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Characteristics of dielectric-loaded coaxial cables. Lau, Ming Chee 1972

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CHARACTERISTICS OF DIELECTRIC-LOADED COAXIAL CABLES by Ming Chee Lau B.A.Sc, University of Bri t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of L L£ CT K11 A L EriSlMttKlfJG The University of British Columbia Vancouver 8, Canada Date Qfjt 5" ' 1^71 i i ABSTRACT The characteristic's of a l l modes i n the dielectric-loaded coaxial cable have been given. Reduction i n attenuation by dielectric loading i s possible for the dominant mode i n these cables, including -cables with optimum-dimensions,. i f lossless or sufficiently low-loss dielectrics are used. For higher losses i n the loading dielectric, higher frequencies are required to achieve attenuation reduction. No significant reduction i n conductor losses can be achieved with cables;whose dimensions are such that only the dominant mode can be supported. Measurements of the attenuation coefficients of several lower order modes were carried out at S-band frequencies using a resonant cavity. Experimental observations agree favorably with theory for the TM modes. i i i TABLE OP CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS " i v LIST OP TABLES . v LIST OF SYMBOLS . . v i ACKNOWLEDGEMENT i * 1. INTRODUCTION 1 2. DERIVATION OF CHARACTERISTICS 3 2.1 C h a r a c t e r i s t i c Equations 3 2.2 C u t o f f C o n d i t i o n s 6 2.3 T r a n s i t i o n Conditions 10 2.4 A t t e n u a t i o n C o e f f i c i e n t and Group V e l o c i t y 12 3. NUMERICAL RESULTS 17 3.1 Phase C h a r a c t e r i s t i c s 26 3.2 A t t e n u a t i o n C h a r a c t e r i s t i c s . ... •.... 27 4. EXPERIMENTAL INVESTIGATION 30 4.1 I n t r o d u c t i o n 30 4.2 R e l a t i o n Between A t t e n u a t i o n C o e f f i c i e n t and Q F a c t o r .. 30 4.3 R e l a t i o n Between Unloaded Q and Loaded Q 31 4.4 Experimental Measurements 32 4.5 Experimental R e s u l t s 34 5. CONCLUSIONS 38 APPENDIX A FIELD COMPONENTS 39 APPENDIX B DEFINITION OF A FUNCTIONS USED 42 k REFERENCES 43 iv LIST OF ILLUSTRATIONS Figure Page 2.1 Dielectric-loaded Coaxial Cable 3 3.1 Mode Spectrum 17 3.2 Mode Spectrum at 3 GHz 18 3.3 Cutoff Conditions 19 3.4 Cutoff Conditions for Optimum Cable 20 3.5 Transition Conditions 21 3.6 Group Velocity 22 3.7 Normalised Attenuation Coefficients 23 3-8 Normalised Attenuation Coefficients of Optimum Cable .. 23 3.9 Attenuation Coefficient of Dominant Mode i n Optimum Cable 24 3.10 Attenuation Coefficient of Dominant Mode in Nonoptimum Cable 25 4.1 Die-lee-tri-c-loaded Cavity used i n Experiment 33 4.2 Q Factor Measurement Circuit 33 V LIST OP TABLES Table Page 4.1 Experimental Results 35 ^4.2 Experimental Results for lower Conductivity 35 4.3 Comparison of Results for different Loss Tangents .. 36 4.4 Comparison of Results for different Permittivities .. 36 LIST OF SYMBOLS A (p. .), B (p..), - functions of Bessel functions °i = V f i C (p. .), D (p..) = functions of Bessel functions E^., E . , E = longitudinal, radial and azimuthal components of electric f i e l d , respectively, in medium i f = frequency f = resonant frequency f , g^ = constants h. = wavenumber of medium i 1 H^ _. , H^_. , EQ^ = longitudinal, radial and azimuthal components of zi ri magnetic f i e l d , respectively, in medium i I (p..) = modified Bessel function of the f i r s t kind n X i i j ' J (p. .) = Bessel function of the f i r s t kind k Q = phase coefficient of free space k. = wavenumber of medium i 1 K (p. .) = modified Bessel function of the second kind L = length of resonator L (p ), M (p. .) - functions of modified Bessel functions n «n m, n = mode subscripts T$ = power flow in medium i p. = h.r or k.r 1 i i p ~ h.r. or k.r . i j 1 3 1 J P^ = power loss per unit length in medium i P - power loss in each end plate i n medium i ' Q = quality factor of a cavity resonator Q as loaded quality factor V l l P (p. •)> Q. •) = functions of Bessel functions n i,i T L in -J r d = (1/2) . (r2+i>3) • -r. = radius of an interface between two different media 1 R . = surface resistance of conductor of medium i mi R = surface resistance of resonator end plate . m S Q = standing-wave ratio of cavity at resonance SA' SB' SAB' TA' S , S , S , T , = integrals of functions of Bessel functions C D CD L SP' SQ' SPQ' V SL' V SLM' V V V' suv' T u integrals of functions of modified Bessel functions tanS^ = loss tangent of medium i T(f ) =» insertion loss ratio of cavity at resonance U (p. .), V (p .) = functions of modified Bessel functions n * i j n i j v , v , ' V = speed of light i n free space, group velocity and ° ^ ^ phase velocity 1/L = energy storage per unit length in medium i Y (p. .) = Bessel function of the second kind n ^ i j ' z = longitudinal co-ordinate Z Q - impedance of free space a attenuation coefficient of loaded cable a = attenuation coefficient of TEM mode in unloaded cable o 6 = phase coefficient = coupling coefficient function of Bessel functions Af = bandwidth e Q = permittivity of free space relative permittivity of medium azimuthal co-ordinate relative permeability of medium conductivity of medium i angular frequency i'x ACKNOWLEDGEMENT The author i s indebted to his research supervisor Dr. M.M.Z. Kharadly for his guidance and encouragement throughout the course of this project. Grateful acknowledgement is made to the National Research Council of Canada for a Bursary for the year 1970-1971, and for support under grant A-3344, and to the British Columbia Telephone Company for a . scholarship in 1971. The author i s also grateful to Dr. B.Chambers for his guidance during the i n i t i a l stages of the project to.June 1971. The author wishes to thank Mr. C.Chubb for building the resonant cavity and Miss Norma Duggan for typing the manuscript. 1 I. INTRODUCTION The subject of dielectric-loaded coaxial cables has received much attention in the last few years. In 1965 Barlow 1 suggested a tech-nique for screening the Goubau surface wave transmission line, with the resulting structure having lower losses than the unloaded coaxial cable. 