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Measurement of the propagation characteristics of shielded and unshielded dielectric-tube waveguides Makino, Ikufumi 1970

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MEASUREMENT OF THE PROPAGATION CHARACTERISTICS OF SHIELDED AND UNSHIELDED DIELECTRIC-TUBE WAVEGUIDES . by IKUFUMI MAKINO B.Sc, Doshisha University, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Research Supervisors • . > «... Members of the Committee Head of Department Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 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 Beefarieg.) tBno^/veerC The University of British Columbia Vancouver 8, Canada Date December 1' , 1310 ii ABSTRACT Accurate measurements of the propagation coefficient of the HE^^ mode on polythene-tube waveguides in air and surrounded by a polyfoam shield are reported. These were carried out at X-band frequencies using a cavity-resonance method. The results obtained confirm previous theoretical predictions although there is an element of uncertainty concerning the exact dielectric properties of the commercial grade polythene tubes used. The measurements also yielded the phase coefficient of the HE^^ mode which was confirmed by measurement of the radial decay of the electric field outside the tube. Enclosing the dielectric-tube in a low-density, low-loss poly foam shield resulted in only a slight degradation of the attenuation characteristics of the waveguides. . -Measurements of the phase characteristics of the higher order TEQ^ and TMQ^ modes on the tube at frequencies close to cutoff are also reported. iii TABLE OF CONTENTS Page ABSTRACT .  ii TABLE OF CONTENTS iiLIST OF ILLUSTRATIONS ... . iv LIST OF TABLES . vi LIST OF SYMBOLS viACKNOWLEDGEMENT iv 1. INTRODUCTION 1 : 2. SURFACE-WAVE PROPAGATION ON DIELECTRIC-TUBE WAVEGUIDES 3 2.1 Field Components 3 2.2 Mode Spectrum 6 2.2.1 Characteristic Equations 6 2.2.2 Cutoff Conditions 9 3. CAVITY-RESONANCE METHOD OF MEASURING ATTENUATION , 11 3.1 Introduction 13.2 Relation Between Attenuation Coefficient and Q Factor ... 11 3.2.1 Relation Between Attenuation Coefficient and Q. Factor for Surface-Wave Resonator 13 3.3 Relation Between Unloaded Q and Loaded Q 16 4. EXPERIMENTAL APPARATUS 18 4.1 Introduction4.2 Surface-Wave Resonator 20 4.3 Mode Exciters 26 5. RESULTS 29 5.1 Dependence of Cavity Q Factor on Size of Coupling Aperture5.2 Measurement of Guide Wavelength 30 5.3 Measurement of Radial Decay of Electric Field 36 5.4 Measurement of Attenuation Coefficient 38 6. CONCLUSIONS 42 REFERENCES 3 iv LIST OF ILLUSTRATIONS Figure . Page 2.1 The Dielectric Tube Waveguide 3 2.2 Mode Spectrum of Polythene Tube, p=0.5 8 2.3 Cutoff Conditions; TEQ1, TMQ1, EH and HE12 Modes 10 3.1 Characteristics of Dielectric Tube Waveguides, HE^ Mode ... 12 3.2 Transmission Characteristics of a Resonant Cavity 16 3.3 .Variation of Input Impedance of a Resonant Cavity 16 4.1 Layout of Apparatus 19 4.2 General View of Surface-Wave Resonator . 22 4.3 Surface-Wave Resonator 23 4.4 Details of Cavity End Plate Showing HE^ Mode Exciter ...... 24 4.5 Details of Cavity End Plate 25 4.6 Diagram of TEQ-^ Mode Exciter and Position of Exciter Relative to Dielectric Tube 26 4. 7 TEQ1 Mode Exciter 28 4.8 TMQ1 Mode Exciter5.1 Measured Dependence of Cavity Q Factor of HE^ Mode on Coupling Aperture Diameter 29 5.2 Measurement of Guide Wavelength of HE^ Mode by Field Perturbing Bead Method 31 5. 3.a Experimental and Theoretical Phase Characteristics of HE.. 1 Mode on Polythene Tube I 3 5.3.b Experimental and Theoretical Phase Characteristics of HE.^ Mode on Shielded and Unshielded Polythene Tube II 34 5.3. c Experimental and Theoretical Phase Characteristics of HE.. Mode on Polythene Tube III 35 5.4. a Radial Decay of Ez3 for HE^ Mode 36 5.4.b Radial Decay of E , for HE-. Mode 7 J r3 11 V Figure Page 5.5.a Experimental and Theoretical Attenuation Characteristics of HE Mode on Polythene Tube I 39 5.5.b Experimental and Theoretical Attenuation Characteristics of 'HE Mode on Shielded and Unshielded Polythene Tube II. 40 5.5.c Experimental and Theoretical Attenuation Characteristics of HE Mode on Polythene Tube III 41 LIST OF TABLES Table 3.1 Sample Output from QFACTOR . 4.1 Details of Polythene Tubes .. 