin ne sinsz) ..3.10 = H /(sin ne sinez) H = H /(cosne sinez), ri  ei  Qi  Qi  Qi  where Ezi., En., E.., ., H •• andoi H . are given by equations 2.2. The oi Hzi ri Q  normalized field components were plotted for several lower-order modes on polyethylene tubes in free space (e 2 2.26, ] 3 l«0)• =  e  r  =e  r  =  r  The plots for eight  representative cases are shown in figures 3.2 to 3.5. cases are listed in Table 3.1.  The conditions for these  In all cases, r /x = 2.0. 2  figure  mode  3.2.a 3.2. b 3.3. a 3.3. b 3.4. a 3.4. b 3.5. a 3.5.b  HE11 HE11 TE02 HE12 EH11 EH11 TM02 EH12  k/B Q  0.1 0.0 0.5 0.5 0.1 0.0 .0.5 0.5  0.6729 0.6701 0.7742 0.7759 0.6885 0.6883 0.8095 0.8146  TABLE 3.1 Parameters for Radial Field-dependence Plots  17  E  cn c at s-  4->  CD  + -> ( /)  4-> tO  "O  0)  r/r.  •I—  r/r.  4-  O S • MO QJ  o  .7  •r+->  /  ai c cn  E/ 7  HH  *  /  FIGURE 3.2.a Radial Dependence of Fields, HE^ Mode, =0.1 P  \ E,  +->  cEn QSi+ -> 4->  cn c  OJ i-t-> to  to  ^ 0 OI  r/r,  r/r. V  /  o QJ  4->  II  crens  /E  /  /  •I  ,1  II-  r  /  0  FIGURE 3.2..b... Radial Dependence of Fields , HE-j -j Mode, p=0.0  FIGURE 3.4.b Radial Dependence of Fields, EH^ Mode, p=0.0  21 Examination of these graphs reveals the following properties of the modes: (i) All of the modes HE , H , E nm  Qm  Qm  or EH  nm  can be supported by the  dielectric tube, where the first subscript indicates the number of harmonic variations of the fields with e. (ii) For H~ Om. and HE, lm modes, m is the number of zeros of zH~ in x  r  l  2 (fi9  < r < r  u r e s  3.2 and 3.3). Here, equality is required for the HE^ modes m  because the first zero of H" in r-j < r < r^ moves to r=0 as r-j->0 (figures 3.2.a 2  and 3.2.b), to become coincident with the zero already existing there. (iii) For EOm and EH, E in Im modes, m is the number of zeros of z n  r  l  2 ^ *9  < r < r  1  ures  ^-4 * 3.5). Equality is not required here since there anc  is no zero shift to the origin as r^-s-0 (figures 3.4.a and 3.4.b). (iv) Strong similarities exist between the H^ and HE-j modes for m  large r A (figures 3.3.a and 3.3.b). This effect becomes increasingly 2  pronounced as p-*l, but is not easily detected as p-»0. The same behaviour exists for Eg and EH-j modes (figures 3.5.a and 3.5.b), thus providing the distinction m  m  between EH and HE modes in the mode designation. Although the above properties are derived from a study of n=0 and n=l modes, there appears to be no reason why they would not be valid for all n.  22 3.3 Cutoff Conditions Lossless surface-wave propagation on the dielectric tube requires that all quantities appearing in equations 2.3 be real and positive. If r3 r3 ^rl r l ' *  y  G  =  e  t  when P320 =  ancl  c^ ^  ien  0  when h^=0 and h^=0, or more generally,  o c c u r s  P"n - These requirements a^e used in Appendix A to derive the =0  cutoff conditions given by equations A2, A7, A9 and A12, for all modes with n=0 and n=l. Letting  Q 22 (p  conditions reduce to  t n e s e  r  r l 21 0 21 " r2 l 2 1 : rl 21 0 P21 " r2 Y l ( f W  e  J  i»  = 1J  p  J  (p  }  2e  p  Y  (  )  2£  J  (p  }  TM modes  )  .3.11.a  21 *V 2T^ " l( 2l)  p  P  Y (p  21  p  22  2J  P  0  2 1  p  TE modes  2Y p )  ) -  l(  21  = 0  HE-j-j mode .3.11.b  HE-j modes m> 1 m  l< 2lj J (P 2  J  p  VP )  M 21> p  21  P  (P 2^ " i CP ) Y  J  1  V 21>  2  2  22  1 21  J  21 r l  p  e  EH-| modes m> 1 m  .3.11 .c  Equations 3.11.a for TE and TM modes yield the same results as obtained by Astrahan 3, while equation 3.11.b for HE modes has been derived by Unger7 , and equation 3.1T.C reduces to Unger's condition for EH modes i f  =  = 1.  For the dielectric rod r-|=0 (or p-j=0), and equations 3.11 reduce to 2  JQ(P ) 2 2  =0  J ( p ) .= 0 T  2 2  P  22  =0  TM and TE modes HE 1 and EH modes, m > 1 1m +  lm  3.12  HE-j -j mode  From these conditions, it can be seen that, for the dielectric rod, the Eg and m  Hg modes have the same value of p m  22  at cutoff for given m. This is also true  23  for E H  1 M  and HE-]  modes.  At cutoff p2 2 is given by J  p  22  =  99  2ir  r —2 /e y r2  r 2  -  e  3  y  r3  ,  3.13  c from which the value of r A 2  c  may be determined. If P  2 2  =0,  then r A 2  c  =  0  and there is no lower cutoff frequency. Solutions of equations 3 . 1 1 to 3 . 1 3 for polyethylene tubes and rods in free space are shown in figure 3 . 6 for the ,  EQ.J ,  E H ^  and HE-j modes. 2  24 3.4 Field Configurations While the field configurations of TM and TE modes on the dielectric 12 14 tube have been fully discussed in the l terature ' , only limited information is available for hybrid modes. It was only recently that the correct field :  configuration in a transverse plane of the dielectric rod for the HE-j-j mode has 22 been given . In this section, an attempt is made to present a clear picture of the field configurations of the first two hybrid modes. This is achieved by obtaining three-dimensional computer plots of the field lines. The method involves solving the differential equations of the field lines using numerical methods. These equations are formed by taking ratios of the field components in cartesian co-ordinates and equating them to the corresponding ratios of differentials. These field components are converted from cylindrical co-ordinates as shown in figure 3.7 by using the transformations. Exi. = -ri E . cose E . sine ei Eyi. = Eri . sine + E ei. cose  3.1 4  Ezi. = zi E . The expressions for these field components in cartesian co-ordinates are given in Appendix B for n=l modes.  x z  FIGURE 3.7 Cylindrical and Cartesian Co-ordinate Systems for the Dielectric Tube Waveguide  25 The computational procedure for obtaining E- field lines is as follows: (i) Given an initial condition ( X  Q 5  y , z ) through which the Q  desired field line is to pass, convert the point to (r , e , z ) Q  Q  and calculate the three field components, (ii) Take the field component which is largest in magnitude, say E , and form the two differential equations  thus assuring that all slopes are less than one in magnitude, (iii) From the initial condition (x , y , z ) determine a new point Q  Q  Q  ( 1» y-j» z-|) on the field line by numerical solution of the two x  differential equations using a fourth-order Runge Kutta method, (iv) Using (x-j, y-|, z-|) as the new initial condition, repeat the procedure until the field line touches a boundary of the medium considered or until the field line closes upon itself, (v) Choose a new initial condition and repeat (i) to (iv). (vi) Repeat the procedure for each of the three regions of the tube, and for H-fields as well. Plots of the electric and magnetic field configurations for the HE-j-j and EH^ modes are shown in figures 3.8 and 3.9, respectively, where the transformations  u s e  d for the conversion from three to two-dimensional co-ordinates  are given by x' = z -(0.7/^")y •y' = x -(0.7//2)y The dotted lines shown in figures 3.8 and 3.9 represent the field lines in medium 2, while the solid lines are those in media 1 and 3.  3.15  26  FIGURE 3.8.b Magnetic Field Configuration for the Polyethylene Tube, HE -j -j Mode, p=0.9, r /A=l.68, A /A=0.8804 0  FIGURE 3.9.a Electric Field Configuration for the Polyethylene Tube, EH^ Mode, p=0.9, r/X=1.68, A/A=0.9654 2  g  28 4. PROPAGATION CHARACTERISTICS In this section, the phase-velocity characteristics are obtained by solving the dispersion equation.  The group velocity and attenuation coefficient  are obtained by integrating the appropriate combinations of field components. The standard perturbation technique is used where it is assumed that the fields are only slightly perturbed by the small losses, thus permitting the use of lossless field components in obtaining expressions for power flow, power loss per unit length and energy storage per unit length. Numerical results are given for the phase and group velocities normalized to the speed of light in free space and for the attenuation coefficient normalized to that of a TEM wave in medium 2. For polyethylene tubes in free space,  e r  -i  = e  3 le 2 2.26, =  r  =  r  tanfi-^tane^O and tan6 is arbitrary. 2  4.1  Phase Velocity Given values of y ., e -, p and r^A, equations 3.2, 3.4 and 3.5 (for n  hybrid, TM and TE modes, respectively) can be solved numerically for k /fi. Q  This also yields the normalized phase velocity and the normalized guide wavelength, since k  o  /e  =  v  p o /V  =  A  g  4.1  A  The solutions for HE^ and  TEQ-J  modes for "thick" tubes (p=0.1 to  0.9) and for "thin" tubes (p=0.91 to 0.99) are given in figures 4.1, and for the  TMQ-J  mode in figures 4.2. By inspecting figures 4.1 and 4.2, the following properties may be  noted: (i) The phase velocity decreases with increasing r^/x from the free space value at cutoff and asymptotically approaches that of a TEM wave in medium 2. (ii) The dispersion characteritiscs improve as r A is increased. 9  1.0  v  o  0.9  0.8  0.7 0.6652 0.65  FIGURE 4.1.a Normalized Phase Velocity of Thick Tubes, HE-j-, Mode, TE Mode Q1  1.0  ^____^99 \<)8  v  o  ^97 0.9 \  \.93  0.8 p= .91^ 0.7. 0.6652 0.65 0  2.0  4.0  6.0  8.0  r A  10.0  2  FIGURE 4.1.b Normalized Phase Velocity of Thin Tubes, HE-J-J Mode, -TE Mode  1.0  0.9  0.8  0.7 0.6652 0.65 FIGURE 4.2.a Normalized Phase Velocity of Thick Tubes, TM Mode Q1  1.0  v  o  -^^99 .98 " . 9 7 ^ ^  0.9 .95 93\  0.8  P=.91\  0.7 0.6652 0.65  2.0  4.0  6.0  8.0  r /X  10.0  2  FIGURE 4.2.b Normalized Phase Velocity of Thin Tubes, TM Mode Q1  31  (iii) The separation of the  TMQ-J  and TEQ-J modes increases as p  increases. (iv) The phase characteristics of the able from tho.~e of the TEQ-J  TEQ-J  HE-J-J  mode become indistinguish-  mode for values of ^ A remote from cutoff of the  mode. This behaviour becomes increasingly pronounced as p increases.  Similar behaviour of group velocity and attenuation characteristics will be noted in Sections 4.2 and 4 . 3 , respectively. The significance of these properties will be considered when dealing with possible practical systems in Chapter 6.  4.2  Group Velocity The group velocity of a given mode in a waveguide is normally found  by taking the slope of the w-B diagram. This may be found graphically, or by differentiating the characteristic equation. In the case of the dielectric tube this procedure is both lengthy and tedious. Alternatively, the group velocity may be found by determining the rate of transport of energy, which for a lossless waveguide is given by 3  where and  3  4.2  = power flow in the i medium w\ = energy storage per unit length in the i^* medium. 1  For hybrid modes, these quantities are given by  4.3 r i-1  and  where r^^O and r^ = °°. Substituting from equations 2.2 yields  33 a a Tr/A (p )N l " ., 4 V2 2  N  2  n  rl o  21  F^ M o o  a a Tr 2  N  S  +  k  BZ  C  £  3 =  l)  + 2 n c  +  _ T  2^  a  A  7  y  k  (  )N  4v  e  o o t n(Pll^ h  Z  ll  *  =  6S  9  r l  P l l  a  a  Tr  )  4.5  ^r2 r2 o AB  2 +  +  e k2)l2f  e  n  r  + 2 n c  .,  3  f  l  k  2.  ( B  +  y  )S  .2^2, ' r3 r o n 32 +  e  k  )K  (p  )  3  2 v  L  2  J  n  T  i-  r2 2 2 h T +B S + (y k Z c ) S + I r ^ y ^ e Z ^ g 4~~ o2o U 2  2  c  r l  h?T Z1 c,) ]S ,k o 3ZnI^p,,) I I + [g +(y rl k oo I - 4nc,y 1 rl o HI T  1  Z  + U  2  C l  /  _ rl 2V/ n P21 c  I - 2n (B  6 S  k  2  p  W  S  C  * 2 a a ,/yp^Y/ r3 o , 2\ , Ih^'lTf )A ' ~ ~ r3 o o 3J K 3 n^ 32' L 2  i  Z  r2 o Po- A V o o 2 B  2  ~ 4h*'  2  k  +  T  2  A  2  A  r2  0  0  2  4.6  B  * 2 e 3a a 7r/A (p )' r  W  2  2  n  22  3K ^  h  3  4Z  o 3 o >32) L h  V  lK  /  T  +  +  ^r3 o o 3 ^ K k  Z  C  )2  S  + 4nc  3^r3 o c n k  6Z  K  <P > 32  The functions Sj, S^, Sg, S^g, S^, Tj, T^ and T^ are integrals of functions of Bessel functions which are defined and evaluated as follows: Pll T  I  =  2 (Pll - ^ ^ P l l ) - P^n (Pll)  ^ l ^ l  J  2  Pll  V  2  V  ^ ' n V d  j  2  \ n(P )P P P2 K  d  3  3  3  " (P 2  +n2)K  3  = -Tj + 2p I (P )I (P )  Pl  11  n 32 (p  )  +  n  P 2 n 32 K  (p  11  n  11  4.7.a  }  3  3  2 [ [ n (P P3 ^ ^32 m  K  S  = 2  K  }  3  +  2 ^slK  =  " K " P32 n(P 2) n P3 ) T  2  K  K  3  (  2  . 4.7.b  34 K  A  T  =  2  2  S n 2 ^ 2 p 2 A,2 = (p2-n 2)A2(p ) + p„ (p ) V 22' 22 n ^22' 2  A  (P  )f  dp  21  nK  22  P  = 2  p  99  n  ((p^-n )A (p ) fp-^ < (p ) 2  2  n  00  21  2]  22  r .2 ^ [ n (P2)P2 ^ n(P )J P21 . • A  +  A  dp  2  = T + 2p A  2  A (p )A ;(p ) - 2p A (p )A^(p )  22  n  22  t  22  21  n  21  21  ,4.7.c  SAME AS S WITH A REPLACED BY B A  P  22  [A (p )B (p ) + B (p )A (p )]dp  AB  n  P  2  n  2  n  2  n  2  2  21  = A (p )B (p ) - A (p )B (p ) n  21  n  21  n  The corresponding  22  n  22  and W.. for TE and TM modes are obtained by using equations  2.4 for the field components in equations 4.3 and 4.4. This yields for TE modes .i  M o o 2 2V 0 21 \ 2h: In(PnV vp k  BZ  b  b  B  i  BZ  )  b  b  u  'B  ' r3 o o 2 2 / 0 22 y  k  BZ  2h  b  b  3  y .e .k N. n n o i W, = 8v M  c  1  -o n  V o o 2 2 2h, k  No =  (p  7T  B  (p  K^32»  4.8  :  "  i=l,2,3  For TM modes the corresponding relations are  4.9  e  r1 g 2 2Yo 21^ k  a  a  (p  0  E  No  =  K^V  o l  2 Z  h  )J  r2 o^ 2 2 k  a  2Z  a  o2 h  TT  4.10  4  r3 o 2 2 / Q P22 V k  e  ga  Tr  a  A  )  (  1  2Z  £  o3  Wo  =  a  a  a  Z  u 0  a  1T  A  Z  T  o ni v  (h T  3S)  2  4.11  2  A+  A  ^0 9L o0 H  a  2  . J> .  (p  V  r3 2 2.  e  =  7r  r2 2 2 2  Wo  P  * 2 r1 2 2 / 0 2lV2  oT e  V 32^ "  h  o  a  h  ' 0 22^  1T  A  (p  K (p V  > o  0  32  o3 o The normalized group velocity characteristics for the same conditions as used for figures 4.1 and 4.2 are given in figures 4.3 and 4.4, respectively. From figures 4.3 and 4.4, it can be seen that for large r /x the normalized 2  group velocity appraoches l//e  r2  , but unlike the phase velocity, may become  less than this asymptotic value at some intermediate r /A. A similar 23 24 2  phenomenon has been observed  '  in the case of wave propagation along  dielectric-coated conductor surface-wave transmission lines.  1.2 FIGURE 4.3.a Normalized Group Velocityy of Thick Tubes, HE-,-, Mode, -TE 01 Mode 1.0 nl  v  .99  o 0.9 \ S-97  0.8 \ V  .98 , ^  \ .95 9  3  \  p=.91 0.7 0.6652  0.6  0  2.0  4.0  6.0  8.0  FIGURE 4.3.b Normalized Group Velocity of Thin Tubes, HE Mode, TE Mode iy  Q1  ,  1.  38 4.3  Attenuation Coefficient The attenuation coefficient of the dielectric tube may be found by  assuming a complex permittivity for medium 2 in the characteristic equation 4 and solving tor the propagation coefficient, y. Jakes has solved the equations for TM and TE n.odes by expanding the resulting complex equation in a truncated Taylor's series about the lossless solution, thus separating the equation into its real and imaginary parts. The real part yields essentially the lossless characteristic equation which is solved for 3> and the imaginary part may then 7 be solved for a. Unger has applied this method to the HE-,-, mode, but his result is only valid for small phase-velocity reductions since he used smallargument approximations of the Bessel functions. A disadvantage of this method is that it involves differentiating the lossless characteristic equation with respect to three variables. Also, losses in regions 1 and 3 cannot be included unless two additional derivatives are obtained. An alternative approach is to use the well-knov/n perturbation method. The small-argument approximation of exp(-2a) is used to obtain an expression for the attenuation coefficient which is valid for small losses. This method is employed in this work, yielding for the attenuation coefficient 3 3 a = P/2N =(1/2) I P. / I N. 4.12 i=l i=l P| = power loss per unit length in the i ^ medium, and is given by P. = utanfi. W. 4.13 1  where  i  i  1  i  The equations for N. and W. are given in section 4.2 for all modes. The attenuation coefficients of the HE  , TE^ and TM  modes,  normalized to that of a TEM wave in medium 2 (a ) are shown in figures 4.5 and 4.6 where a is given by a = 0/2) /TT~ k tan6 o r2 o c Q  9  4.14  41 From figures 4.5 and 4.6, the following points are noted: (i) The attenuation characteristics of all modes become more slowly varying with r /A as p increases. 2  (ii) For any particular value of r A> the attenuation coefficient 2  decreases with increasing p. (iii) For large r A the attenuation coefficient approaches that of 2  a TEM wave in medium 2.  42 POWER  5.  CONCENTRATION AND  FIELD  EXTENSION  Of interest in any open surface waveguide is the degree of power concentration in the vicinity of the structure. A measure of this concentration can be taken as the fraction, C , of total transmitted power which is contained within a given radius r . This s given by i-  oo V  - k j  (E  r3 83 - i W O " H  1  -  5  J  Conversely, it might be desirable to know the value of rp for some fixed value of Cp. This may be determined numerically. An alternative measure may be taken as the ratio, C^, of the field strength, either E ^ or H ^ , at some radius r^ to that at the surface of the guide. This is given by f n( 3 f) n(P 2 ' Again, this equation can be solved for r^, given a specified value of C^. C  =  K  h  r  /K  )  5  2  3  Solutions of equation 5.1 for the HE-j-j mode are shown in figures 5.1 for C = 0.999. P Similar curves for C^=0.001 are shown in figures 5.2. If maximum permissible values of rp and r^ are set, then a lower operating frequency is established for a given tube size. This will be considered in Chapter 6. The behaviour of the dielectric tube waveguide can be further understood by examining the division of transmitted power among the three regions. Since the variations of N^/N with r A for all modes are very similar, only 2  those results for the HE-J-J mode are plotted in figures 5.3 and 5.4. From these plots, it is seen that the dominant feature is the power shift from outside to within the walls of the tube as r^j\ increases from zero, until for very large r^/x, essentially all power is transmitted within the tube walls. This accounts for the asymptotic behaviour of the propagation characteristics discussed in Chapter 4.  46  47 6. POSSIBLE PRACTICAL SYSTEMS The results of Chapters 4 and 5 suggest the possible use of dielectric tubes as low-loss waveguides in regions vhere conventional waveguide systems are too impractical.  In an attempt to show possible superiority of the dielectric  tube waveguide over other types, a series of tube sizes is proposed corresponding to the standard rectangular waveguides in the /arious frequency ranges. The performance of the dielectric tube is adversely affected by accumulation of snow, ice and moisture, and by the presence of nearby obstacles which may cause radiation. Also, supporting the guide presents a problem. A method is proposed for overcoming these difficulties by embedding the tube in a low-density, low-loss material of sufficient extent that a negligible portion of the wave is carried outside this medium. The effects of such a shield upon the propagation characteristics are also investigated in this chapter. 6.1 A Series of "Standard" Dielectric Tube Waveguides The choice of size of dielectric tube for a given frequency range is an arbitrary one depending upon which restrictions are more important in a particular application, i.e. attenuation, dispersion, field extension or higherorder modes. The choice given here is by no means optimum. The emphasis is on improving the dispersion characteristics over those of rectangular guides and achieving a lower attenuation for guides used above 30 GHz. The propagating mode is the HE-^ (fundamental) mode, which requires the smallest diameter tubes. A moderately thin wall tube (p=0.9) is chosen since the attenuation is smaller and the dispersion characteristics are better. The same bandwidth as for rectangular waveguides is achieved by using r^/X from 0.6 to 0.9.  This range provides a reasonable compromise between excessive attenua-  tion and excessive field extension. The normalized characteristics for these condi-  48 tions are given in Table 6.1. r A  0.6  0.9  v  p o  0.9839  0.9590  v  g o  0.9399  0.8839  a/a  0.0836  0.1950  p 2  3.81  2.34  9.05  4.60  2  /v  /v  Q  r  /r  r /r f  TABLE 6.1  2  Normalized Characteristics of Polyethylene Tubes in Free Space, HE-j-, Mode, p=0.9. The outer radius of each tube is given by v r = 0.6 ^ 2  6  j  where f is the recommended lower operating frequency of the corresponding rectangular waveguide. For the conditions given in Table 6.1, the is cut off, but the  TEQ-J  mode has its cutoff at r A = 0.66. 2  of each band, r A = 0.9 and v /v = 0.9654 for the 2  p  o  TEQ-J  TMQ-J  mode  At the upper limit  mode, compared with  0.9590 for the HE -j mode, indicating that the two modes still have an appreciable difference in phase velocity.  It is assumed that the  TEQ-J  mode is not excited,  either at the source or due to bends and irregularities in the tube. Straight line approximations to the attenuation characteristics of the rectangular waveguides and the corresponding tubes are shown in figure 6.1.  The  tube characteristics are based upon a loss tangent for polyethylene of tan6 = 0.00050. There is a range of published loss tangents for polyethylene 2  from 0.00030 to 0.00060. The corresponding shift in attenuation for these values is also shown in figure 6.1 for the smallest tube size. Because of the normalization, the curves of figures 4.5 and 4.6 are valid for any value of tan6 , 2  and the attenuation coefficient is readily found for any loss tangent.  49 1000  o o CO  -o +->  c  O  o o •r—  4->  fO C D + > 4J  1000  100  Frequency, GHz FIGURE 6.1 .Comparison of Rectangular Waveguide and Dielectric Tube Attenuation Characteristics " ~ " ' ' ' ' Polyethylene tubes in free space, tan6 =0.0005; -.Polyethylene tubes in free space, tan6 =0.0003 & 0 . 