2 3 Later Barlow ' completed an approximate analysis for dielectric-lined cables, revealing the poss i b i l i t y of obtaining significant reductions of attenuation at some combinations of dielectric thicknesses. Subsequent experimental observations by Barlow and Sen^'^'^'^'^ on lined coaxial ca-bles indicated attenuation reductions of up to 30% for dielectric linings on either the inner or outer conductor. The validity of Barlow's theore-9 10 11 t i c a l analysis was questioned by Brown and Millington ' ' who showed no such reduction in attenuation using a quasistatic transmission line 12 13 theory. Their conclusion was substantiated by Lewis and Kharadly ' who solved the exact equations involved in the analysis. Further theor-14 e t i c a l work was then carried out by Barlow on a cable structure with a minute air gap between the dielectric lining and the inner conductor, with the results appearing to support the experimental observations, but again approximate expressions were'used. At about the same time Chambers and 15 Kharadly questioned the calculation performed by Barlow and Sen in the conversion from Q factor measurements to attenuation coefficient, and they showed that the correct calculation gave no conclusive evidence of reduc-tion in attenuation. Other workers X^' X^' X^ have shown the possibility of attenuation reduction by dielectric loading, at least in principle, but under different 18 circumstances. In particular, Bach Andersen and Arnbak concluded that a high-permittivity low-loss dielectric shell placed between the conductors 2 may reduce attenuation for cable structures which are not optimum. Em-ploying a perturbation analysis with no restrictions on cable or dielec-19 t r i e tube parameters, Chambers and Kharadly have shown the dominant mode in these cables to be a TM mode, and have derived expressions for this mode. Computations indicated that attenuation reductions of at least 20% are possible with a high permittivity dielectric tube placed between but not adjacent to the conductors. Some results for the loading of optimum 20 cable structures have been given by Albrey and Gunn , showing no reduc-tion in attenuation for the range of parameters considered. Very recent-21 ly, Lewis and Sarkar have carried out a study of the complete mode spec-trum of the loaded cable. Using a numerical optimisation procedure to de-termine optimum dielectric tube dimensions, they show that attenuation re-ductions of up to 75% are attainable in certain highly overmoded cases. From the above review i t can be seen that there have been no ex-perimental results for the dominant mode of the dielectric-tube-loaded coaxial cable, nor a study of the propagation characteristics of higher order modes. The purpose of this work is to provide such an investigation. In Chapter 2 the derivation of the equations for phase co e f f i -cients, cutoff conditions, group velocity and attenuation coefficients i s given. The attenuation coefficients are derived using the perturbation method, and the other characteristics- are derived assuming a lossless struc-ture. These equations are solved numerically for some specific cases of cable and dielectric parameters and the results are given and discussed in Chapter 3. Experimental measurement of the attenuation coefficient of the dominant mode with the cavity resonance method is described in Chapter 4. Conclusions from this investigation are contained in Chapter 5. 3 2. DERIVATION OF CHARACTERISTICS 2.1 Characteristic Equations The configuration of interest is shown in figure 2.1. Media 1 and 5 are conductors. Media 2, 3 and 4 are dielectrics with relative per-mittivity e . and relative permeability u . (i=2,3,4) so that y „e „ > J n v J r i ' r3 r3 u „e „ u .e ,. To obtain the characteristic equations of the cable, the r2 r2, r4 r4 conductors and dielectrics are assumed to be perfect. Assuming lossless propagation in the z-direction, t-0-z depend-ence i s of the form exp j (cot-n0-Bz) . The radial wavenumber of the wave equation of medium i i s 2 2 2 ' k. = y .e .k - B for i = 2,3,4 2.1 l r i n o 2 k_^  is always positive for medium 3, but may be positive or negative for media 2 and 4. The propagating mode may be referred to as a waveguide 2 or fast-wave mode i f k (i=2,4) i s positive, and as a surface-wave or slow-2 wave mode i f k. (i=2,4) is negative. Figure 2.1 Dielectric-loaded Coaxial Cable 4 The axial f i e l d components are linear combinations of Bessel 2 functions. If k. is positive, ordinary Bessel functions J (k.r) and i n l 2 Y (k. r) are used. If k. is negative, modified Bessel functions I (h.r) and n i l n l K (h.r) are used, where h. is defined by n i l J h 2 = ( - l ) 1 ^ 2 - y .e .k2) for i = 2,3,4 l r i r i o 2.2 Defining p. = h.r = k.r and p.. = h.r.= k.r., h. and k. being real, the l l l * i j x 3 l j i ' i 2 2 axial f i e l d components of a l l modes for k^ < 0 and k_^  > 0 respectively are given by E z 2 = f2 VV> = f2 {W " W } n 21 = f 2 A n(k 2r) - f 2 { J n ( p 2 ) - ^ 2 2 l Y n ( p 2 ) } H z 2 = S 2 V h2 r ) = *2 { W' - g 2 B n ( k 2 r ) g 2 U n ( P 2 ) I n ^ 2 1 ) K > 2 1 > j;<p2i>  Y ; ( P 2 I > V ? 2 ) } r L < r * r 2 E z 3 " f 3 P n ( h 3 r ) H z 3 = § 3 Q n ( h 3 r ) 3 n 3 s 3 U n ( p 3 ) r2 * r . * r 3 E = f U (h.r) z4 4 n 4 f4 C n ( k 4 r ) lz4 - H V n ( h 4 r ) = « 4 { W " K ^ T V'*4 Jn (P44 } TOW r3 *~ r- * r4 2.3 5 The transverse f i e l d components are given in Appendix A. The characteristic equations are obtained by matching the tangential f i e l d components at x=r^ and r=r^ and eliminating the arbitrary constants f^, g^, i=2-5 from the resulting equations. Matching tangential f i e l d com-ponents at r=r^ gives f2 Pn ( p32> Pn ( p32> f3 Ln ( p22 ) A n ( p 2 2 ) g2 . ' W Qn ( P32 ) g3 M n ( p 2 2 } B n ( p 2 2 ) 83 P n ( p 3 2 ) V" j V ^ ? ^ 3 ' 3 + E^A7)/(ZoAl> P n ( p 3 2 ) A l Q n ( p 3 2 ) Z0 r 3 5 r 2 8 " j C3 = £3 £3 £g = _ f2 " f 3 f2 g 3 " " 2 where the are functions defined in Appendix B. Matching tangential f i e l d components at r=r^ gives f4 P n ( : p 3 3 ) Pn (- P33 ) 2.4 f3 U n ( p 4 3 ) Cn (' p43 ) H Q n ( P 3 3 ) W g3 " V n ( p 4 3 ) V P 4 3 } 6 83 P n ( p 3 3 ) OTPS < er3 A4 + ^V^OV P n(p 3 3) A 2 Qjt^) ( y r 3 A 6 + yr4 A10 ) 2.5 - j C3 f 4- = f l f l £4 _ f4 " f3 f4 83 " " 4 Equating the two expressions for from equations 2.4 and 2.5 yields the following two equations . ( E r 3 A 3 + er2 A7> < vr 3 A5 + "r2A8> = A l 2 ' 6 ( £ r 3 A 4 + er4 A9> ( y r 3 A 6 + 'WV " A2 2 ' 7 The above equations contain the ratios f^/f^ and g^/g^, which have yet to be determined. By equating the tangential f i e l d components at r=r 2 and r = r3> eigh't linear homogeneous equations can be obtained in the eight un-knowns, f.