5.1 Details of Coupling Apertures vii LIST OF SYMBOLS a., b . = constants A (p..), B (p..) = functions of Bessel functions c. = jb±/a± ^zi' ^rL' ^fli = ^-on^^u^na^-» radial, azimuthal components of electric field, respectively, in medium i f = frequency f = resonant frequency h.  wave number of medium i 1 H ., H ., H . = longitudinal, radial, azimuthal components of magnetic zi ri 61 field, respectively, in medium i I (p..) = modified Bessel function of the first kind n *ij J (p..) = Bessel function of the first kind n *ij kg = phase coefficient of free space K (p. .) = modified Bessel function of the second kind I = number of half wavelengths in resonator L = length of resonator m, n = mode subscripts N, = total power loss per unit length and power loss per unit length in medium i, respectively N = power flow g N , N = total power loss in each end plate and power loss in P P each plate in medium i, respectively P*4 h.r. ij i j Q = quality factor Q = loaded Q factor 1 Q = unloaded Q factor u r = radial co-ordinate r^, = inner and outer radius of tube respectively viii R = normalized resistive component R^ = resistive component of wave impedance of a metal S>A' Sg, S^B> = integrals of functions of Bessel functions S_, S„, T , T„ = integrals of functions of modified Bessel functions 1 K. 1 K tan5, = loss tangent of medium i VQ , v , v = speed of light in free space, group velocity and phase ^ ^ velocity, respectively W, W. = total energy storage per unit length and total energy storage per unit length in medium i, respectively Y (p..) = Bessel function of the second kind z = longitudinal co-ordinate Z = impedance of free space a = attenuation coefficient of tube 3 = phase coefficient of tube 3^ = coupling coefficient A f = bandwidth e .  relative permittivity of medium i ri 0 . = azimuthal co-ordinate X = free space wavelength X , X , X = cutoff, guide and resonant wavelength,respectively c g r u = relative premeability of medium i P - r±/r2 ui = angular frequency iv ACKNOWLEDGEMENT The author is deeply indebted to his research supervisors Dr. B. Chambers and Dr. M.M.Z. Kharadly for their encouragement and guidance throughout the course of this project. Grateful acknowledgement is made to the National Research Council of Canada for support under grants A3344 and A7243 and to the University of British Columbia for the award of a University of British Columbia Graduate Fellowship during the academic years 1968-1970. The author is also grateful to Mr. C.G. Chubb, Mr. D.G. Daines and Mr. J.H. Stuber for building the precision equipment and to Mr. H.H. Black for the photographic work. The author also wishes to thank Miss Linda Morris for typing the manuscript and Mr. B. Wilbee, Mr. F. Scholz and Mr. S. Graf for their careful proofreading of the final draft. 1. INTRODUCTION During the last forty years or so, many investigators have considered the problem of surface-wave propagation along dielectric tubes. In 1932, Zachoval"'" obtained the characteristic equation for TMQ^ modes and solved this graphically for a range of tube parameters. 2 Two years later, the existence of these modes was verified by Liska , whose measurements of guide wavelength showed good agreement with Zachovai's theory. In 1949, Astrahan obtained the characteristic equations for TE and hybrid modes and measured values of guide wavelength for the HE^, TMQ^ and TE modes which agreed very well with theory. At 4 about the same time, Jakes gave expressions for the attenuation coeffi cients of TM„ and TE„ modes and measured the attenuation of the TM' Om Om 01 and TEQ-^ modes on polystyrene tubes. 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 6 ^11 moc*e on tubes with small diameter to wavelength ratios. Mallach made a rough estimate of the attenuation of the HE-Q mode by measuring the radius at which the magnitude of the electric field fell to 1/e of its value at the tube surface. In 1968 Kharadly and Lewis'' completed a comprehensive study of the possible usefulness of the dielectric tube as a low-loss waveguide. They concluded that a moderately thin-walled tube propagating the dominant mode could have propagation characteristics greatly superior to those of conventional metallic waveguides at millimeter-wave frequencies. Also, they proposed a method for overcoming the problems of supporting the tube and the degradation of performance due to adverse weather conditions or nearby obstacles. This consisted of embedding the tube in a layer of low-density, low-loss dielectric of sufficient radial extent that a negligible portion of the wave was carried outside this dielectric. So