0 0 0 6 ; Polyethylene tubes in polyfoam, tan6 =0.0005, tan6 =0.00015; Standard aluminium rectangular waveguides, a = 3 . 5 4 - 1 0 ^ mhos/cm; Standard silver rectangular waveguides, a = 6 . 1 7 - 1 0 mhos/cm 2  2  2  3  5  50 6.2  Comparison of Dielectric Tubes and Standard Rectangular Waveguides In this section, the characteristics of rectangular waveguides are  compared with those of the dielectric tubes chosen in Section 6.1.  The fac-  tors considered are dispersion, attenuation, size and support. The normalized group velocity of standard rectangular waveguides varies from 0.6 to 0.848 over the recommended frequency range. The dielectric tubes have corresponding values of 0.9399 and 0.8839. Hence, not only is the velocity of energy propagation higher for the tubes, but also the change in group velocity over each band is only 23% of the change for rectangular waveguides. The attenuation characteristics for dielectric tubes (tan62=0.0005) are given in Table 6.2, along with those for rectangular waveguides. From these results, it can be seen that there is a steady improvement of attenuation limits above 10 GHz of the proposed tubes over the metallic guides. These results are also shown graphically in figure 6.1 for those guides operated above 1 GHz. The dimensions of both types of guide are given in Table 6.2, from which it can be seen that each tube diameter is approximately twice the broad dimension of the corresponding rectangular waveguide. This is an advantage for the tubes at very high frequencies, where the guides are small and a strict dimensional tolerance is required. Dielectric tubes, being open waveguides, cannot be approached too closely, since a major portion of the energy is transmitted in the space surrounding the structures. It is noted from Table 6.1 that, at the lower frequency limit, the field strength falls to 0.1% of its surface value at r^ = 9.05 T^-  These values of "field radius" for the tubes considered are  given in Table 6.2.  In order that the tubes may be shielded from adverse  weather conditions and supported without causing reflections or radiation, it is suggested that they may be embedded in a polyfoam medium (e =1.03,  51  Frequency (GHi)  Rectangular Waveguide Inner Dimension* (Inches)  Dielectric lube Diameter (Inches)  Rectangular Waveguide Iheoretlcal Attenuation Lowest to Highest Frequency (d3/100ft)  Dielectric Tube Theoretical Attenuation l c ~ e s t to Highest Freoency (do/IOOU)  Olalectrlc Tube Field Radius for C -0.COl (Inches) f  0.320-  0 490  23 000-II 500  44.291  0.039  0.027  0.056-  0.195  0.350-  0 530  21 000-10 500  40.495  0.046  0.031  0.O61-  0.213  183.  0.410-  0 625  18 000- 9 000  34.569  0.056  0.038  0.071-  0.250  156.  200.  0.490-  0 750  15 000- 7 500  28.925  0.069  0.050  0.065-  0.299  131.  0.640-  0 960  i : 500- S 750  22.146  0.128-  0.075  0.111-  0.390  100.  0.750-  1 120  9 750 4 875  18.898  0.137-  0.095  0.V31-  0.457  0.950-  1 500  7 700- 3 850  14.919  0.201-  0.136  0.165-  0.579  67.5  1.120-  1 700  6 500- 3 250  12.655  0.269-  0.178  0.195-  0.683  57.3  1.450-  2 200  5 100- 2 550  9.775  0.388-  0.255  0.253-  0.834  44.3  1.700-  2 600  4 300- 2 150  6.337  0.501-  0.330  0.295-  1.04  37.7  2.200-  3 300  3 400- 1 700  6.442  0.669-  0.466  0.383-  1.34  29.2  2.600-  3 950  2 840- 1 340  S.451  0.940-  0.641  0.453-  1.58  24.7  3.300-  4 900  2 290- 1 145  4.295  1.190-  0.845  0.575-  2.01  19.4  3.950-  5 850  1 872- 0 872  3.588  1.770-  1.22  0.688-  2.41  16.2  4.900-  7 050  1 590- 0 795  2.892  1.840-  1.42  0.854-  2.99  13.1  5.850-  8 200  11.0  85.5  1 372- 0 622  2.423  2.45 -  1.94  1.02 -  3.57  7.050- 10 00  1 122- 0 497  2.010  3.50 -  2.74  1.23 -  4.30  8.200- 12 40  0 900- 0 400  1.728  5.49 -  3.83  1.43 -  5.00  7.62  10.00 - 15 00  0 750- 0 375  1.417  6.45 -  4.50  1.74 -  6.10  6.4!  12.40 - 18 00  0 622- 0 311  1.143  6.14 -  5.36  2.16 -  7.56  5.17  15.00 - 22 00  0 510- 0 255  0.945  8.37 .  6.10  2.61 -  9.14 •  18.00 - 26 50  0 420- 0 170  0.787  13.3 -  9.5  3.14 . 11.0  3.56  22.00 - 33 00  0 340- 0 170  0.644  16.1 - 11.2  3.83 - 13.4  2.92  26.50 - 40 00  0 280- 0 140  0.535  21.9 - 15.0  4.62 - 16.2  2.42  33.00 - 50 00  0 224- 0 112  0.430  31.0 - 20.9  5.75 - 20.1  1.94  40.00 - 60 00  0 188- 0 094  0.354  38.8 - 27.2  6.97 . 24.4  1.60  8.71 - 30.5  1.28  50.00 - 75 00  0 148- 0 074  0.283  52.9 - 39.1  eo.oo - 90 00  0 122- 0 061  0.236  93.3  75.00 - 110 00  0 100- 0 050  0.109  9.10  4.28  • 52.2  10.5  - 36.6  1.07  100.  - 70.4  13.1  - 45.7  0.66  90.00 • 140 00  0 080- 0 040  0.157  152.  - 99.  15.7  - 54.9  0.71  110.00 - 170 00  0 065- 0 0325  0.129  163.  , 137.  19.2  - 67.0  0.58  140.00 -220 00  0 051- 0 0255  0.101  308.  -193.  24.4  - 65.3  0.46  170.00 -260 00  0 043- 0 0215  0.083  384.  -254.  29.6  -103.  0.33  220.00 - 325 00  0 034- 0 0170  0.064  512.  -348.  38.3 -134.  0.29  • The tppor frequency of each dielectric tube 1s taken as 1.5 times tho lower frequency. The upper f r e q u e n d u only. The f i r s t nineteen rectangular waveguide! t r * aluminium, the remaining fifteen are s i l v e r .  shown apply to the rectangular waveguides  TABLE 6.2 Rectangular Waveguide and Dielectric Tube Characteristics  52 tanS^O.OOOl5) which extends to r=r^. The modified characteristics of these shielded tubes are considered in Section 6.3. The rectangular waveguides are closed structures, and obviously no such problems of support or radiation exist. 6.3 The Dielectric Tube in a Polyfoam Medium The characteristics of the HE-^ mode on polyfoam-shielded polyethylene tubes are shown in figures 6.2 to 6.5, where the normalized quantities Vp/v , g/ ' a/a , a^/a and a^/a are plotted for the same ranges of p and v  Q  v  0  Q  r^/x as considered previously for thick tubes in free space. Inspection of these graphs yields the following information: (i) The HE^i mode still has no lower cutoff frequency, (ii) The phase-velocity reduction has a non-zero lower limit dependent upon e^,^, and an upper limit which is unchanged from the case of the unshielded tube. (iii) The attenuation is increased because of the outer lossy medium. The increase is most significant at low phase-velocity reductions. (iv) Since at cutoff, all the power is carried in the outer medium, there is a lower asymptote to the normalized attenuation given by  53  54 .1.0 a a  0.75  0.5  /  /  ^8  p=.i//.3/';  //.  JJJ  0.4  0.8  1.2  1.6  2.0  FIGURE 6.4 Normalized Attenuation Coefficient of Thick Shielded Tubes, HE-,-, Mode 1.0  0.75  \W \ > V i \ /> N  hi //\/ Hi// / / \ /  / y A  /  0.5  /.8  1  I'I'/V  0.2!  V  '\  /\  \  \ /  A  y N 0.4  h  \  \ / \ / \ /  / 0  \  \ \ \  \  \  \ ^  •s. v. • v  0.8  1.2  1.6  2.0  r /x FIGURE 6.5 Division of Attenuation Between Media 2 and 3 of Thick Shielded Tubes, HE-,-, Mode a /a, a^/a, a-,=0.0 2  2  55 The characteristics of the HE^ mode for p=0.9 and r /x of 0.6 and 2  0.9 are listed in Table 6.3. A comparison with the data in Table 6.1 reveals little significant change from the case of the tube in free space, except for the increase in attenuation, particularly at the lower frequency. r A 2  p  0  g o a/a  Q  0.6  0.9  0.9746  0.9528  0.9392  0.8826  0.2116  0.2852  TABLE 6.3 Normalized Characteristics of Polyethylene Tubes Surrounded by an Infinite Polyfoam Medium, HE^ Mode, p=0.9. The straight line approximations to the attenuation coefficients of the shielded tubes are shown in figure 6.1, where it is noted that, although the attenuation is increased, its range is reduced below that of the unshielded tubes. The proposed shielded tubes also have lower attenuations than the corresponding rectangular waveguides for frequencies above 30 GHz.  56 APPENDIX A Mode Cutoff Conditions for the Dielectric Tube (1)  0m  E  0m  a n dH  M o d e s  From equation 3.4, the characteristic equation of Eg modes m  can be written •  ^2 0 P22 _ r3 l 3 2 ^ r3 P TJJ^J - p KQCP^OT " 2 (__!__) A  (  )  e  K  22  as p  (p  )  e  32  p  Al  2l n  0- At cutoff P3 0, d equation Al is satisfied if A Q ( P =  32  a n  2  2 2  = 0.  )  It can be shown by applying small-argument approximations to the lefthand side of Al, that p process.  + 0 is an unacceptable solution to the limiting  22  Now A (p ) =Q^22^ 4 ' 2 V 22^ J  Q  p  + a  /  a  p  s a n db y 1 e t t i n  22  in equation 3.8, it can be proven that the condition  9 P-ji +  0  AQ(P  2  2  )  = 0 for  EQ^  modes corresponds to 0 (P ) /e P 0  Y  22  0 22 (p  rl  21  J (P ) " 2e Q  21  J-,(P )\  r2  \ r l 21 0 21 " r2 l 2 1 A  }  F  P  Y  (p  )  2e  Y  ^  21  (p  )  0 m  Similarly, it can be shown that the cutoff condition for given by equation A2 with e . replaced by HE  (ii)  ]m  modes is  .  Modes From the theory of Bessel functions, it is readily proven that  /A (P )B (p ) n  22  n  /Vl( 22>Yl( 22\ n p  22  ^P A (p )B (p )/ 22  n  22  n  p  B  p*, Vp2 A (P )B (p ) / p^  22  2  n  22  n  22  n-l 22 n 22>  B  2  (p  (p  }  A  n-l 22 (p  )  "yP2? A3:,.a  and  K  n( 3 ) 32 n 32 p  p  K  (p  , ./ n-1 32J n l P32 ' 4 Po K (p„) 32 '32'V 32' 2  2  K  (p  |(  (  )  +  )  2  p  9  K  Multiplying out equation 3.2.a and using equations A3 yields  A3.b  57 j ^ / V l ^  Y2V  +  p  n-1 22 \  A  (p  \ n 2 2 " n P22  3  B  2  ( p  r  A  (  / n-1 22 n-1 P22>  }  A  /  }  (p  )B  (  VP2 n P22 n 22 '\ A  (  )B  (p  )  2  / n-l P32 n l P32^ K  Y3V3  (  )K  {  +  P32 n 32 K  K  +  nlP2 (  +  / '  (p  )  / n-1 P22 \, / n-1 22 Y2 r3lp A (p )j r3 r2lp B (p 7 A  }  3  p?S?  3 2  1  2 k o \P  22  32  r2 r2 4 P  n  e  y  22  For small p  B  (p  )N  -f-(c y +e y )  p  22  22  n  22  r2  /  r3  r3  22  -n  2 P>  +  }  e  22  e 1  (  y  n U y  r3 r3 4 P32 y  + e y ) 2 2 22 32  r 2  r 2  P  r3  r3  A4  P  32  n+l 32 „ 2n_ P32 n P32 " p K  (p  K  (p ) 0K ^32' P32 l 32  )  (  )  K  a n d  2  V  Q  32  K  32  (p  In  )  1 0.89 p. 32  A5  Applying these approximations, letting n=l and multiplying throughout by p  yields  32  /B (p )  J L p  32 r2 r2 p e  £r2  0  y  0  2  22  +  £r3  A  22  " l 22 . A  (p  r2  Q  ( P  2  2  ) B  0  ( ' P  2  2  )  \  vP A (p )B (p )/ 22  1  22  1  22  22  B  2  (y  (o-S^Pop \0.89 p ) j  M  0  P 2 1 ^22^  P  r3 r3  }  ' A  y B (p )  22 1^ 22^  P  Q  W^?  