£>g^ > i=2-5. Applying Cramer's rule to these equations yields T3 = A12 [ A2 ( A11 ~ A12> + A l l ( £ r 3 A 1 3 + er2V <MA16 " y r 3 A 1 4 ) A 2 M l " ( £r3 A14 + £ r 4 A 9 ) {vr3^U ^ r 3 A 1 4 A 1 2 + 1 J r 4 A 1 0 ( A l l " A 1 2 ) } ] t A 2 ( A l l - A12 } + A12 ( Er3 A15 + e r 2 V ( yr3 A16 - y r 3 A 1 4 ) A 2 M l ( er3 A16 + £ r 4 A 9 H y r 3 A 1 6 A l l " 1 Jr3 A14 A12 + y r 4 A 1 0 ( A i r A 1 2 ) } ] 2.8 7 The ratio gij/gg i s obtained from 2.8 by interchanging a l l and e ^ , A ? and A g, Ag and A 1 Q. The above equations are similar to those given by Lewis and Sar-21 kar. The characteristic equation for hybrid modes can be obtained by sub-stituting f^/f-j a n d gi>/§3 i n t o either equation 2.6 or 2.7. Alternately i t may be obtained by setting the determinant of the system of eight l i n -ear homogeneous equations equal to zero as given below C A1 A2 ( A11 " A12 ) + A2 A11 ( Er3 A13 + £ r 2 A 7 } ^ r 3 A 1 6 " " r f V " A l ( er3 A14 + £ r 4 A 9 } { ^ r 3 A 1 6 A l l " \ 3 A14 A12 + V A10 ( A11 " A 1 2 ) } ] x [ ^ r 3 A l A 2 A 1 2 ( A16 - A14> + A2 { " r 2 A 8 ( A11 " V " " r 3 ( A15 A12 " A13 A11> } " { e r 4 A 9 + e r 3 A 1 6 H l i r 2 \ 4 A 8 A 1 0 ( A11 " A12 ) + yr2^r3 A8 ( A16 A11 " A14A12> " yr3 Pr4 A10 ( A15 A12 " A13A11> + 4 ( A13 A11 A16 ~ A15 A14 A12> }] + ^ r 3 A l A 2 A l l <A16 " A14> + A2 {^r2 A8 ( A11 " A12> " "r3<A15A12 " A 1 3 A 1 1 ) } " { £ r 4 A 9 + er3 A14 H^r2^r4 A8 A10 ( A11 " A12 ) + V r 3 A 8 ( A16 A11 " A 14 A12 ) " Vr4 A10 ( A15 A12 " A13 A11 } + 4 ( A13 A11 A16 " A 1 5 A 1 4 A 1 2 ) } ] x [ A lA 2 ( A n - A 1 2) + A 2A 1 2 ( e r 3 A 1 5 + e^Hv^ -" A l ( £r3 A16 + £ r 4 A 9 ) { ^ r 3 A 1 6 A l l " »r 3 A14 A12 + Pr4 A10 ( A11 ~ A12 ) }^ = 0 2.9 The characteristic equations of the circularly symmetric TM 0m and T E ^ modes are obtained by setting n = 0 in equations 2.6 and 2.7, each of which separates into two factors, a l l equal to zero: 8 £r3 A3 + £r2 A7 = 0 2 - 1 0 yr3 A5 + y r 2 A 8 = ° 2 ' 1 1 £ r 3 A 4 + £ r 4 A 9 = ° 2 ' 1 2 ^3 A6 + \ 4 A 1 0 = 0 2 ' 1 3 Equating the two expressions for f^/f-j obtained from equations 2.10 and 2.12 yields the characteristic equation for TM modes T3 " A H ( £r3 A13 + £ r 2 A 7 ) > <£r3A15 + £r2 V = A12 ( £r3 A14 + E r 4 A 9 ) 1 ( £r3 A16 + £ r 4 A 9 } 2 ' 1 4 Equating the two expressions for g^/g-j obtained from equations 2.11 and 2.13 yields the characteristic equation for TE modes g^ = A H ( Pr3 A13 + y r 2 A 8 ) 1 ( Mr3 A15 + "rtV = A12 (^3 A14 + 1 ( Mr3 A16 + 2 ' 1 5 For fast-wave modes, i f the thickness of region 3 tends to zero when regions 2 and 4 are f i l l e d with the same dielectric, the limiting form of the characteristic equation 2.9 for hybrid modes reduces to ( £ r 2 A 7 " £ r 4 A 9 ) ( y r 2 A 8 " yr4 A10 ) = 0 This equation is satisfied when either factor i s zero. Setting the f i r s t and second factors equal to zero and simplifying yields respectively the following two equations. Jn<k2rl> J n ( k 2 r 4 ) Wl> ' V k2 r4> 9 WW WW •WW WW 2.17 Equations 2.16 and 2.17 are the same as the characteristic equations of the TM and TE modes respectively of the homogeneously f i l l e d coaxial nm nm & cable. The limiting forms of the characteristic equations 2.14 and 2.15 for TM and TE modes are equations 2.16 and 2.17 with n = 0, which are i d -entical to the characteristic equations for the TM- and TE„ modes res-n 0m 0m pectively of the homogeneously f i l l e d cable. 2.2 Cutoff Conditions Lossless propagation for any mode in the cable requires a real phase coefficient g. Cutoff occurs when 8 becomes zero. The radial wave-number at cutoff i s given by .2 ,2 k. = u . s . k x rx rx o i = 2,3,4 For hybrid modes (n £ 0), as 8 vanishes, the equations obtained by matching the tangential f i e l d components can be separated into those involving only E -. and those involving only H . . The cutoff condition for b zx b J zx HE nm modes is obtained by eliminating the arbitrary constants g. from the equations involving only H ^ , J n ( p 3 2 ) K 3 J n ( p 3 2 ) ' Jr2 Bn ( p22 )"i Yn( p32 ) LP 32 Jn ( p32 ) p22 B n ( p22 ) j ^r3 Yn ( p32 ) \lW^22) J>33> Yn (P 33> u _J' (p 0 0) p ,D' (p . _) r3 n 33 r4 n 43 P33 Jn ( p33 ) P/.^^P-) P32 Yn ( p32 ) P22 Bn ( p22 ) ^3Y>33) ^r4 Dn ( p43 ) 43 n^43 _ L 33 * 33 PQI y„(PQI> P4 3 Dn (P43 ) -1 -1 2.19 The cutoff condition for EH modes is obtained by eliminating the arbi-nm trary constants .f from the equations involving only E ^ , 10 Y n ( P 3 2 } V ! 3 3 l £ r 3 J n ( P 3 2 ) _ f r Z ^ V P 32 Jn ( p32 )- P22 An ( p22 ) £ r 3 J n ( P 3 3 ) £ r 4 C n ( P 4 3 } >33 Jn^ P33^ P43 Cn ( P43 } _ . L £ . 3 Y : ( p 3 2 ) 32 n 3 2 £ r 3 Y ; ( p 3 3 ) £ r 2 A n ( P 2 2 )  P22 An ( p22 )' £ r 4 C n ( P 4 3 )  P43 Cn ( p43 )' -1 2.20 The cutoff conditions for TM and TE modes are obtained by sub-stituting 2.18 into the characteristic equations 2.14 and 2.15. The cut-off condition for TM modes is similar to 2.20 with n = 0, and that for TE modes is similar to 2.19 with n = 0. 2.3 Transition Conditions The characteristic equation for any mode becomes indeterminate at the points where wave propagation changes form a slow-wave to a fast-wave mode or vice versa. For a cable f i l l e d with the same dielectric in regions 2 and 4, transition occurs when b.^ and h^ equal zero. The trans-i t i o n conditions can be obtained by letting h^ and h^ approach zero and applying small argument approximations to the modified Bessel functions in the characteristic equations. For TM modes, as h^ - h^ -> 0, equation 2.14 reduces to W _ J 0 ( P 3 3 )  Y 0 ( P 3 2 ) " 2.21 This equation is satisfied i f h^ vanishes, which means there is no lower cutoff or transition frequency for the dominant TM surface-wave mode. The equation is also satisfied when approaches r^. Thus as the thickness of region 3 vanishes, the phase coefficient of the domi-nant TM mode becomes equal to that of the TEM mode of the unloaded cable. By applying small argument approximations to equation 2.15, 11 the following t r a n s i t i o n condition f o r TE modes i s obtained r2 2 2 y r 3 J i ( P 3 2 ) + y R 2 P 3 2 | i - J 0 ( P 3 2 ) 2pr3j;cp33) + y R 4 P 3 3 ( l - ^ ) J 0 ( P 3 3 ) 2 y r 3 Y 0 ( p 3 2 ) + P r 2 P 3 2 I1" ^) V p32> 2"r3 Y0 ( p33> + W 3 3 I1" ^ ) Y 0 ^ 3 3 ) \ r2 V r 3 2.22 For hybrid modes n = 1, applying small argument approximations to equation 2.6 and the r a t i o s f ^ / f ^ , g^/g 3 a n d taking l i m i t s y i e l d s the conditions 2 2 / r2 " r l "r2 er3 A3- L 2 ... 2 J + y r 3 £ r 2 " 5 [ 2 r2 + r l 2 j . 2 A. & 1 1 r — r 2 1 y „e + u „e „ rz r2 r j r j '32 . 4 r2 4r, In — y 0 e « / 1 r- \ r2 r2 / -, , 1 4 4 r2 " r i 2.23 J 1 ( P 3 3 )  Y 1 ( P 3 3 ) V ! r 4 + P r 3 £ r 3 / A ^ ^ l . ^ 33 11 12' *r3 r2\- 2 2 v 16 14' 11 denominator same as numerator except A 12 -y 2 2 :3 ~ r4 r4\ 2 2 r 3 + r 4 ' , f l , r4 l n r4  £ r 4 + 4 + 7 1 O (r: 2 2, f + £ r 3 A 1 4 2 ( r 3 ~ r p ^ ( A i rA 1 2 > denominator same as numerator except 16 2 2 r 3 + r 4 'r4\ 2 2 r3" r4 v r 4 ( A l l " A L 2 ) 41 3 X 1 2, 2 . 2-r 4 l n Th + i V V V ' 2 272 ( r 3 + r 4 ) + e r 3 ( A l l A 1 6 " A 1 2 A l 4 ) denominator same as numerator 2.24 12 g^/g^ i s obtained from 2.24 by interchanging a l l u and er^> inter-changing the factors in curly brackets, and inverting a l l ratios of squares of r a d i i . 2.4 Attenuation Coefficient and Group Velocity The group velocity of a given mode in any waveguide is defined as doi/d3. The group velocity may be found by determining the rate of trans-port of energy, which for a lossless waveguide i s given by 4 4 v = E N. / E W. 2.25 8 i=2 1 i=2 1 where is the power flow in the ith medium, and the energy storage per unit length in the it h medium. The losses i n the cable are calculated assuming a f i n i t e con-ductivity (i=l> 5) for- the conductors, and a loss tangent, tan (i=2, 3, 4) for the dielectrics. The attenuation coefficients are derived 13 by the procedure used by Lewis and Kharadly , which has proved to be very accurate for low-loss loaded coaxial lines. It has the advantage of not requiring the evaluation of complex-argument Bessel functions. The attenuation coefficient of any mode is given by 5 4 a = (1/2) E P./ E N. 2.26 i=l 1 i=2 1 where P^ is the power loss per unit length in the it h medium. The quan-t i t i e s ftL, W\ and P^ are given by r. N i - i r f * ^ ^ - ^ H y r d r , i - 2,3,4 2.27 J r i - 1 r. 'W. = ne .e f 1 (E .E*. + E .E*. + E„.E* ) r dr, i=2,3,4 2.28 i n o zx zx r i rx 6x 0x' J r i - l 13 P. = to W. tan <S. , i = 2,3,4 1 i x P. = (1/2) R . x mi 2ir (H .H*. + H..H* ) r d9 2.29 2.30 r=r 1 i f i=l, j=2 /wu .y where R . = — — mx \ 2a. r^ i f i=5, j=4 i = 1, 5 For hybrid slow-wave modes, substituting the f i e l d components from Appendix A into equations 2.27 through 2.30 and integrating yields 2 h 2 4 Ln<P"22>" Z ° ST + y 0k gZ C 2 S M + 2nC9(g2+y e 0 k 2 ) S T M ] L r2 o o 2 M 2 r2 r2 o LMJ N3 = *~Zri c f £T^ SP + "r3koBZoC3 SQ + 2 n C 3 ^  + "rfSsO SPQ 3 2h„ o N, = Trf f* P'(p„) e .k & 0 : —^-f- " [ ? S T T + y ,k (3Z C,S„ + 2nC, (fT+y ,e ,k*)S T N,] „, 4 2. . Z U r4 o o 4 V 4 r4 r4 o UVJ 2 h4 U n ( p 4 3 ) ° • TTE £ f f* P (p ) r2 o 3 3 n u 32; 2 2 h. 4 ,2 L n ( p 2 2 ) [tit T '+ 0 2S T+(y _k Z C„)2S +4nC„y _k BZ S ] 2 L L r2 o o 2 M 2 r 2 o o LM 7re 0 E f„f* r3_o_3_3 [ h2 T p + ^ + ( ^ z ^ ) 2 S Q + A n C ^ ^ Z ^ ] 2 h. "r^ oVS Pn(p33) r, .2 2 h, T T2, ^ th4 TU + + ( ^ 4 k o Z o C 4 ) 2 s v + 4 n V r 4 k o B Z o SUV ] Un(p43) irr.R , f , f * P (p- 0) 0 0 2.2 • /e 0 k L ' ( p 0 1 ) P, = 1 ^ 3 3 ._^__32_. [ c2 ( h2 + n j ^ M 2 ( ? 2 i ) + r2 o n 21 Ln ( p22 ) '21 2nC e „k 6 o 21 14 R f f* P 2(p ) 2.2 0 /e ,k U' ( p . . ) \ 2 P 5 - — - T — — rc4(h4 + _ ) V (p )+( V U n ( p 4 3 ) P44 1 ° ' 2nC, e. ,k g Z p., ° U ^ p 4 4 ) V n ( p 4 4 ^ 2 ' 3 1 o *44 The expressions for TE slow-wave modes are given below * g 3 s j - y r 2 y z o Q ^ P 3 2 ) N 2 = — r n > -JL,—r SM . 2 h 2 M 0(p 2 2) • . ^ 3 M k o g Z o q N y r 4 k o B Z o Q^P33> Q 4 _ 2 ^ / V 2 ( p43> V 4 0 . V . e . k T 7 r i r i o „ W. = N. 1 « 1 P v o 2 M 0 ( P 2 1 ) P l ° ' r l Rml g3 g3 Q 0 ( P 3 2 ) 7I~— . M Q ( P 2 2 ) P5 = 11 r4 Rm5 g 3 g ^ Q^ P33 ) 77 A 4 , 2 ' 3 2 v Q(p 4 3) .... The expressions for TM slow-wave modes have been given by 19 Chambers and Kharadly . These expressions may be obtained from equation 2.31 for hybrid modes by setting n = 0, and discarding the irrelevant terms involving functions associated with H .. The expressions for fast-& z i wave modes are similar to those for slow-wave modes with h. replaced bv 1 k^, and with the modified Bessel functions and their integrals replaced by the corresponding Bessel functions and their integrals for media 2 or 4. The functions S , T , S , S , etc. are integrals of expressions ij Li r l LiiXL of Bessel functions and are given in their evaluated forms below: 15 <*\l^ Ln<P22> - P22 L;2<*22> + 4 Ln 2 ( p21> " T L + 2 P22 L„< P22> L i ( P 2 2 ) P22 M; 2 ( P22> " ^ 2 2 ^ M n <P22> + 2 p22V P22> M n< P22 ) + < p L ^ M n< P2 -L n(P 2 2) Mn<p ) ( p22 " n 2 ) A n < P 2 2 ) + P22 A n 2 ( p 2 2 ) ~ p " L^^o^ 21 n ^21' T A + 2 P22 V P22>- A n ( p 2 2 ) P22 B n 2 ( p 2 2 ) + ( p 2o" n 2) B?(Poo) + 2 Poo B.