3  y A (p ) r3  A (p )'  2 2  r2" r3 r2- r3 y  32  )(£  e  )  A6  '22  From this equation, it can be seen that as p the right-hand side becomes infinite, and p A-|(p ),'a zero of i ^22^' B  22  o r m u s t  "i  t s e  22  32  approaches zero,  must approach a zero of  ^ approach zero. Hence, the  cutoff condition for the HE-,-, mode is given by P  22  =0  A7  r2  58 If A-j(p ) 0> then from equations 2.2 a^/a = - y ^ — J " <  A8.a  Ji(P ) Similarly, if B-,(p ) = 0, then b^/b = y — ^ — j = a^/a  A8.b  =  -  22  2  22  22  2  2  Hence, at cutoff for all n=l modes, A-|(p ) = B^(p ) = 0. By letting 22  P  0 and p-ji  32  22  0 in equation 3.3, it can be shown that equations A8  are satisfied if J (P 2) 1  2  J  ]( 21^ p  A9  which is the cutoff condition for HE^ modes, m>l (iii)  EH-| Modes m  By ..applying a similar limiting procedure on equation 3.2.b as was used on equation 3.2.a, the following result is obtained ^B (p ) ^(p ) r2M^p A (p ) /' rl r2\p (p ) 21 l 1  2 1  e  r e  21  1  21  e  21  y  y  B  21  D  r2 r2 r l r l 2 P y  + e  y  £  rl rl y  .A10  21  But from equations A8, B^(p ^) = A^(p ^), and equation Al0 becomes 2  2  A-,(p ) Ip^T 21  J_ ( r2 r2 rl rl ) ^ r2 • • + r l r••2 ' rl £  + E  y  y  c  21  y  E  P21 rl rl r2 rl 'r^T? e  )  2 ( e  y  y  y  .All  +  Applying equation A8.a yields J  l(P lj 2  V^2]l  J-|(P]) i( 21^ (P ) " ^7^2? Y  p  e  2  22  P21 rl rl  r2 r2 rl rl y  +e  £  y  y  P21 V r2 rl rl r2/ " r 2 r l r l r 2 e  y  +e  y  which is the.cutoff condition for EH^ modes, m > 1. m  2(e  y  +e  y  )  A12 "  59 APPENDIX B Dielectric Tube Field Components in Cartesian Co-ordinates Using equations 3.14 and 2.2, the field components for the dielectric tube in carvesian co-ordinates are given by 0 < r < r. 1  E  xl  = sinez^-j^cosVd-j sine)I (p-,) + (sih e-cos e)(l-d ) -L-  E  yl  = sine  2  2  2  G  .  /o  cose sinpz^g-jO+d,)  1  / 21 (p ) ^ ( ) -— 0  P l  E i = cose cosgz I (p )  rn n  1  ft)  21,  H •, = sine cose singz(£—  H  l E r  < r  2  p  -2 sin^eB ) + (sin^e-cos^,)^-(A (p )+d B (p ))J = -singz|-)(cos e A,(p )-d~-~" '-(p ^~--„ __2, l r  Q  Ey2= -cose sine 0  2  1  I-|(P-J)  2  x2  2  Q  i = sine cosgz  H  Bl .a  = singz^j|(sin e-e1 cos e) I (p ) + (cos e-sin e) (He-,) 2  yl  (P,r  N J 2  2  2  2  0  2  1  x  2  2  1  2  sinez^-j|A (p )+d B (p ) - ^(A (p )-d B (p )) 0  2  2  Q  2  1  2  2  ]  2  E  z2  = cose cosgz A-, (p )  H  x2  = -.cose sine sinez  H  y2  = - singz^sin e B (p )-e cos e A (p )+(cos e-sin e)^-(B (p )+e A (p )^  H  z2  = sine cosgz B-, (p )  2  e A (p )+B (p ) - ^ ( e ^ (p )+B (p ))j 2  Q  2  2  0  2  2  2  Q  2  2  2  2  Q  2  ]  Bl. b  2  2  1  2  2  1  2  60 r  2  -  r  _  sinez^—-(cos e-d sin e)K (p )+(sin e-cos e)(l+d )K (p )' 2  "x3  E  y3  2  2  3  0  2  3  3  = -sine cose singz||-J(l+d ) ^K (p ) + 3  Q  1  3  2K (p ) 1  3  3  P3 E  z3  H  = c o s e  c o s g z  K  i(P3) IB  x3  2K  /  1  = -sine cose sinez^j(l+e ) ^ ( p ) 3  Q  3  +  (p )\ j  Bl.c  l(P )' o 2 2 2 Hy = sin&z^—W-(sin e-e cos e)K (p ) + (cos e-sin e)(l+e ) P3 K  10 > /  3  3  H  z3  3  3  3  = sine cos3z K-j (p ) 3  , . A where d. .llo)— Z0 i = yn. c i\B k  7  and  Q  A r  p. = h.r  ,  A ri o e. l=— c.Z 8 E  k  B2  l 0  B3  •  PART II SCREENED SURFACE WAVEGUIDES  62 7. INTRODUCTION in 1965 Barlow suggested a technique for screening the Goubau surface25 wave transmission line  . The: proposed waveguide consisted of a coaxial cable  with thin dielectric layers on both conductors.  The prediction was made that  the dual surface wave supported by this structure would have a lower attenuation than that of the coaxial cable with no loading. In a later paper , Barlow completed the exact analysis for quasi-TM modes, and discovered that one of these modes had no lower cutoff frequency.  Calculations also revealed that,  for a fixed frequency, it was possible to obtain a minimum of attenuation at some combination of thickness of the dielectric layers, this minimum being significantly lower than the attenuation of the bare coaxial cable. The validity of Barlow's technique of obtaining numerical data has since been 27 questioned by Millington whose calculated results show a very gradual increase in attenuation in the case of the lined stripline. Subsequent experimental 28 29 30 31 results ' ' ' on lined coaxial cables apparently indicated that a minimum in attenuation existed for loading on the inner, outer, or both conductors. 32 In a recent paper , Barlow outlines the approximations used in obtaining his numerical results. Barlow was not, however, the first to consider slow-wave modes propagating in such structures. The lossless characteristic equations for TM 33 modes in these three coaxial cables had been derived earlier by Prache , but no numerical results were given. The screened Goubau line had been studied by Kaplunov ' 34  35,36  , and by Yoshida ' 37  38  as early as 1952.  Both derived the  expression for the attenuation coefficient by perturbation for the TM modes, 35 and Kaplunov has given some design data In this work, a perturbation analysis is given for these configurations for TM, TE and hybrid slow-wave modes. It is shown that the analysis is easily extended to include five additional waveguides which have been studied  63 previously to various degrees of d e t a i l by other workers.  These structures are  the shielded and unshielded d i e l e c t r i c rod, the d i e l e c t r i c rod in a lined c i r c u l a r waveguide, the d i e l e c t r i c - c o a t e d conductor and the c i r c u l a r waveguide with a d i e l e c t r i c l i n i n g . The previous work on these waveguides w i l l now be outlined.  The  d i e l e c t r i c - c o a t e d conductor is the most common form of surface waveguide and i t has been quite thoroughly studied. Harms  39  in 1907.  The original analysis was carried out by  It was not u n t i l 1950 that Goubau  40  demonstrated that the  device had potential as a p r a c t i c a l transmission l i n e .  He undertook extensive  41 experimental work  to determine the effects of bends and weather conditions 42 upon the propagation c h a r a c t e r i s t i c s , and published a design chart v a l i d for small phase-velocity reductions.  Extensive experimental work followed by  43 44 others  '  to v e r i f y the c h a r a c t e r i s t i c s of the TM^ mode.  approximations  24 45 ' for large phase-velocity reductions.  Other workers gave  In the present work,  a design chart is given which has no r e s t r i c t i o n s imposed upon the phase 46 velocity  24 47 .  Semenov  '  has analysed the TMQ^ mode exactly, but, in order to  solve the equations, he made approximations for thin coatings, for thick coatings and for low frequency.  The c h a r a c t e r i s t i c equation f o r hybrid modes 48 49 50 has been obtained by several authors ' ' . The theory of bends in the 51 surface-wave l i n e has been developed by Suzuki , and experimental v e r i f i c a t i o n 52 has recently been obtained  .  The d i e l e c t r i c rod has been studied intensively because of  its  usefulness both as a waveguide and as an antenna. The c h a r a c t e r i s t i c equation 53 was obtained by Hondros and Debye in 1910, and i t s attenuation constant was 54 derived much l a t e r by Elsasser . Experimental confirmation of his results 55 was simultaneously given by Chandler . Measurements on lower-order TM, TE 56 and hybrid modes followed in 1952 by Horton and McKinney . Detailed theoretical 19 21 57 studies of higher-order hybrid modes were l a t e r published by several authors ' ' 58 The theoretical aspects of bends have recently been investigated by Bohme .  64 The case of a dielectric rod in a waveguide has been considered in 21 59 detail by Clarricoats and his colleagues ' , both for fast and slow-wave modes. Their work includes the derivation of the characteristic equat'ons for all modes and of the attenuation coefficient for all hybrid modes. Extensive 33 60 numerical results are given in these publications. Other workers  '  have  given results for TM and T E modes in this structure. The case of a dielectric rod inserted in a circular waveguide with a dielectric lining has not been mentioned in the literature. The problem of propagating slow waves in a dielectric lined circular waveguide has received only limited attention by a few authors. Beam and Wachowski^ have derived and solved the characteristic equations for TM and TE modes, while Loshakov  has derived cutoff conditions for the TMQ-, mode and  has given additional numerical data. The general case of a coaxial cable with dielectric linings on both conductors is analysed in Chapters 8-11.  The lossless characteristic  equations are derived, and expressions for the attenuation coefficient and group velocity are obtained for all slow-wave modes. The results are extended to seven other configurations as shown in Chapter 8. In Chapter 12, numerical results are given for four of the surface waveguides considered.  65 8.  FORMULATION OF THE PROBLEM  The most general form of surface waveguide considered in this work consists of a coaxial cable with dielectric linings on both conductors as shown in figure 8.1.  Media 1 and 5 are conductors, media 2 and 4 are  FIGURE 8.1  The Dual Surface Waveguide  dielectric layers with relative permittivities  and e ^, respectively, and  medium 3 is a dielectric region of relative permittivity e^, such that r2 r3 e > e E  and  >  r4  e  8.1  r3  Additional surface waveguides can be obtained from this configuration by removing one or more of the five media shown in figure 8.1.  The eight  structures shown in figure 8.2 are some of the possible configurations, with the common characteristic of having at least one interface between two dielectric media of unequal permittivities. These cases have been chosen because the radial dependence of the z-components of electric and magnetic fields can be determined by inspection. This is possible when the following conditions are satisfied by the dielectric media:  (1) (2) (3) (4) (5) (6) (7) (8)  FIGURE 8.2 Types of Surface Waveguides Coaxial cable with two dielectric linings Coaxial cable with dielectric lining on inner conductor Dielectric-lined conductor Coaxial cable with dielectric lining on outer conductor Dielectric rod in a lined circular waveguide Dielectric rod in a circular waveguide Dielectric rod Lined circular waveguide  67 (i) extension to r=0 of medium 3 in case 8, and of medium 2 in cases 5, 6, 7; or (ii) extension to r=°° of medium 3 in cases 3, 7; or (iii) adjacency to a conducting boundary of medium 4 in cases 1, 4, 5, 8, and of medium 2 in cases 1, 2, 3, and of medium 3 in cases 2, 4, 6; or  (iv) existence of some radius within the region at which a conducting  boundary could be placed (i.e. tangential electric fields are assumed equal to zero at r=rm in cases 1 and 5). The analysis of each of the eight waveguides may be carried out independently as a separate boundary-value problem. However, it is also possible to obtain the solutions for waveguides 2-8 from that of 1 by using the step(s) shown schematically in figure 8.3, which involves allowing a radius to assume a limiting value. In addition, certain cases may be obtained from others which are less simple by using the same procedure, e.g. case 5 reduces to case 8 by letting r =0 in the solution for case 5. o  FIGURE 8.3 Steps for Obtaining Solutions for Cases 2-8 from Case 1  68 9. FIELD COMPONENTS In this chapter, the field components of hybrid, TM and TE modes are given for the eight configurations shown in figure 8.2. Lossless propagation is assumed in the z-direction, with t-e-z dependence of theorm p  exp j(wt - ne - gz). The radial variation of the z-components of electric and magnetic fields are described by Bessel functions of the first and second kinds in media 2 and 4, and by modified Bessel functions of the first and second kinds in medium 3. The remaining field components are found using Maxwell's curl equations. Omitting the factor exp j(wt - ne - gz), these are given by z2 2 n 2  E  = a  A  (h  r)  ft -Jfr 2 V 2 > 1  E  r2  =  a  h  r  nv  k_ 1. V? n n T2~~ 2 h r k  Z  b  2  r2 o o ng f — a A (h r) + j h r y  •02  2  n  k  Z  2  b B (h r) 2  n  2  2  z2 2  H  \  = b  r-, < r < r  ^  2  ne „ k r2  'e2  z n r Q  2  o 2  t  h r 2  9.1 .a  69 z3 3 n 3  E  = a  C  (h  i  j  T3  r)  a C (h r) + 3  n  riy T  o k Z %.2 °- b, "3D„(h,r) n "3' h r 0  v  3  3  7^ 3 h r  W  a  "63  - J -  3  H  ^ 3 n< 3 >  r  b  3  D  h  r  <r< r  z3 = 3 n<V> b  D  ne k r ° 3 n< 3 > Zo h, 3 r  3  r3  V3  a  h  r  j E f r 3 >3 ' 3 o  'e3  2 a  z4 4 n 4  E  C  = a  F  (h  C  r  +  J t 3 °n< 3 b  r)  .9.1 .b  3 „( 3 ' h r  +  b  D  h  r  3  r)  1  R  n  T4  V l nn k  Z  4  h  2 4 n< V h r  •64  h  3  a  F  +  *  r  "TT"  5  4  b  4  H  z4 4 n<V> = b  r4  r  G  3 ?