(Poo)BKPoo) 22 11^22' 22 n^22' n^22' (P21-n ) B n(p 2 1) - A N ( P 2 2 ) B n(p 2 2) P33 F i 2 ^ 3 3 > + (P33- 2) P n ( P33 } " P 3 2 P ; 2 ( P 3 2 ) - P P>32> T P + 2 P33 Pn<p33> P i ( p 3 3 ) " 2 P32 V P32> P i (p32> P 2 3 3 Q; 2(P 3 3) + (p 2 3 3-n 2) Q 2(P 3 3) + 2p 3 3 (^33) Q.;(P33) P32 Qn 2 ( p32> " ( p32- n 2) <P32> " 2 p32 V P32> Qn ( p32> " Pn ( p33 ) Qn<P33) + P n ( p 3 2 ) V P32> 2 2 - p i / . u : <P ) + PL u;2(p,,) - ( p 2 3 + n 2) U 2( P /,,) 44 n ^ 4 4 ' ^43 n ' - 4 3 43' "TU " 2 P43 U n ( p 4 3 ) " (P44^ 2 ) Vn (p44> " p43 + ^ « ^ > Vl^3> 2 p43 V p43> V>43> Un (P43 ) V p43> 16 TC = P44 C ; 2 ( p 4 4 ) " P43 <p43) ~ ( P43- ^ Cn<P43> SC = TC - 2 p43 C n ^ 4 3 ) SD = ( p44- n 2> Dn <P44> " P43 " ( p 4 3 " n 2 ) D n ( p 4 3 > " 2 P 4 3 V P 4 3 > D;<P43)  SCD = C n ( p 4 3 ) Dn (P43 ) 2 ' 3 3 It i s of interest to examine the limiting form of the attenua-tion coefficient of the dominant mode as the thickness of region 3 vani-shes when regions 2 and 4 are f i l l e d with the same dielectric. It has been shown that under these conditions and h^ approach zero and the phase coefficient becomes equal to that of the TEM mode in, the homogeneous cable. Replacing the functions P 0 ( p 3 2 ) / L Q ( p 2 2 ) and p 0^ p33^ ^ 0^43^ w ± t n their derivatives P 0^ P32^ /' L0^ P22^ a n d P0^ P33^ U0 ^ P43^ u s i n § equations 2.10 and 2.12 and then applying small argument approximations to the modified Bessel functions, i t may be shown that the expression for a of TM modes reduces to -. -, R 1 R /e , 1 ; . . 1 ml r2 . 1 m5 r4 9 0 / , a = -r Ve k tan6 0 + — + — -^34 2 r2 o 2 2 r, 2 r, Z r, lri — Z r, In — o 1 r^ o 4 r^ which is the same as the expression for a of TEM mode in the homogeneous cable derived by the perturbation method. 17 3.0 0 2 4 6 8 10 12 Frequency, GHz Figure 3.1 Mode Spectrum r, = 0.157 cm, r0 = 0.9 cm, r_ = 1.9 cm, r. » 2.5 cm 1.0, e _ = 9-2 r2 r 4 r 3 18 Frequency, GHz Figure 3-3 Cutoff Conditions Conditions as for Figure 3.2 20 0 1 2 3 4 5 6 7 Frequency, GHz Figure 3-4 Cutoff Conditions for Optimum Cable r^ = 0.697 cm, r^ = 2.5 cm, r^ =• O.^ir^+r^) = 1.6 cm e _ =» 9.2, • e _ - e - 1.0 r3 r2 r4 21 0 1 2 3 4 5 6 7 Frequency, GHz Figure 3-5 T r a n s i t i o n C o n d i t i o n s C o n d i t i o n s as f o r Figure 3.2 0.9 0 2 4 6 8 10 12 Frequency, GHz Figure 3-6 Group Velocity Conditions as for Figure 3.1 23 1.2 a a. l . Q h 0.8 h 0.6 0 4 6 Frequency, GHz Figure 3-7 Normalised Attenuation Coefficients Conditions as for Figure 3«1 0 4 '8 . 1 2 16 . 18 Dielectric thickness, r^ - r ^ , mm Figure 3.9 Attenuation Coefficient of Dominant Mode in Optimum Cable r^ =» 0.697 cm, r^-~ 2.5 cm, r^ = 1.64 cm (a) f » 3 GHz, tari« « 6 x 1 0 ~ 5 . ' ( b ) f » 3 GHz, tan$, =* 0 (c) f » 4 GHz, tan5, = 6 x10 5 3 (d) f a -4 GHz, tan5,, • 0 25 0 4 8 12 16 20 22 Dielectric thickness, rj - r2» 111111 Figure 3.10 Attenuation Coefficient of Dominant Mode in Nonoptimum Cable r = 0.157 cm, r =« 2.5 cm, .. r .a. 1.37 cm ^ c- d (a) f » 3 GHz, tan5 =. 6 x l 0 ~ 3 (b) f = 3 GHz, tan 6: = 0 (c) f = 4 GHz, tan 6: = 6x10 5 (d) f = 4 GHz, tan<r » 0 (e) f » 2 GHz, tan5 - 0 26 3. NUMERICAL RESULTS 3.1 Phase Characteristics The curves representing solutions to the characteristic equations for n = 0 and n = 1 are shown i n figures 3.1 and 3.2. The dimensions of 3 12 the cable are the same as those used by many of the previous workers ' ' 13,14,17,18,19,21 . Regions 2 and 4 are air and region 3 a high permit-t i v i t y dielectric. In figure 3.1 the normalised phase coefficient 3/kQ is plotted as a function of frequency. It i s seen that the dominant TMQQ mode is the only mode with no lower cutoff frequency and exists only as a slow-wave mode. As the frequency increases, 3/kQ increases towards the asymptotic value of VE~\ and the phase coefficients of the TE-. and HE. , also the TM- .. and EH. modes become equal. Om lm 0,m-l lm Figure 3.2 shows the normalised phase coefficient as a function of dielectric thickness at 3GHz, similar to the case considered by 19 Chambers and Kharadly . ^/kQ decreases for decreasing thickness as expected. It is noted that four other modes, HE^, ™Q 1 ' T E o i a t K* E H11 can exist in the range of thickness where attenuation reduction i s possible for the dominant TMQQ mode. The cutoff conditions are shown as a function of frequency and dielectric thickness in figures. 3.3 and 3.4. It is noted that the cutoff frequency increases as the dielectric thickness decreases, and the i n -crease is particularly sharp at small thicknesses for modes other than the HE modes. The transition conditions are given in figure 3.5 as a function of frequency and dielectric tube thickness. The normalised group velocity characteristics for the same conditions as for figure 3.1 are shown in figure 3.6. It can be seen that except for the dominant mode, the normalised group velocity increases from zero at cutoff and approaches the value „ at high frequencies. For the TMQQ mode, the normalised group velocity becomes equal to the reciprocal of the normalised phase velocity at low frequencies, and i t approaches unity as the thickness of region 3 vanishes . For thinner dielectric shells, the normalised group velocity of any mode is higher, and approaches ~L/JE~^ at higher frequencies. 3.2 Attenuation Characteristics Attenuation characteristics were calculated for loaded cables whose conductor and dielectric properties are the same as those used by 20 21 7 Chambers and Kharadly and others ' ; namely a = 1.43 x 10 S/m, = 9.2, tan 5 = 6 x 10 \ e „ = e ,=1.0, tan 6„ = tan 6 , =0. Figure 3.7 3 r2 r4 2 4 shows the attenuation characteristics of various modes, normalised with respect to the attenuation coefficient a of the TEM mode in the a i r - f i l l e d o cable, as a function of frequency for the same parameters as for figure 3.1. From this figure i t is seen that a l l modes have a minimum attenuation at a certain frequency for fixed radial dimensions, this minimum may be below a . The' dominant TM_. mode has the lowest attenuation coefficient o 00 at the lower frequencies since i t has no lower cutoff, and interest on the subject has centered on i t s attenuation characteristics. 18 Bach Andersen and Arnbak made the comment that attenuation reduction was possible only because the cable was i n i t i a l l y more lossy than necessary, i.e. when the ratio of outer to inner conductor r a d i i 20 was well above the optimum value of 3.59. Albrey and Gunn found no re-duction in attenuation for the optimum cable (a cable with r^/r^ = 3.59) at 3GHz for a wide range of relative permittivities up to 50. A cable of similar dimensions was investigated at a higher frequency of 10 GHz 28 and the results shown in figure 3.8. It is seen that a reduction of 18% can be obtained for the dominant mode with a dielectri'c shell thickness of 3.1 mm. It is noted that under these conditions the hybrid mode EH^ has a lower attenuation coefficient. Figure 3.9 shows the variation of the attenuation coefficient of the dominant mode with di e l e c t r i c shell thickness for both perfect and imperfect dielectrics at 3 and 4 GHz for the optimum cable. An increase in attenuation results by loading with the imperfect dielectric at either frequency. But a thick shell of perfect dielectric w i l l reduce the at-tenuation significantly by 44% at 4 GHz. However the conductor losses increase with the same perfect dielectric loading at 3 GHz. For comparison, the attenuation characteristics of.the non-optimum cable are shown in figure 3.10. By comparing curves a and c in figures 3.9 and 3.10 i t is seen that although loading the nonoptimum cable with the imperfect dielectric reduces the attenuation by 20% as i n -19 dicated by Chambers and Kharadly , the resulting value is s t i l l above the attenuation coefficient of the unloaded optimum cable. From curves b and d i t is seen that for perfect dielectric loading the nonoptimum cable has a much lower attenuation than the optimum cable. It is noted that a substantial reduction in conductor losses can be achieved by loading the nonoptimum cable at 3 GHz while an increase results for the optimum cable. Similar computations at higher frequencies indicate that for a given r^, leading cables with r^/r^ well above the optimum value of 3.59 generally results, in lower attenuation than loading optimum cables. The results indicate that at the lower frequencies a dielectric shell placed between the conductors enhances the fields on the conductors 29 and thus increases the ohmic loss e s . The energy becomes concentrated i n the d i e l e c t r i c and away from the conductors above a c e r t a i n frequency depending on the cable parameters. With the l a r g e r inner conductor of the optimum cable, the spacing between the conductors decreases and hence the frequency has to be increased i n order to reduce the concentration of energy near the conductors. Thick loadings of very low-loss d i e l e c t r i c s may r e s u l t i n large attenuation reductions. However the losses i n thick lossy d i e l e c t r i c s h e l l s may be excessive, and thin s h e l l s within large cable conductors may be used to advantage at higher frequencies as i n d i -21 cated by Lewis and Sarkar I t i s evident that s i g n i f i c a n t reduction i n attenuation i s p o s s i b l e only f o r the range of frequency and cable parameters which can support at l e a s t the higher order modes and TMQ^, even for perfect d i e l e c t r i c , loading. Contrary to what C u l l e n X ^ indicated,, loading with a p e r f e c t d i e l e c t r i c can r e s u l t i n a much lower attenuation than that of the TE-.mode i n the absence of the inner conductor and the d i e l e c t r i c . 30 4. EXPERIMENTAL INVESTIGATION 4.1 Introduction Laboratory methods for measuring the attenuation coefficient of a low-loss waveguide usually require measurement of either the standing-wave pattern or the resonance Q factor of a section of the waveguide. Since more than one mode propagates in the dielectric-loaded cable in the frequency range of interest, standing-wave-pattern measurements of the dominant mode w i l l be very d i f f i c u l t . Hence the cavity resonance method was used. 4.2 Relation Between Attenuation Coefficient and Q Factor The Q factor of an inhomogeneous d i e l e c t r i c - f i l l e d cavity 22 is given by , 2av- 2 (P + P + P ). Q w uL (W2 + W + W"4) th where = energy storage per unit length in the i medium th P . = power lost in each end plate in the i medium Pi a = attenuation coefficient of the waveguide section v = group velocity L = length of resonator Hence a is given by / ' 2 ( P „ + P „ + P.)v a = —!»- I L 2 2 Pj 2*1 4 2 2 v Q wL (W 0 + W. + WJ j g \ 2 3 4 ' Expressions for v and are given by equations 2.25 and 2.31, and those for P . are given below pi TM modes p _ V f l !^32> • f ! i o V 2 q p2 . 4 , 2 , , I Z / ' bL h2 L 0 ( P 2 2 ) ^ ° ' 31 \ i. JL " / E K. v 2 P = m 3 3 r3 o-PP3 ^ 4 V Z Q j S P P **» f3 £3 •P0(P33).: / *r£o.\ c p4 .4 2 • I z U h4 U 0 ( P 4 3 } . ° ' Hybrid modes . R ^ f * P ; ( P 3 2 ) f E r 2k 0,2 £ r 2 k PP2 " h 4 [\T^r) S L + C 2 e SM + 4 n C 2 g ( ^ ~ ) SLM ] 2 "n 22 TTR f „f * ,e „k . 2 e k h 3 2 TTR f f* P (p ) ,e k ,.2 0 0 e ,k m 3 3 n 33 r / r4 o\ „ , „ 2„2„ , , „ , ri o, pP 4 - -rt^-tr— . i s u + c4 * s u + 4 n V ( - f ^ > V 4-4 h4 U n ( p 4 3 ) The above expressions 4.3 and 4.4 are for slow-wave modes. For fast-wave modes h. is replaced by k., and the modified Bessel functions and . i i integrals replaced by the corresponding Bessel functions and integrals. 4.3 Relation Between Unloaded Q and Loaded Q The loaded Q factor, , of a cavity resonator is given by 4 - 5 where f i s the resonant frequency and Af is the bandwidth at the half-power points of the transmission characteristic. From measurement of and the coupling coefficients and g^, the unloaded Q factor Q can be obtained from the relation Q = (1 + B]_ + 6 2 ) 4.6 If the normalised resistance of the input impedance of the cavity at resonance is less than unity, the cavity is undercoupled. If the normalised resistance i s greater than unity, the cavity is overcoupled. 32 For an undercoupled transmission.cavity, the coupling coefficients 23 are given by 4S - T(f ) (S +1) o o o B 0 = B, S - 1 2 1 o where = input coupling coefficient $2 = output coupling coefficient S Q = input standing wave ratio of the cavity at resonance T ( f Q ) = insertion loss ratio of the cavity at resonance 4.4 Experimental Measurements Resonance measurements were made using the cavity shown in figure 4.1. The brass cavity i s 5.87 cm long, with an outer conductor of 5 cm. inside, diameter, and an inner conductor of 0.317 cm outside diameter. The inner conductor is clamped tightly by the end plates, which are attached very tightly to each half of the outer conductor by screws. The cavity opens longitudinally as i t was originally intended to solder the end plates to the outer conductor. This discontinuity should not have much effect on TM modes since i t is along the lines of current flow. The input and output coupling