-4 rK  r  .0 4  h„ 4 10  '64  4 n "4 v  ..2 . r.„4 n 4 b  G  (h  r)  9.1 .c  where, from the wave equation 2 2 2 . 2 r2r2 o " h  = y  e  k  B  -B 11 k 3 " r3 r3 0 ? ? ? h = y - e« k - B 4 r4 r4 0 2  2  2  e  B  A  p  e  9.2  70 The functions A (h,,r), B (h r), F (h^'r) d G^h^r) are linear combinations a n  n  n  2  of Bessel functions which are chosen in such a way that the following conditions are satisfied: (i) fields are finite at r=0 (ii) E , E zi  Qi  and H  ri  vanish at r=r-,, r=r and r=r^. m  Also C (fur) and D (h^r) are linear combinations of modified Bessel functions satisfying the same requirements. These functions for the eight waveguides can be determined by inspection, and are listed in Table 9.1, where Pi and  V  =  p.. = h^.  MEDIUM GUIDE E NUMBER NUMBER 2  9.3  A  zi  DESCRIBING FUNCTIONS  1,2,3  zi  DESCRIBING FUNCTIONS  " ? \ V.P> 2  n21 r  5,6,7  2  B (p )=J (p )=A (p )  A (p H (p ) n  2  n  n  2  2  n  2  n  2  n P3m  K  3  H  1,5  W=W  (  }  W  -rrpfr  K(Po/i) n  2,6 3,7 K (p ,)  4  KJPo-,) n<P > n< 3> " ? \  D  =K  P  3  VP^  3  8  W  c (p )=i (p ) n  3  n  3  1  4  1,4,5, 8  F„(P )=J (P ) 4  N  4  -  Y  ^ y  v (P ) N  4  G  N  ( P  4  H „ ( P  4  )  -  ?  VP  TABLE 9.1 Functions Describing E . and H . 3  zi  z1  4  V„(P )  4  4  4 4  >  Upon setting n=0 (no e-variation), equations 9.1 separate into the two independent sets of circularly symmetric modes designated TMQ and TEg . The m  m  corresponding field components are givtn by TM MODES  TE MODES  z2 2 V 2 )  E  E  = a  r2  =  . V  h  z2 2 0 2  r  H  ' h^ 2 0 2 j  a  A  (h  r)  H  _ r2 Z B r2 E  H  a  EJ 3 0 3  j  a  C  (h  r)  r3. o Z B r3  =  b  '83  z4  T4  = a F (h r) 4  V3  ^  j  a  H  4  • h^ 4 0 4 e  H84  Q  F  (h  r4 o Z B r4 Q  b  r  2 - - 3  3  r  r3  G 4  k  oo  ( 4 h  0  r  .. 9.4.b  r3  r )  r4 oo k  r  Z  r  y  "84  . 9.4.a  r2  r4  k  F  =  b  z 4  r)  Z  Q  "63  o  k  b D (h 3)  y  F  r2 b o  3  r-, < r < r^  B  z3  !  E  r)  r2 "J ?V 2 0V>  "82  z3 ~3  :  (h  y  Q  T3  B  k  0  :  = b  Z  Hr4  3  4  < r < r  .... 9.4.C  If n/0, equations 9.1 describe inseparable combinations of TE and TM modes which are designated hybrid modes. The hybrid modes may be of the EH or HE type and a mode designation could be devised for each of the waveguides under consideration.  72 10. CHARACTERISTIC EQUATIONS The characteristic equations are obtained by matching the field components at r=r and r=r,'and eliminating the arbitrary constants from 2  3  the resulting equations.  Matching tangential field components at r=r  2  using equations 9.1 yields four homogeneous equations in the four'unknowns a , b , a and b^, one of which is arbitrary. These are expressed in the 2  2  3  following convenient form *2 n 32 3 ' C  (p  )  a  b b  2  D (p n  3 2  )  3 " A< 32> p  :  P  •  A (p )  r2  n  y ^ y  2 2  e  22  p  i  r3  3 2  C (p ) n  c (p n  neZ,  1 2, P /J  32  3 2  )  '22  32  -1 ' _r2 n 22 P B  v  22  (p  } +  M  n 32 p r j ^ T D  32  (p  }  1  nB  2 P  k L Z oo  22  1 +  2, P /J 32  b3 a3 b2 A . 3 2 3 0  a  0  a  Q  10.1  b  In a similar manner, matching tangential fields at r=r , one obtains 3  73  l± _ V 3 3 p  a  }  3 " n P43 F  (  )  V^S*  4  ' n P 3^ fr4 n^43 * n 33 > 43 ri 43 C  (  F  3  D  (p  p  }  F  (p  C  + )  P  C  (  }  (  }  G  (p  J  D  1  (p  ^3 ^3 ^4 34 3 a  a  A  G  {  P  ]  D  2 43 J  +  yp  p  }  1  +  p  1  2 33  lp  • / n P33 M n 43 1V3 n 33 43 n H3 33 n 33 C  ngZ,  fr3 n P33 33 n 33^  )  (p  0 0  }  1  > 2 ^33  +  2 43  P  10.2  -JC/  b  Equating the two expressions for c from equations 10.1, and those for 2  from equations 10.2, yields the following two equations 1  1  r2 V 2 2 1 22 V ^  e  p  1  . r3 n 32 32 ^3?'  }  e  p  C  (p  }  p  1  r2 , S3 n 32 P 2 ¥ ^ 2 ? P32 n P32 D  +  D  (p  (  )  2  )  1 1 2 2 k0 , ^p p 22  32  10.3 1 e  H  n^A3  F  ] +  43 y P i ?  p  2  1  ^33  fr3  33 n P33  P  C  T  T  M n P43 P43 n 43 G  ]  G  (  (p  }  v +  }  _r3 n 33 P33 ^33 D  (p  }  D  ]  k L 0  1  2  P33  1  2.  +  P  43  J  10.4 Cases 1 and 5 require the simultaneous solution of both equations for 8 and r  m<  However, r does not exist for the remaining cases and only equation m  10.3 is required for cases 2, 3, 6 and 7, while equation 10.4 is applicable to cases 4 and 8. In all cases, the relationships among the various wavenumbers are given by equations 9.2. Upon setting n=0, each of equations 10.3 and 10.4 separates into two factors, each of which equals zero. These correspond to TM and TE modes,  74 The equations for TM modes are given by  r2 V 22 22 V 22  r3 o( 32  e  P  e  p  P  P  C  p  C^TP  3 2  r4 0 43 + J £ l V 33 P„ _43 ^ 4 3 '33cTp~~ "0 33 e  F  (P  P  p  =0  10.5  =0  10.6  32  V,  and for TE modes \ P  B  Z  2  Q  ( P  2  %(P 22  2  _r4 0 43 43 " 0 43 G  P  (p  G  (P  v  2  r+  r3 0 32  P  D  3  (P  D ^  2  3  10.7 2  r3 0 33 33 ^ 3 3  v  p  D  (P  0  10.8  For TM modes, solutions in cases 2, 3, 6 and 7 are obtained by using equation 10.5, while equation 10.6 is applicable to cases 4 and 8. The simultaneous solution of both equations is necessary in cases 1 and 5. A similar situation exists when dealing with equations 10.7 and 10.8 for TE modes.  11. GROUP VELOCITY AND ATTENUATION COEFFICIENT In this chapter, expressions for group velocity and attenuation coefficient are derived by employing the same techniques as were used in Part I. The phase coefficients for hybrid, TM and TE modes can be found by solving the characteristic equations obtained in Chapter 10 for 8, from which the phase velocity is readily found. 11.1  Group Velocity Expressions for group velocity of all slow-wave modes are found by  integrating the appropriate combinations of the lossless field components to obtain the power flow and energy storage per unit length, which determines the rate of transport of energy given by 4 v = N/W = I N./ 9  1 i = 2  4 I W. , i=2  ....11.1  1  where N^ andto\are the power carried in medium i and the energy storage per unit length in medium i, respectively.  These quantities are given by  r. "ri  en  ei  n  and r  W. 4 1 l  —)  \  \Z v Z y o  i \  o  r  {l . E * + l . E * + l . n  zi  zi  n  ri  0i  E *)r dr, i = l,2,3 ..11.3 er  '  '  '  i -1  For hybrid modes, by substituting equations 9.1 into 11.2 and 11.3, one obtains  _  N  A l / n 32 \  a  C  (p  e  }  a a Tr 3  N  3  '•T— C S  +  r2 o •Z o k  e  SA +r2 v„ok„BZ o^o 2 c s + 2nc (e + y n  + y  K  »r3 o o 3 0 k  eZ  S  c  +  c  2 n c  rt  9  R  3 ^  2  e r2  k r2  o AB )s  "r3 r3 o CD  +  k  E  )S  3  .  a  4  3 3 / n 3S^  "  1,  a  C  (p  S  *JIF*(P„> 4 n 43 u  VK  r2 3 3 / n 32 ^ 4Z h \ \A (p2 )i  e  W 2  a  a  ir  C  (p  W =  2  M o o 4 G k  2A  h  T  ^ A  +  S  +  hT +BS 2  a  a  4Z  h  V  S  4^  + 2 n C  M r4 o FG  +  e  k  11  )S  (y  r2 o o 2 B k  Z  C  )2s  +  4nc  2 r2 o o AB y  k  6Z  S  2  C+  u  C  r3  0  (p  4  4  c  (p k Z c ) S + 4nc y k 6Z S  2  c  „ _ r4 3 3 / n 33^ h T H S " o 4 o ^ 43^ e  BZ  2  r-r o3 o  3  +  )  2  o  F  J  p  p  +  0  3  D  3  r3  0  o  CD  ( k Z c ) S + 4nc k BZ S M  o  o  4  G  4M  0  0  FG  P  11. The integrals S^, Sg, S^, S^, Sp, Sg, S^g, S^, Spg, T^, T^ and Tp are functions of Bessel functions which are defined by 22  p  ' 22 ' '2,L ..+. . n 2 \ _ , C .2 2 n 2> 2 —PT^-J 2' A J V 2> 2 2 p  9  n A  S = 2  A  (p  (p  }  p  dp  T  T =  p  p  dp  21  21  p  J  S = SAME AS S WITH A REPLACED BY B B  A  ^22 ^AB  ^ ( n(P ) n 2 n 2> n 2>) 2 A  B  2  21  p  (p  )+ A  (p  B  (p  dp  11.6.  / ,  3 3  s  c  =  2  \ (n C  2  C (pA  n  2  (P )P3  +  3  2  3 3  V  — p j ^ - j P> d  3  2  ^  3i?  =  D  SAME  ^  'CD p  3  J  S  P  =  S  c  WITH C  REPLACED  ( W W  BY  3  3  D  C> )D (p )jdp  +  3  2  3  A;  > )P dP 3  p  p  S  C  n  3  .11.6.b  3  2  44  2 2  J  2  T  F=  44  2  ^ J  4  F  n  ( p  4  ) p  4  d  P4  3  Sg = SAME AS S WITH F REPLACED BY G F  44  FG  f  ( n( 4) >4) F  p  G  +  ¥  ^^Wj  d  P  .11.6.c 4  P43  All of the integrals defined by equations 11.6 can be evaluated in closed form. The limits of integration for the individual structures are determined from figures 8.2 and 8.3. These are summarized in Table 1 1 . 1 , where an asterisk shown in a certain case denotes the absence of a region in that case. CASE  LIMITS OF INTEGRATION P  1  22  P2  p  21  •P22  P2  p  P  P2  2  P  3  21  P  21 *  4  5 6 7 8 TABLE 11.1  p  21  0 0 0  p  P  22  22 *  3  3 3  3  3  P2  22  P2  P22  P2  P4  22  *  *  *  p  4S  P44  P  43 *  00  P P3  P  P44 *  4  3  44  4S *  p 3 3  p  P  3  l  p 3  p  P4S  3  3  P2  33  3  3  P  44 * *  00  3  0  P  P  33  4  3  P44  Limits of Integration for the Eight Waveguides  78 All of the integrals defined by equations 11.6 vanish at r=0 at r=°°, and in addition, at metallic boundaries, certain functions contained in the evaluated integrals also vanish. These can be determined by inspection and result in obtaining the simplified integrals listed it; Tables 11.2 and 11.3 for the eight waveguides. CASES  EVALUATED INTEGRALS  1,2,3  T =P  5,6,7  T  1,2,3  S  A ; (p ) + (p -n )A (p )- p  2  A  2  2  2  [  22  2 2 A 22 n ^22^ A  B 22 n 22 = p  B  2(p  2  A^p^)  2  22  21  2 2 2 ^22"" ^ n^ 22^  1  = P  2  22  +  A  } +  P  ^ Z Z ^ l ^  +  2p  22 n 22 n 22 B  (p  )B  (p  }  - (p -n )B (p ) 2  2  21  21  5,6,7  S  1,2,3  S = T + 2p  1,4, 5,8  B A A = S  = T  A  T  22 n 22 n 22  +  2p  A  A  22  (p  )A  (p  }  A (p )A^(p ) n  22  p  F  2  p  p  +  S = (p^ -n )G (p ) - (p G^p^) 2  G  2p  44  =  3  2 " n 22 A  (p  )  - n 22 n 22 A  (p  )B  (p  ^ " ^ n ^ ) ( p ^ G  +  2  ^ )  43 n<P43 n P43| G  F F ' 43 = T  AB  2  4  +  S  2  =  S  22  F - 44 n ( 44) " ( 43  AB  S  2p  )G  (  ^ s K ^  FG  S  =  F  n 4 (p  )G 3  n 4 (p  TABLE 11.2 Evaluated Integrals for Media 2 and 4  } 3  )  79 • CASES  EVALUATED INTEGRALS  1,5  T T  =  c  ( p ^ n  C = "PL  ) ^ )  2  >34>  c  " PL  -  ((P 2 2  i <P33>  c  +n2  -  2  ) n(P32)  faW^l^  "  P ^ C ^ )  " P32 ' t f ^ )  C  3,7 4  T  8  T  C = ^P 3  1,5  S  D  C  ^ 3  =  2  S  + n 2  S  D  =  4  S  D  =  D  =  8  S  S  1,5 1,5  S  CD  2,6  S  4  S  C C  =  ' l ^  ^  i  V  W  2  ^  2  2  2  +n2  ^  +  P 3  2  v  3  ) n(P32) D  +  2  ^  K  ^  P 2 n<P32> n <P 2>) D  D  3  3  -(p 4 n )D (P3 ) 2  +  2  2  3  S  n  S  4  -(P32 n (P32) * (P32  +n2  C  ) C  2  C  T  2  2  " C  =  T  D  (  C  (  +  2  T  C  +  2 p  +  2  C  P 2  }  S  +  2  n  33  32  2p  3 3  TABLE 11.3  C  n  ( p  32  r  n^ 32^  C  D  S  p  (  D  CD  - S P33  =  C  (  J  2p3 C (-p3 )C^p ) 2  n  2  3 2  WtzK^  +  ) C  n  ( p  32  )  C^PggJC^Cpgg)  Evaluated Integrals  The equations f o r  =  3  C  C (p )C ;(p33) -  CD  2>)  3  P 3 n P33> r>33>  3  3 3  ^32^P  3  P33 n ( P 3 ) > 3 3 >  D  " C "  (  ~ ^ ^ K ^  ^  + 2p  C  C  " n P33> n<P33>  = -T  ^ n(P32^  " C " P32 n P 3 2 n P 3 2  =  P33 V  =  C  ^ - ^  D  C  " P33 n <P33>  (P32 n ( P 3 2 ^ - ( P 3 2  = -T  C  - ^ 3  C  2  D  3,7  K ^  2  ) n(P33)  2  =  D  n  p3 K  =  2,6  +  S  CD  =  S  CD  =  C  n^32J n P32 D  (  )  - n P33^n(P33^ C  (  f o r Medium 3  and W. f o r TE and TM modes (n=0) are obtained  by using equations 9.4 f o r the f i e l d components in equations 11.2 and 11.3.  y This y i e l d s f o r TE modes  80 y N  r2 o o 3 3Y 0 32 V k  "  2  BZ  2h  b  b  D  (p  )  Vo^ZzV  2  B  D  2h. *  2  ^r4 o o 3 3 / 0 3 J vk  11.7  2h, y .e .k N. ri n o i  11.8  For TM modes, the corresponding relations are V2 o 3 a^TT 3 /C (p )' pu  2Z  4  11 2 W  0  h  (p  c  2Z  r  11.9  \TT,—rl F \ 6(P4 V  „ ,4 o4  =  32  W) 22>  o2  _ W V W 4 2Z h? o 3  3  J  N  u  h  F  3  r23 3 / 0 32 \ ,,2. 2 - , „,4 ['Z, J 2 A * S ) o 2 o \ 0 P 2>/  £  a  a  1T  C  (p  )  /  =  r + +  Q  u  A  2Z  h  v  A  (  2  £  •  w J =  •j W  4  =  r\  3 Q 3 Q 7T  r\  ^ (h^ T + 3 S ) 2Zoh^ 3v o  r d  J  J  £  ur  L r  r4 3 V 0 33 \, 2 . Z - . 4 H> ( /i F V a  a  C  (p  )  3  n  u  T  T  +  P  Q  11.10  o 4%\ 0<P43>/  2Z  h  F  11.2 Attenuation Coefficient The attenuation coefficients of the eight waveguides are found by using the perturbation method employed in Part I. The attenuation  81 coefficient of any mode is given by a =  5  P 2N  4  (1/2) J P. / J N.,i=l ' i=2  where the dielectric power loss Q  1=2,3,4  i  The expressions for  11.12  per unit length is given by  P. = k v tan6 W. , o  •  , Vl^, w" and  11.13  are given in Section 11.1.  3  The power  loss per unit length in the i ^ conducting medium is given by -2TT  Pi =0/4)R ^ (H . H*. + H . H*.)r I dG i  0  "r-| if i = l r^ if i=5  ft=l or 5 and j is the dielectric medium number bounded by ^  11.14  or r^, respectively  r  and R.  =  /(k l )/[2a.)  11.15  Q o  Substituting equations 9.1 into 11.14 yields for hybrid modes  Pi  1  . i M v f f c ] |( | c  h  sV) 2  +  B  Vl(P h  2h?  P  Z2  ( P 2 i  ,  +  o  21  2nc eA e i 0  /  CASES 1,2,3  Tl^-VP2l' n<P l> o ^21 B  2  2 r  P, =  l l 3 3 2h R  a  a  Tr  c  2  3 3 • — Pn^Sl) +  h  1  +  ^31  o  2nc e k 3  3 r3 o C (Po-i)D ( P O T ) -0 31 0  0  CASE 4  p  r R a a3,/C (p 3)' 2  4  P  5  5  3  3  2 L 2 , V \ 2 , 4 4 ~2 J n 44 P44  2h  4  \n<P43>> F  h  2nc  +  G  4 r4 o o 44 £  Z  p  k  (p  / r4 o n(P44 T . E  C  C  }+  w  B  F (P 4 n 44 )G  n  4  (p  )  k  F  )N  CASES 1,4,5,8  82 and r R a a TT 4  5  3  'z0k C (p VK 0 „) y  2 L2 , n 8* \ .2,  r3 o n 34  2  3  2h  2  _  \ 2nc  P  /  34  3 r3 o n(P^/l) n(P^) Z0 p-. 34 "n 34 rT 34' e  k  f  C  vp  y  D  CASES 2,6  K  ..11.11  The corresponding equations for TE modes are obtained from equations 9.4 and 11.14. /B 0 ^ B  p  i  ' i W a * o(p 2Hi  =  D  3  0  r  P  5=  l l R  b  (p )/ 22  CASE 4  V 0 P31 D  3  CASES 1,2,3  (  }  r R b b , D (p ) 4 VG (p )v  CASES 1,4,5,8  r R b b , D (p )  CASES 2,6  4  5  3  3  0  33  0  43  2  4  5  3  3  34  11.17  and for TM modes * / r2 o 0 32 " 0 21 P = r ^ a ^ "n(Poo) o2 V22' e  ]  k  C  Z  h  (p  )  A  (p  )  A  CASES 1,2,3  2 * / r3 o 0 31 rRaa Zoh 3 e  P = 1  P  5  =  1  1  3  k  C  (p  )>  CASE 4  3lT  0  r  * 4 5 3 3* R  a  a  / r4 o 0 P o4 e  k  Z  C  (  )F 33  h  * / r3 o 0 34 r R a a TT 4  5  3  3  k  C  {p  o 3  (p  ?  *'  e  0 44 0^?I  )  )  CASES 1,4,5,8  2  CASES 2,6  11.18  83  12.  C H A R A C T E R I S T I C S OF PARTICULAR STRUCTURES  In this chapter, numerical results are given for four of the eight waveguides described previously. These are the Goubau line (case 3 ) , a.nd the three configurations of the dielectric-lined coaxial cables (cases 1, 2 and 4). Calculations were made on the Goubau line with no restrictions on its radial dimensions, and the results are displayed in the form of design charts. Detailed calculations were also carried out on the three coaxial waveguides studied by B a r l o w ' 25  26,28  '  29530  ' ' , since they, like case 3 , 31  32  have no lower cutoff frequency when operated in the  TMQ-J  mode. Accurate  evaluation of a and B was made in order to ascertain Barlow's prediction of a reduced attenuation below that of the unloaded coaxial cable. Results for the unshielded and shielded dielectric rod, cases 7 and 6, are not given, since these structures have been studied in detail by 21 Clarricoats  . Also, no numerical results are given for the remaining two  waveguides, cases 5 and 8 , because all modes in these two structures have a lower cutoff frequency. 12.1  Goubau's Surface-wave Transmission Line  12.1.1 Design Charts The object of the presentation of design data is to provide a graphical solution for a and B of the Goubau line. The chart given here is applicable in the case of polyethylene-coated copper conductors in free space (e 2 2.26, =  r  e  =i ,o,' tane^O, a-| = 5 . 8 • 10° mhos/cm), for arbitrary values  of tane^. It displays four parameters: 2 ^ l ' r  r  A / 2 r  i' P  reduction, Av , and a line impedance, Z, defined by Z = 2N/II*,  erce,rf:  phase-velocity  p  12.1  where I is the total peak conduction current. The evaluation of Z requires  * the determination of a^a^, which appears in the expression for N, in terms of  84  I. This is achieved by applying Ampere's circuital law at r=r-j, where  is  given in equations 9.4.a. Thus, -j2Tr k a p A (p ) £r2  I=  o  2  21  0  21  12.2 h  2. o Z  from which a can be determined in terms of I. Using equations 1 0 . 1 , which 2  relate a and a^, one obtains 2  h Z A (p ) 2  a  o  0  22  12.3  II  33 a  2 rc k r A (p )C ( )_ 1  r2  o  1  0  21  0  P32  Substituting in equation 1 2 . 1 from 1 2 . 3 and 1 1 . 9 and making use of the relation  z=  CQ.(P  r  ez.  3 2  )  ^ 0 ^ 3 2 ^ '  =  ^  R  2 / 2  4nk  'rl  _  r  0  l \  A  2  (  P 2 1  )  A  2 o^ 22^  r3  :  A  p  \r Ao(p ) 1  9.1  E  0  2 1  2  (  P  2 1  1  2 2  0  P 2 2  )  K (p ) 0  > the line impedance becomes  2A (p )A (p )  22  2r r  A  A Q (p )  2 2  0  " ^ ^  M  '2  r /A (p ) 2  O  A  0  2  (  Q  P  2 1  2 2  )  2K (p )  32  Q  32  12.4 i  K  0  (  P  32  }  3 2  P  K  0  (  P  3 2  )  The percentage phase-velocity reduction of the^line is given by  Av = ioo(e-k)/e p  12.5  o  Substituting in 1 0 . 5 from Table 9 . 1 , and performing the differentiations, the characteristic equation for TM modes becomes ¥ 22> p  _r_2 22  i ^ j j h ^  + r3 l|P32 P K (P ) £  K  }  =  p  32  J  0  (  P  2 2  }  - ( Y ^ T J  Y  0  (  P 2 2  0  12.6  0  32  )  It can be proven that the parameters e , e n, — and ^ — uniquely define Z 0  from equation 12.4, and AVp from equations 12.6 and 12.5.  Fixing e ^ and e ^ , 2  a design chart can be constructed by plotting r /r-j against Z, first for 2  constant AVp, and then for constant X/2r-j.  Figure 12.1.a shows such a chart  3  85  0  50  100  150  200  250 300 350 400 450 Z, ohms FIGURE 12.1.a Surface-wave Transmission Line Design Chart  with the following ranges of parameters: Av = 0 to 33% and x/2r-j = 1 to 10 . 4  p  figure 12.1.a for r  / r 2  ] f  r0IT1  500  Z = 0 to 500 ohms, r /r-| = 1 to 100, 2  An expansion of that portion of  0 to 5 arid Z from 0 to 120 ohms is shown in  figure 12.1.b, which provides better accuracy for lines operated at millimeterwave f requenc i es.  86  20  40  60 80 100 120 Z, ohms FIGURE 12.1.b Surface-wave Transmission Line Design Chart for the Millimeter-wave Region  87 The design charts shown in figures 12.1 can be used to determine the phase characteristics of the lossless line. The attenuation of the line is calculated by considering losses in the conductor and in the dielectric layer. The attenuation coefficient due to conductor losses, a-,, is given by a-, = (1/2) P /N = P-,/(ZII )  12.7  1  5 Using equations '"1.18 and 12.3, and letting a-j = 5.8 • 10 mhos/cm for copper yields a-, = 95.17/(r /F Z)  dB/100 ft ,  1  12.8  r-j and X in centimeters, and Z is obtained from figures 12.1. The attenuation coefficient due to dielectric losses, a , is given by a =(1/2) P/N = P /(ZII*) 2  2  2  12.9  2  It is convenient to normalize a to the attenuation coefficient of a TEM wave 2  in medium 2, ao, where a = (l/2)/r7 k tan6 12.10 o r2 o 2 / a , and using equations 12.9, 12.10, 12.3, 11.13 and 11.9, one 0  w  Letting  a - a  2  Q  obtains 2 / 0^ 22^ r \A (p ) r  M/i"^)  3  k z jf (  ] +  2  A  P  2  h) 2  2  Q  21  A  0 ^ 22^ A (p ) 2  23 r  P  2  Q  21  h  /  2  /A (p )A (p ); 0  22  0  22  2 21 l ^ 0 ^ 21^ P  r  A  P  12.11 It can also be shown that a is uniquely defined by A'V  and Z. A chart of a  against A V with Z as parameter is shown in figure 12.2. and 1 2 . 1 1 a  2  = (}/zVe~^  k tan6  2  a  = 4168  f tan6  2  a  From equations 12.10  dB/100  ft  12.12  5  where f is expressed in GHz. The total attenuation is given by a =  95J7_  + 4 1 6 8f t a n  r,/lZ  -  dB/100 2  ft  12.13  88  FIGURE 12.2 Surface-wave Transmission Line Dielectric Attenuation Characteristics  89 12.1.2 Accuracy of the Design Charis 40 can be derived by The approximate relations obtained by GoubaiT u  using small-argument approximations for the Bessel functions pertaining to medium 2 in equations 12.4 and 12.11. Th?se are given by . BZ A 2irk0  /K ( 2  e  1 r2  2  £r3  P32  \ 0 32 K  (p  2K (p )  ) }  0  32  12.14  P32 o' 32 K  D  )  and /8.Z„ tan6 & o 2 47re k Z A  0  r2  In  12.15  o  The approximate attenuation coefficient of the surface-wave transmission line is then given by a  g 5  a-j  + a  1 7  =  2  /l66,347 8 f tan6 \ ,r \ dB/100 ft In 2 ~ E o k Z r2 o 2  2  12.16  r-| ,x in centimeters, f in GHz. It can be shown that P  22  2TT  r 2  1 (1-0.01 Av )  1/2 12.17  2  p  from which it can be seen that p  22  is small if r^/x or AVp, or both, are small.  