is by means of small coaxial line loops projecting through the end-plates. The planes of the loops l i e along radii at right angles to each other. The alumina dielectric tube has an inner diameter of 1.32 cm and an outer diameter of 4.4 cm. It is supported by two thin polyfoam discs resting on the outer conductor. The nominal relative permittivity of the dielectric is 9.30 and the loss tangent 0.00009 at 8.5 GHz. The setup for swept frequency measurements i s shown in figure 33 Fi g u r e 4-1 D i e l e c t r i c - l o a d e d C a v i t y used i n Experiment HP 8690B Sweep O s c i l l a t o r 8692B RF U n i t scope or chart recorder v a r i a b l e a t tenuator c r y s t a l d e t e c t o r tuner resonant c a v i t y F i g u r e 4.2 QFactor Measurement C i r c u i t 34 4.2. The resonant frequency f was determined by the frequency at which power transmission through the cavity was maximum. The measured f Q was accurate to within 0.005 GHz. The level of the half-power points was determined withthe variable coaxial attenuator. The sweep width Af measured had an uncertainty of about 0.03 MHz, giving a maximum uncertainty of 5% in the measured Q factor. Measurements were made at different power levels at each resonance and with the input and output couplings interchanged and the results averaged. The sum of the coupling coefficients and g^ w a s found to be less than 0.06 for the frequency range concerned. Some resonances were observed which could not be predicted from theory. The power transmission levels at these resonances were much lower than those of the predicted resonances and i t was believed that these spurious resonances were caused by harmonics in the sweep oscillator. A-t some of the resonances of the hybrid modes, the resonance curve did-not appear to be smooth and symmetrical, and the Q factors at these resonances were not measured. 4.5 Experimental Results The experimental results are shown in table 4.1, where the TEM mode refers to the unloaded cavity. It was assumed that e » = e , = 1.0, J r2 r4 and tan 6^  = tan 6^ = 0. The experimental results agree reasonably well with theory for the TM modes, but agreement is not as good for the hybrid modes. It may be noted that a given error in the measured Q factor results in a larger error in the attenuation coefficient subsequently determined. The exact properties of the conductor and dielectric were not known. The effective conductivity was lower than the nominal value of 1.2 x 10^ S/m due to imperfect surface finish and joints between conductors. Calculated Measured Mode f 0 GHz Q a .., xlO~ 2 N/m f o GHz Q x 10~ 2 N/m TEM^ . 2,55 2220 0.932 2.55 2040 1.05 TM 002 3.01 4780 1.02 3.01 4420 1.17 TM u003 3.83 5640 1.35 3.82 5470 1.43 ™011 2.84 2090 1.98 2.86 1910 2.21 E H111 2.87 3650 2.16 2.90 3080 2.74 E H112 3.59 5050 1.79 3.60 4590 2.07 H E112 2.59 4460 1.57 2.51 H E113 3.32 4980 1.58 3.29 Table 4.1 Experimental Results r3 • 9.31, tan6 3 = 0.0001, a = 1.2 x 10 7 Mode f o GHz Calculated Q a x 10~ 2 N/m f o GHz Measured Q a x 10~2 N/m TEM^ ^ 2.55 2120 0.974 2.55 2040 1.04 TM 002 3.01 4630 1.04 3.01 4420 1.13 ™003 3.83 5500 1.36 3.82 5470 1.38 ™011 2.84 2010 2.06 2. 86 1.910 2.19 E H111 2.87 3540 2.21 2.90 3080 2.71 E H 1 1 2 3.59 4930 1.82 3.60 4590 2.02 Table 4.2 Experimental Results for Lower Conductivity e n = 9.31, tanfi^ = 0.0001, a = 1.1 x 10 r3 3 36 Mode TM 002 TM 003 TM, O i l EH 111 EH 112 Calculated tan 6^ = 6xl0" 5 Q « . xlO N/m 5250 6720 2090 4010 5850 0.824 0.905 1.95 1.84 1.38 Measured lO" 4 4420 5470 1910 3080 4590 1.13 1.38 2.19 2.71 2.02 Calculated 1.4 x lCf* Q a 4140 4660 1920 3170 4250 1.26 1.82 2.15 2.59 2.26 Table 4.3 Comparison of Results for Different Loss Tangents r3 = 9.31, 1.1 x 107 S/m Calculated Measured Calculated Mode r3 9.0 9.31 9.6 f o f o -L o CL GHz x 10~ 2 N/m TM 002 3.04 1.04 3.01 1.13 2.98 1.04 TM 003 3.88 1 1.35 3.82 1.38 3.79 1.38 TM 011 2.86 2.08 2.86 2.19 2.82 2.03 E H111 2.91 2.25 2.90 2.71 2.83 2.19 E H112 3.63 1.82 3.60 2.02 3.54 1.82 H E112 2.64 2.51 2.56 H E 1 1 3 • 3.37 3.29 3.27 Table 4.4 Comparison of Results for Different Pe rmi t t i v i t i e s tan5 3 = 0.0001, a = 1.1 x 10 7 37 Table 4.2 shows the r e s u l t s f o r a conductivity of 1.1 x 10^. I t i s seen that agreement between t h e o r e t i c a l and experimental values improves some-what. Table 4.3 shows the r e s u l t s f o r the loss tangent tan 6^ of 6 x 10 ~* -4 and 1.4 x 10 . The e f f e c t of v a r i a t i o n of tan 6^ i s much more pronounced on the higher resonances since d i e l e c t r i c losses are dominant at higher frequencies. The r e s u l t s f o r the v a r i a t i o n of r e l a t i v e p e r m i t t i v i t y are shown i n table 4.4, where i t i s seen that the resonant frequencies are lowered noticeably with the increase i n e „, but the attenuation c o e f f i -r3 cients are not much af f e c t e d w i t h i n t h i s range of v a r i a t i o n . Experimental values of the attenuation c o e f f i c i e n t f o r the do-minant mode are about 9% above that of the unloaded cable. No reduction i n attenuation was obtained because the d i e l e c t r i c used i s not s u f f i c i e n t l y low-loss. I t may be noted that the- measured Q factors for the dominant mode of the d i e l e c t r i c a l l y loaded cavity i s about twice that of the un-loaded c a v i t y , even though the attenuation i n the waveguide section i s not reduced. With d i e l e c t r i c s of higher p e r m i t t i v i t i e s or lower loss tangents higher Q f a c t o r s can be obtained. 38 5. CONCLUSIONS It has been shown that attenuation reduction by d i e l e c t r i c loading i s pos s i b l e f o r the dominant mode i n optimum coaxial cable s t r u c -tures i f l o s s l e s s or s u f f i c i e n t l y low-loss d i e l e c t r i c s are used. For d i e l e c t r i c s with higher losses higher frequencies or l a r g e r cable dimen-sions are required to achieve attenuation reduction. No s i g n i f i c a n t reduction i n conductor losses f o r the dominant mode can be achieved below the cut o f f frequency of the f i r s t hybrid mode. The propagation c h a r a c t e r i s t i c s of a l l higher order modes have been given. Experimental r e s u l t s of attenuation c o e f f i c i e n t s by the cavity resonance method generally agree with theory. However, a reduction i n attenuation has not been observed due to the d i e l e c t r i c used not being s u f f i c i e n t l y low-loss. Such cable structures may f i n d a p p lications i n the construction of resonant c a v i t i e s with high Q f a c t o r s , i n addi t i o n to t h e i r possible usefulness as low-loss waveguides with no lower c u t o f f . 