Thus equations 12.14, 12.15 and 12.16 are quite accurate for small phase-velocity reductions or for cases where the radius of the guide is much smaller than a wavelength. The cases of very thin dielectric coatings, i.e. r /r-| - 1, result 2  in small phase-velocity reductions and the two conditions required for equation 12.16 to be valid are r < 2  or  < X  r * ^ 2  12.18  A particular case will now be considered to ascertain the accuracy of Goubau's relations for the attenuation coefficients. The parameters used are  90 f = 5.906 GHz, E = 2.26, e r2  r3  = 1 .0,  tan5 = 0.0005, tan6 = 0.0, 2  3  o = 5.80 • 10 mhos/cm, 5  1  r-j = 0.0254 cm and r varies from 0.0 to 2.0 cm 2  The variation of a, a, a-], a-j, a and 'a^ 2  a s a  function of dielectric thickness  is shown in figure 12.3.b, from which it can be seen that Goubau's equations • are quite accurate for very thin dielectric linings.  91 12.1.3 Comparison with Experimental Results The results given in this section, which are obtained from the pertur bation theory, can be-shown to agree better with published experimental data than those obtained from Goubau's approximate equations. The results chosen for the purpose of comparison are given by Schiebe et. al., because they are well documented. In their experiments, the attenuation coefficients of several surface-wave transmission lines were obtained by measuring the guide wavelength and Q of the resonant waveguides, at a frequency of 9.375 GHz.  The attenuation coefficient was calculated from  the approximate formula 12:19 where Q is the unloaded quality factor of the resonator after correcting for end-plate losses. The comparison of the two sets of results is shown in Table 12.1. The "corrected measured" values were obtained by using a more accurate relation than equation 12.19, ir  v  -P-—  a = H  12 20*  g g  It may be noted that Schiebe's calculated results were obtained by using Goubau's approximate semi-graphical method given in reference 42.  * This result has been derived for the dielectric tube waveguide by Mr. T. Bourk as part of a Masters Thesis project at the University of British Columbia, Electrical Engineering Department. In Appendix E, it is shown that equation 12.20 is a general result valid for any waveguide, regardless of the number of homogeneous dielectric media.  • PARAMETERS  2.26  2.26  2.10  2.10  2.10  0.0003  0.0003  0.0003  0.0C03  0.0003  r., (cm)  0.1295  0.04015  0.0455  0.0705  0.1295  r (cm)  0.1660  0.1500  0.1490  0.2365 '  0.4165  f (GHz)  9.375  9.375  9.375  9.375  9.375  £  r2  tan6  2  2  ORIGINAL RESULTS (reference 44)  Q meas.  8790.  2750.  3030.  3470.  X (cm) c a l c . g  3.12  2.82  2.88  2.79  2.61  X (cm) meas. g a(dB/100 f t ) c a l c .  3.12  2.82  2.92  2.85  2.79  3,17  12.32  9.72  9.29  11.87  a(dB/100 f t ) meas.  3.03  11.55  10.35  9.61  8.59  % diff. i n a MODIFIED RESULTS  2550.  v /v c a l c . g P X (cm) c a l c . g a(dB/100 f t ) c a l c . a(dB/100 f t ) corrected measured % diff. i na  + 4.6  + 6.7  - 6,1  - 3.4  + 26.5  0.991  0.952  0.960  0.942  0.910  3.12  2.82  2.88  2.81  2.70  3.16  11.93  10.39  9.94  10.44  3.06  12.12  10.79  10.20  9.44  + 3.2  - 1.6  - 3.9  TABLE 12,1 Measured C h a r a c t e r i s t i c s of the Goubau Line  - 2.6  + 9.6  93  12.2 Dielectric-lined Coaxial Cables In this section, the propagation coefficients are calculated for the TMQ-J  mode in a coaxial cable with a dielectric lining on the inner, outer or  both conductors. The propagation characteristics obtained are compared with those of other workers ' ' ' ' 26  27  28  29  35537  .  The attenuation coefficients of two cables with only one dielectric lining were calculated using the perturbation method. The relations used are developed in Appendix C from the analysis given in Chapter 11 . The results show a gradual increase in attenuation as the thicknesses of the dielectric 27 linings are increased. This behaviour is similar to that found by Millington" in the case of the loaded stripline, and it is also supported by previous 35 37 work '  in which the case of a lined inner conductor was analysed.  Millington's comments regarding the accuracy of computations in reference 26 appear to be equally valid in the coaxial case. To check the validity of the perturbation method, the exact characteristic equations, given in Appendix D, were solved to a high degree of accuracy using an IBM 7044 computer. In these equations, the fields in the outer conductor were described by Hankel functions of the second kind, instead of the Bessel functions used in reference 26, since the fields must decay with increasing radius. No approximations were made when solving these equations; the large and small-argument approximations, used in reference 26 to replace the complex-argument Bessel functions, were avoided by employing accurate computer subroutines to generate these functions. The results for thin dielectric linings are given in Table 12.2 from which it can be seen that the attenuation coefficients obtained by the two methods differ by less than 0.1%. This ascertains the validity and establishes the accuracy of the perturbation method, which has the additional advantage of requiring less than ten percent of the computing time taken for the solution of the exact equations to the same degree of accuracy.  Dielectric thickness r  2~ l' r  c  a Np/m  m  3 rad/m  EXACT RESULTS a Np/m  3 raJ/m  0.01  0.0095956  63.227306  0.0096041  63.236745  0.02  0.0098491  63.608605  0.0098577  63.618141  0.03  0.0100988  63.977169  0.0101076  63.986799  0.04  0.0103452  64.334212  0.0103543  64.343936  0.05  0.0105888  64.680779  0.0105980  64.690595  0.06  0.0108298  65.017776  0.0108391  65.027682  0.07  0.0110686  65.345995  0.0110781  65.355990  0.08  0.0113055  65.666131  0.0113151  65.676213  0.09  0.0115406  65.978798  0.0115504  65.988965  0.10  0.0117743  66.284539  0.0117842  66.294792  0.11  0.0120067  66.583844  0.0120i68  66.594181  0.12  0.0122379  66.877145  0.0122482  66.887568  0.13  0.0124682  67.164837  0.0124787  67.175342  0.14  0.0126976  67.447271  0.0127082  67.457858  Dielectric thickness r  PERTURBATION RESULTS  4" 3' r  c m  PERTURBATION RESULTS a Np/m  8 rad/m  EXACT RESULTS a Np/m  8 rad/m  0.1  0.0094505  63.096932  0.0094588  63.106277  0.2  0.0095842  63.387834  0.0095924  63.397186  0.3  0.0097426  63.707592  0.0097508  63.716955  0.4  0.0099297  64.059459  0.0099379  64.068833  0.5  0.0101500  64.446891  0.0101582  64.456278  0.6  0.0104082  64.873532  0.0104163  64.882930  0.7  0.0107092  65.343178  0.0107172  65.352587  0.8  0.0110579  65.859742  0.0110659  65.869160  0.9  0.0114589  66.427222  0.0114669  66.436645  1.0  0.0119160  67.049687  0.0119240  67.059110  TABLE 12.2  Comparison of Propagation Coefficients Obtained from Perturbation Theory and from Exact Theory for Coaxial Cables with One Dielectric Lining e =1, e ,=e „=2.26, r =0.157 cm, r,=2.5cm, f=3.0 GHz, r3 ' r4 r2 '1 '4 ' ' tanS =0.0, tan6.=tan6 =0.0005, a =0=1/1 ' 10 mhos/cm  95 The computations using the perturbation method were extended for increasing dielectric thickness until the coaxial cables were completely filled.  The results are shown in figure 12.4, where the conductor, dielectric  and total attenuation coefficients are plotted against dielectric thickness on the inner or the outer conductor.  The end-points of the curves were  obtained from the well-known equations for the attenuation coefficient of the • quasi-TEM mode in a coaxial cable. These fall in line with the intermediate points obtained from the  TMQ-J  surface-wave mode analysis.  The parameters of the cavity used in the experimental work reported 32 in references 28 and 29, have been given in a most recent paper , and these differ only slightly from the parameters used in obtaining the characteristics given in Table 12.2 and figure 12.4.  Computations using these new parameters  give attenuation characteristics which are very similar to those shown in figure 12.4. Similar calculations were carried out in the case of thin dielectric linings on both conductors, using both the exact and the perturba-. tion methods. These results are shown in Table 12.3, from which it can be seen that the accuracy obtained is comparable with that found in the previous cases, and that the attenuation characteristics exhibit the same general behaviour.  96 0.040 •0.0376 i l i  f-  0.032  i l i i i  4->  C QJ  o 0.024  •  /  QJ O O  /  sor * 0.016 c  — a  y  1  ^ S  1 1  1 1  •I—  / a  1^5^-^"  f 0,0140  «-*  #-  QJ +J +J  .-»*  a l 5 a  0.0093 0.003 a  + a  4  -  1.2 0.8 Dielectric thickness, 2~ \  0.4  T  r  o r  1 .6 4~ 3'  r  r  2.0 c m  FIGURE 12.4 Attenuation Characteristics of Dielectric-lined Coaxial Cables f=3.0 GHz r2 r4 ' r3 r-j=0.157 cm r=2.5 cm tan<5=tan<5=0.0005 , tan6=0.0 e  =£  =2  26  4  2  4  3  o-j=a=l/7 • 10 mhos/cm lining on inner conductor — l i n i n g ' on outer conductor 6  5  2.343  97  PERTURBATION RESULTS  D i e l e c t r i c thicknesses  ~ i>  EXACT RESULTS  c m  r . - r . cm 4 3  0.001123  0.003538  0.0169096  209.891927  0.0169165  209.908721  0.001544  0.003110  0.0169492  209.984919  0.0169561  210.001727  0.001948  0.002474  0.0169841  210.064185  0.0169910  210.081007  0.002298  0.002105  0.0170169  210.140735  0.0170238  210.157569  0.002597  0.001765  0.0170446  210.204980  0.0170515  210.221826  0.002913  0.001439  0.0170744  210.274270  0.0170813  210.291127  0.003097  0.001176  0.0170907  210.311392  0.0170977  210.328256  0.003298  0.000948  0.0171094  210.354507  0.0171163  210.371377  0.003500  0.000702  0.0171279  210.397074  0.0171348  210.413951  0.003655  0.000508  0.0171421  210.429490  0.0171490  210.446373  0.01123  0.03538  0.0185315  214.043534  0.0185388  214.060792  0.01544  0.03110  0.0189336  214.931926  0.0189411  214.949332  0.01948  0.02474  0.0192863  215.665516  0.0192940  215.683060  0.02298  0.02105  0.0196278  216.383928  0.0196356  216.401595  0.02597  0.01765  0.0199188  216.981286  0.0199267  216.999059  0.02913  0.01439  0.0202354  217.624527  0.0202434  217.642415  0.03097  0.01176  0.0204096  217.962262  0.0204179  217.980215  0.03298  0.00948  0.0206111  218.358138  0.0206194  218.376166  0.03500  0,00702  0.0208124  218.746546  0.0208208  218.764646  0.03655  0.00508  0.0209673  219.041201  0.0209757  219.059361  TABLE 12.3  Comparison of P r o p a g a t i o n C o e f f i c i e n t s Obtained from P e r t u r b a t i o n Theory and from Exact Theory f o r a C o a x i a l C a b l e w i t h Two D i e l e c t r i c Linings  T  2  v  a  Np/m  8 rad/m  a  Np/m  e =1.0, e ,=e =2.5, r =0.13 cm, r.=0.80 cm, f=10.0 GHz, r3 r 4 r2 1 4 o  0  tan6 =0.0, t a n 6 = t a n 6 =0.0004, o =a =6*10  5  mhos/cm  8 rad/m  98 APPENDIX C ATTENUATION COEFFICIENTS OF DIELECTRIC-LINED COAXIAL CABLES BY THE PERTURBATION METHOD dsing equations 11.9, 11.10, 11.13 and 11.18, and Tables 11.2 and 11.3, and normalizing the arbitrary constant a^a^ to the peak current in the inner conductor as in Section 12.1.1, the power flow and power loss per unit length in the various media of the coaxial cable with two dielectric linings can be obtained.  i ez. 2  N  ^P A (p ) 22  0  N  \ ( P  22  2 ~ 8ire..