39 APPENDIX A. FIELD COMPONENTS EZ2 = f 2V h2 r ) n'u „k Z Er2 " J f2 ^ <V> + T f ^ S2Mn<V> 2 h„ r „ jy „k Z E 6 2 = - 2 - f 2 L n ( h 2 r ) - h, 8 2 M n ( h 2 r ) h 2 r 2 HZ2 " S 2V h2 r ) ne k H r 2 * " ~ff f2 VV> + J hf g 2 M n ( h 2 r ) Z h_ r 2 o 2 H82 • ^ i r f2 Li<v>+ rf- g2 VV> o 2 h2 r EZ2 = f2 A n ( k 2 r ) E =--!£ f 0 A ' ( k r ) S=_2_2. g B (k r) rz K„ ^ a z i ^ z n z 2 k 2 r o jy 0k z 82 = ~ ~2~ f2 A n ( k 2 r ) + k 9 g2 B n ( k 2 r ) k 2 r 2 HZ2 " g2 V k2 r ) ne „k H r 2 = 7 7 ¥ f2 W> " j f c 82 B;< k2 r ) Z k 0 r 2 o 2 lI92 " " £2 *i< V> " ft" % W> o 2 k 2 r EZ3 B f3 W ny oknzn E r 3 • " *h £3 P; ( h3 r> " *3 VV> 3 h 3 r E 9 3 " - r f f 3 W> - 83 Q » ( h 3 r ) h 3 r 3 8 3 V h 3 r ) ne k o 3 3 E „k - ^ _ 2 . f p . ( h r ) _ nB Z L 3 n v"3 ' , 2 6 3 xn v"3* o 3 h 3 r f4 VV> R ny .k Z . P c nl/u \ L r4 o o J h7 f4 U n ( h 4 r ) + T~2 4 h, r 84 VV> N P c T T / u N r4 o o f/. U (h.r) — 84 v;(v> . 2 '4 n v"4 h4 r g 4 VV> n £ 4 k Z n. r 4 o 4 TTT1 h»;<v> - -ff- s 4 v n ( h 4 r ) o 4 h. r 4 f C (k r) 4 n . 4 - j 7-2- f. C'(k.r) k^ 4 n v 4 f, C (k.r) -2 "4 n v~4 J  k4 r ny ,k Z r4 o o jy ,k z J r4 o o 84 D n ( k 4 r g 4 KS\r g 4 V k 4 r ) ne ,k jff f4 V k 4 r ) " j k^  84 ° ; ( k 4 r ) o 4 j £ ,k „ " Z T ~ f4 C n ( k 4 r ) " — g4 D n ( k 4 r ) o 4 k4 r 41 The f i e l d components f o r TM to zero i n the above expressions; obtained by s e t t i n g n and f. to-zero modes are obtained by s e t t i n g n and the f i e l d components for TE modes are 42 APPENDIX B. DEFINITION OF A1 FUNCTIONS USED k A l k ^ 2 2 ; k ^ 2 2 J o p p o p p -32 22 32 *22-2 k k 2 2 ; k ^ ° ° ; A3 A5 A7 A8 A10 A l l A13 A15 o p p, *33 ^43 ° P33 P43 p ; ( p 3 2 ) A - ^ P 32 Pn ( p32 ) 4 P33 Pn ( p33 } P32 Qn ( p32 ) A -6 P33 Qn ( p33 } ^n(p22> An<P22> P22 Ln ( p22 ) P 2 2 A n ( p 2 2 ) M;< p 2 2) Bn<P22> P 2 2M n(p 2 2) P22 Bn ( p22 ) n 43 n 43 43 n 43 P43°n ( p43 ) v ; ( p 4 3 ) c; ( p43> p43 Vn ( p43 ) P43 Dn ( p43 ) Jn<p32> A J n ( p 3 3 ) Y n ( p 3 2 ) 1 2 ' V P33> j; ( p 3 2> J n ( P ^ } A -P32 Jn ( p32 } 1 4 P33 Jn ( p33 } YA<p32> Y>33> P32 Yn ( p32 ) 1 6 " p33 Yn ( p33> 2 2 where A. ,A~ ,A,,An,A_ and A,_ are defined "for k. < 0 and k. > 0 1' 2 7 8 9 10 l i respectively. 43 REFERENCES . Barlow, H.M., "Screened surface waves and some possible applications", Proceedings IEE, 1965, Vol. 112, No. 3, p. 477-482. 2. Barlow, H.M., "New features of wave propagation not subject to cutoff between two paral l e l guiding surfaces", Proceedings IEE, 1967, Vol. 114, No. 4, p. 421-427. 3. Barlow, H.M., "High-frequency coaxial cables", Proceedings IEE, 1968, Vol. 115, No. 2, p. 243-246. 4. Barlow, H.M., and Sen, M., "Use of the hybrid TEM-dual surface wave in coaxial cables and resonators to reduce attenuation", Electronics Letters, 1967, Vol. 3, p. 352-353. 5. Barlow, H.M., and Sen, M., "Further experimental observations on the behaviour of the hybrid TEM-dual surface wave in a coaxial resonator", Electronics Letters, 1967, Vol. 3, p. 389. 6. Barlow, H.M., and Sen, M., "Experimental investigation of the hybrid TEM-dual surface wave", Electronics Letters, 1967, Vol. 3, p. 450-451. 7. Barlow, H.M., and Sen, M., "Additional experimental information about the behaviour of the hybrid TEM-dual surface wave in coaxial and para-llel-wire resonators", Electronics-Letters, 1967, Vol. 3, p\ 494-495. 8. Barlow, H.M., and Sen, M., "New experiments on the behaviour of the hybrid TEM-dual surface wave in a coaxial cable", Electronics Letters, 1968, Vol. 4, p. 523-524. 9. Millington, G., "Hybrid TEM waveguide mode", Electronics Letters, 1967, Vol. 3, p. 310-311. 10. Millington, G., "Surface waves in waveguides", Marconi Review, 1967, Vol. 30, p. 202-228. 11. Brown, J., and Millington, G., "High-frequency coaxial cables", Elec-tronics Letters, 1968, Vol. 4, p. 135-137. 12. Lewis, J.E., and Kharadly, M.M.Z., "Propagation characteristics of the TM m surface-wave mode in dielectric-lined coaxial cables", Electronics Letters, 1968, Vol. 4, p. 228-230. 13. Lewis, J.E., and Kharadly, M.M.Z., "Surface wave modes in die l e c t r i c -lined coaxial cables", Radio Science, 1968, Vol. 3, p. 1167-1174. 14. Barlow, H.M., "Hybrid TEM-dual surface wave in a coaxial cable", Pro-ceedings IEE, 1969, Vol. 116, No. 4, p. 489-494. 15. Chambers, B., and Kharadly, M.M.Z., "Attenuation coefficient of d i e l -ectric-lined coaxial cables", Electronics Letters, 1969, Vol. 5, p. 65-67, 44 16- Millington, G., and Rotheram, S., "Riccati approach to the propaga-tion of axially symmetric waves in a coaxial guide", Proceedings IEE, 1968, Vol. 115, No. 8, p. 1079-1088. 17. Cullen, A.L., "Some pos s i b i l i t i e s and limitations of graded dielec-t r i c loading", Proceedings IEE, 1968, Vol. 115, No. 6, p. 759-761. 18. Bach Andersen, J., and Arnbak, J., "Losses in die l e c t r i c a l l y loaded coaxial cables", Proceedings IEE, 1969, Vol. 116, No. 10, p. 1665-1672.' 19. Chambers, B., and Kharadly, M.M.Z., "Attenuation characteristics of dominant mode in inhomogeneously f i l l e d transmission lines", Proceedings IEE, 1970, Vol. 117, No. 5, p. 897-902. 20. Albrey, I.J., and Gunn, M.W., "Attenuation reduction in coaxial cables", Proceedings IEE, 1971, Vol. 118, No. 12, p. 1749-1751. 21. Lewis, J.E., and Sarkar, T.K., "Characteristics of dielectric-tube-loaded coaxial cables", Proceedings IEE, 1972, Vol. 119, No. 5, p. 523-528. 22. Bourk, T.R., Kharadly, M.M.Z., and Lewis, J.E., "Measurement of waveguide attenuation by resonance methods", Electronics Letters, 1968, Vol. 4, p. 267-268. 23... Sucher, M..and.. Eox., J. ,. "Handbook of microwave measurements", Vol.. 2,. Polytechnic Press, 1963. 

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