„k r2 o \p A (p ) 0  21  2A (p )  )  2  0  vA (p )  K  21  2  /  Q  22  p A (p )  22  22  Q  P21 0 P21 A  + 1  (  }  P A (p )  22  22  0  22  /  CI .a i ez. 2  33 0^ 3 ^  P  C  P  8  "r3 o \p Ao<P ) k  21  21  \ P  /  3  C  2  0  ( P  3  )  2  )  C  (p  33  8lTt  0  21  32  P  2 0  (p  Q  )  3 2  3 2  /P 4 Q(P44 F  )  32  32  c (p Q  2 )>  \P F (p ) 43  0  43  33  C  (p  + 1 )  32  4  y  Q  }  c  3  0  P33 0 33  (p  2C (p )  k  21  0 33  33  C (p )  /P22 0 P22 P 0 3 ' -r4 o \p A (p )p C (p ) (  2C (p )  C  /  0  A  C (p ) Q  3  /  3 2  CI .b  + •1  )  'F^(p )  2F (p )  ,F (p )  P F (P )  43  0  43  Q  43  43  0  43  Cl.c  1  C2.a  4Trr  n  I2h22 Zotan62 Q  8lTe  0  r2 o k  /  P  2  VP  2  A  2 1  0  ( P  2  A (P 0  2  ^ (p ) + l T2 A  ) ^  2 1  0  1_VA  V  22  ( P  Q  2  21 0 2i ,P A (p ) P  A  22  (p  0  22  r  2  )  2  23 A (p ) Q  22  P h A (p V 2  y  22  0  22  C2.b  99 I h Z tan6 / P Ap(P ) 2  2  /  o  3  22  8-rre k  r3 o  33 0^ 33^  P  22  ^P A (p )  0  21  0  C  P  C  0 33 (p  }  •C (Poo) 2  2]  k  P  c (p  3 2  )  'c (p  3 2  )  3 2  Q  2  n  C/(P„) '0 32 VK  2 2B  33 0^ 33^  P  ^ P  C  3  2  C  0  ( P  3  2  0 33  C  P  (p  C  }  0  ( P  )  yp  •P33 0 33 C  (p  }  3 2  c (p 0  C2.c 2h2Z tan5  f  u  4  o  4  8-rre „k  r4 o  /P  2 2  AQ(P  ^P  2 1  A (P 0  2 2  2 ]  )P33C (P3 ) 0  2  ,p  3  )P3 C (P 0  3 2  1 +  )  44 0 44 F  (p  )  v 43 0 43 > p  F  ' F  (p  2  .F  0  ^  F  }  ( P  4  3  (P  )  4 3  + 1  )  Q( 4 ) p  C2.d  3  ^ 4 43 0 43 h  P  F  (p  )y  I R /P A (P )P3 C (P3 )p F (p ) 4TT r / 4 \P ( 21 32 0 32 43 0 43 , 5  22  0  A  22  p  3  )p  0  C  3  (p  )p  44  F  0  (p  44  C2.e  )  21 0  where the functions A , C and F are defined in Table 9.1  The attenuation  coefficient is given by  j i=l where  and  j  j  a. =0/2) P. / ' i=l ' i=2  C3  1  are given by equations,,CI and C2.  For the case of the lined inner conductor only, P and N do not 4  4  exist, and Pp P and N are given by equations C2.a, C2.b and CI.a, 2  2  respectively, while N , P and Pj. are given by 3  3  3  2  )  i  3 2  )/ J  TOO I BZ 8lT i r3 o 2  /p Ao(p )'  0  2 2  k  \P  2 1  /P 4 0 34 C  2 2  AQ(P  C^(p )  )  2C (p )  32  L\P Co(p )7  )/  2 1  (p  0  32  +1  3  32  \ C P32 2(  32  }  0  C4 22 3 r22"0 I h Z tan6, /Po A '22' (p ) * r3 o \p (p ) 1  _ P 4 0 34 \ /  M  3  8  Q  9  e  k  n  C  ^p  A  21  0  (p  _0 P 2  )  C  3  09  21  f  c (p  H  Q  3 2  3 2  c (p 0  3 2  (  }  3  )/  c  (P  Q  + 3 2  I  )  )  C5.a  ^32 3 0 32^ h  P r  5  =  C  (p  I R /p A (p )p C (p )^ 5  22  0  22  34  0  34  C5.b  4-irr 4 \P A (p )p C (p ) 21  0  21  32  0  32  The attenuation coefficient of the cable is given by C6  a =(l/2)(P + P + P + P )/(N + N ) 1  2  3  5  2  3  where P-j, P , P » P^, N and N are defined by equations C2.a, C2.b, C5.a, 2  3  2  3  C5.b, CI.a and C4, respectively. For the case of the lined outer conductor only, P and N do not 2  2  exist, and P^ is given by equation C2.a, while N , N^, P , P and P^ are 3  3  4  given by  i ez. 2  ^sV^S^  r3 o  1  _ \ P  1  C  0  ( P  3  1  o /P33 0 P33 r4 o \P C (P V  3Z  87re  3  C  (  )N  )  0  C  )  3  C  /  0 33 (p  2 r/P44 0 44 F  k  31  / QtP 3 ,  2  >  31  4 3  (p  F (p 0  4 3  )N  ),  2C +  0 33 (P  ) +  C7.a  r  }  2  ^F (p ) 0  2 0  43  (p ) 43  2F (p ) Q  43  + 1  P F (P ) 43  0  43  C7.b I h Z tan6 /p C (p ? 2  2  Q  Po =  8  3  -r3 o k  33  0  33  VP C'(P3 ). 31  3  ^33L_,) Vc/( )  +  ^p c (P y 31  P 3 3  v  0  21  31  P C (p )y 33  0  33  C8.a  101 Ih?Z^tan6„ 4o 4 2  ^44 p(P44 F  8ire ,k  r4 o  iJ  )  L\P43 0 43 >  h  F  (p  )  ' 0- 43 F  -'2  (p  )  + 1  0  ^ V 43> p  C8.b  P 3 4 0 43 h  F  (p  )y  4  IR 4-rrr, 4 2  33 0^ 33^ 44 0^ 44^  P  r  C  P  P  F  P  C8.c  \P C (P )P43F (P 3), 3 1  0  3 1  0  4  The attenuation coefficient of the cable is given by a =(1/2)(P + P + P + P )/(N + N ) , 1  3  4  5  3  4  where P-j, P , P , P^, N and N are given by equations C2.a, C8.a, C8.b, 3  4  3  4  C8.c, C7.a and C7.b, respectively.  C9  APPENDI/ D THE EXACT SOLUTION OF TM SURFACE-WAVE MODES IN DIELECTRIC-LINED COAXIAL CABLES Omitting the common factor exp(jwt - yz), the exact expressions for the field components of. case T for the quasi-TM surface wave modes in a coaxial cable with dielectric linings on both conductors are given by E  = C^^r)  2l  0 < r < r-j  rl  h7 l l l  ^1  °1 J' o o , "rl  C  J  ( h  +  k  r )  Dl .a  /Z  C (J (h r) + F Y (h r)  "z2  2  0  2  2  fcC^J^r)  r2  +  Q  2  F Y h r) 2  l(  2  r-j < r < r  2  Dl .b  r  3  Dl .c  r2 o '62 = 0 - ^ p 0 - J'tan5 )E 2  "z3  C  H  3  0  1)(  V) 3 0 ( 3 ) + F  X-C3^ )(h r)  H  1  "r3  3  j !r|_o  '63 "z4 E  c  4 0 4 J  (h  r2  { 1  +  2)  h  ,(2), r) F H)-(h 3  3  . an6 )E jt  r  3  e4  j  Y  (h  r)  - "fp- (1 - Jtan6 )E o 1  <r<r  40 4 i  r) + F  +F  =  2  r3  r4 " S ^ V V * 4 W > )  H  y  4  r4  3 t  r  r  4  < r  '  Dl. d  103 Z5= 5 0 V >  E  E  C  r5  2  H  H  hT 5 o  =  C  i (V) 2)  r  4  <  Dl .c  1  r  (° +J k / Z ) r y r5 5  -65  tf  0  where 2  2  L  - j 'k Z a, 0  2 2 i =Y  e  °  0.1  0  r2 n-Jtan6- ) k  o  2  c  2  r3 (l-Jtan5 ) k 2 0  2  2  e  3  r4 n-J'tan6 ) k2  4  k -j kZ a o oo5 2  J  c  Matching tangential field components at r=r-j and r=r yields the following 2  two equations  2  7 ¥1^11M hT  2  T  U  /Vll'V  (l-jtan6 )\  /Pn^k  ./Pll r2 o  ,  V  k  £  /P z 0-3t™s )\  ( T = 7 I i n 6  2 V  ,  )  /  D3  "  /J (p ) * F Y (p )^ _ /H< (p ) + F H^ (p )^ . .D4 ^p e (l-jtan6 )/ \ j p ) + F Y^p^^ " ^ ^ v 127 (p )/ 22  32  r3  1)  Q  3  r2  2  l(  22  2  22  2  Q  2)  22  32  3  +  32  H  32  Similarly, matching fields at r=r and r=r yields 3  4  ./P54r4 ( -Jtan6 )\ E  k  1  0  4  J (P )  l 44 o V k„/2 >—; 1 P44 p  Z  (  j  J  o o  -i^P(p )N 64  (27  0  (P )/ 54  ) 44  )  0  M  D5 P  Y (P  (  rt  54 r4 G  k  ( 1 0  ^  t a n 6  P74^ V V o T  r  Z  4 )  ) N  y  Yi(P ) 44  104 /P43^(l-JtanS )\ /J (P ) + F, Y f p ) \ /H^ (P ) + F lp tT-Jtan6 )j I j ^ J T + T ^ T f T i ^ / " ^ D ^ ) ^ )  3  33Er4  0  43  Q  33  4 3  g  33  ^(p^)) '  +  4  H^(p )\  The characteristic equation for the cas^ of dielectric linings on both conductors can be found by eliminating the arbitrary constant, F , from equations D4 3  and D6, with the arbitrary constants  and F^ defined by equations D3 and  D5, respectively. If there is a lining on the inner, but not the outer conductor, then the fields in media 3 and 5 must be matched at r^r^. In this case, the following equation results  .(^wK ^ i k y ^ 1  F  1  U  D7  b 4  3 =  The characteristic equation of a coaxial cable with a lined inner conductor is given by equation D4, with the arbitrary constants F and F defined by 2  3  equations D3 and D7, respectively. If there is a lining on the outer, but not the inner conductor, then matching fields in media 1 and 3 at r=r-j yields  " lww)'  (P3l)  J  D8  (P3,)  The characteristic equation of a coaxial cable with a lined outer conductor is given by equation D6, with the arbitrary constants F and F^ given by 3  equations D8 and D5, respectively. Although the exact characteristic equation for the unlined coaxial cable is not solved in this work, it can be easily obtained by equating  105 the two expressions for  given by equations D7 and D8.  From equations D2, it can be seen that  hi  and  '54 are very large. The Bessel and Hankel functions with these arguments are extremely large, in fact, too large to be calculated accurately. However, the ratios of Bessel and Hankel functions with these trguments, which occur in equations D3, D5, D7 and D8, are approximately equal to unity in magnitude. For computational purposes, it is convenient to obtain the asymptotic series for these ratios, which are readily shown to be i  J (p -|) 2p 11 Q  1  1+  64p  D9.a  2 1  and  H'0n (P 54') (  2, vp  KA  1 2p54 + i 1 + "~ \ "^54/  D9.b  r  Only the first three terms of each series is required, since for the conductivities and high frequencies considered.  1 3 P54  1 3 < 10" Pll  106 APPENDIX E  THE Q OF RESONANT WAVEGUIDES  The q u a l i t y factor of a resorant low-loss waveguide is given by v k W Q =  :  2P P  El  p +  where Pp = power loss in one end-plate, = power loss in the waveguide  P and  W = energy storage in the resonator.  The l a t t e r two quantities are related to the corresponding quantities of the terminated waveguide by P = 2LP = 4aLN W = 2LW = 2 L N / v  ,'  g  E2  where v^ is the v e l o c i t y of energy propagation, which is equal to the group velocity.  Using equations E l and E 2 , the following expression f o r 1/Q can be  derived: ,  1 x  P„ P  =  • • 2av  P  +  P_ +  9_ =  n  o o o o  o o  2av  n  9_  E3  p  The f i r s t term in equation E3 becomes smaller as L increases, and i t s contribution to 1/Q can be made negligible by assuming a very long resonator. case, the Q of the waveguide is given by Q  "  •  from which equation 12.20 is obtained.  • • -  In this  107 CONCLUSIONS The author considers the main contributions of this work to the field of surface waveguides to be: I. The dielectric tube waveguide 1. A mode designation based upon the properties of the field components has been shown to be consistent with a previously given designation for the dielectric rod, and the distinctions between HE and EH modes have been observed. 2. Three-dimensional field configurations, calculated and plotted by computer, have been obtained for the HE-j -j and EH^ modes. 3. Expressions for the group velocity of all surface-wave modes on the dielectric tube have been derived. 4. Expressions for the attenuation coefficients of all modes have been derived, where the tube and its surrounding media are not loss-free. 5. Accurately computed numerical results for the propagation characteristics, covering a wide range of parameters, have been given. 6. The proposal has been made that moderately thin-walled dielectric tubes may be used to advantage at the higher frequencies where standard rectangular waveguides are too impractical. II. Screened surface waveguides 7. A unified analysis of eight waveguides has yielded the following new results: (i) Characteristic equations for hybrid modes in all cases but the shielded and unshielded dielectric rod and the Goubau line, which have been derived previously. (ii) Expressions for the group velocity of all modes except TM modes on the Goubau line, which have been given also by other workers, (iii) Expressions for the attenuation coefficient of hybrid modes in  108 all cases but the shielded and unshielded dielectric rod, which have been derived by previous workers. 8. Extensive and more accurate design data for the TMQ-J mode on polyethylenecoated copper conductors in free space, with unrestricted radial dimensions, have been given. 9. In the three cases of lined coaxial cables, accurate numerical results for the TMQ-J mode with no restrictions on lining thickness have shown that the attenuation cannot be reduced below that of the unlined cable.  109 BIBLIOGRAPHY 1. 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