far as is known, no accurate measurements of the attenua tion characteristics of the dominant HE mode on dielectric-tubes have been made. This seems surprising in view of the fact that this mode is the one most likely to be used in practice. The objectives of the investigation reported here were therefore: (i) to obtain experimental data on the attenuation and phase coefficients of the HE mode on commercially available polythene tubes from direct measurements, using the cavity-resonance method. (ii) to ascertain experimentally the effect of shielding the tube with low-density, low-loss polyfoam. Chapter 2 reviews briefly some of the features of surface-wave propagation on dielectric-tube waveguides. The theory in this chapter is drawn from reference 7. In Chapter 3, the theory underlying the cavity-resonance method for measuring the attenuation coefficient of low-loss waveguides is discussed. This is followed in Chapter 4 by a description of the experimental apparatus used. Experimental results for the propagation characteristics of the HE^ mode on-polythene tubes in air and surrounded by a polyfoam shield are given in Chapter 5, together with results for the phase coefficient of the teQ^ and TM ^ modes at frequencies close to cutoff. Conclusions drawn from this investigation and suggestions for further work are contained in Chapter 6. 3 2. SURFACE-WAVE PROPAGATION ON DIELECTRIC TUBE WAVEGUIDES 2.1 Field Components The tube configuration of interest is shown in figure 2.1. It consists of two coaxial dielectric regions of infinite length and relative permittivities e , and e „ embedded in a third infinite dielectric v rl r2 of relative permittivity er3> where and er2 > erl e _ > e -r2 r3 _ .2.1 In all cases, it will be assumed that the relative permeability of the region, u ., is unity. Propagation is assumed in'the z-direction, ri' with t-9-z dependence of the form exp j(wt-n0.-6z) in the lossless case. "r3 Figure 2.1 The Dielectric-Tube Waveguide 4 Under these conditions, omitting the factor exp j(wt-n6-$z), the field components are given by Ezl = al Vhlr) R ' nyri knzn Erl = J hT al W> + \ \ InV) 1 h, r 01 2_ "1 n^l hlr Hzl = bl Vhlr> H rl 1 n 1 n e , k — ai W)+^bi Vhir) hl r Z0 e k Hei = Jhfi~ai i^+f~h W> 10 1 r Ez2 = a2 J Our) H Y (h.r) n Z n 2 « a2 An(h2r) o i • ' nu _ k Z Er2 = ll a2 Vh2r) " [l b2 VV> 2 h2 r • - n6 * /v \ i yr2 k0Z0 . .» . • '82 " " TT a2 Vh2r) + J h~ b2 Vh2r) 2 ^ ^ Hz2 = b2 J (h r) + r1 Y(h.r) n 2 b2 n 2 = b2 Bn(h2r) \2 = ?7a2Vh2r)-jfb2 VV» h2 r ZQ 2 £ k H62 = "J zfhT a2 An(h2r) " X" b2 Bn(h2r) 0 2 h2 r 0 < r < ,2.2.a rl ± r ± r2 ,2.2.b Jz3 r3 a3 Kn(h3r) j K' (h„r) +' n yr39 ° ° b„ K (h,r) h„3ni ,2 J n 3 3 h« r n^ v tu \ - yr3 K0Z0 , ' , . a_ K (h r) -3 b0 K_(h,r) z3 .2 3 nv 3 r b.K (h„r) 3 n 3 3 n 3 H = -r3 a3 Kn(h3r) + J IT b3 Kn(h3r) h3 r ZQ 3 H 63 J 1^a3K>3r)+7|--b3Kn(h3r) 3 0 h^ r r2 < r < «> .2.2.c where, from the wave equation 2 2 2 h- = 3 - u , * e , kn 1 "rl rl 0 2 2 2 h2 = yr2 er2 k0 " B 2 2 3 - u 0 e _ k_ r3 r3 0 ,2.3 The symbols appearing in equations 2.2 and 2.3 are defined in the list of symbols. Upon setting n=0 (no 6-variation), equations 2.2 separate into two sets corresponding to the circularly symmetric modes designated TMQ^ and TE„ . For n^O, equations 2.2 describe inseparable combinations of TE and TM Om modes which are designated hybrid modes. In general, one or 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, nm it is termed EH . The nature of TE or TM dominance and the significance nm of the subscript m in the mode designation is discussed fully in reference 7. 6 2.2 Mode Spectrum 2.2.1 Characteristic Equations By matching the axial and tangential field components in media 1 and 2 at r=r^ and those in media 2 and 3 at *=*2> el8nt homogeneous equations in eight unknowns, a.^,.b^, i=l-4, are obtained. These may be solved to give the following characteristic equations for the hybrid modes: « r i /£r2An(p22) , £r3Kn(p32¥r2Bn(P22) + ^_n_^_ \P22An(p22) P32Kn(p32)AP22Bn(p22) P32Kn(p32) yr3Kn(p32) 32yJ ,2.4.a and ^£r2An(p21) , Wn^' lP2lAn(p21) PllIn(pll)^ fyr2Bn(p21) + 'p21Bn(p21) PllI„<Pll>- '21' 2.4.b Equation 2.4.a is applicable for EH modes and equation 2.4.b is applicable for HE modes. The ratio a./a„ is given by equation 2.5. The ratio b./b„ nm 4 2 <\ L • is obtained from equation 2.5 by interchanging e and ur^- The character istic equations for the TErt and TM_ modes are obtained by setting n=0 in M Om Om equation 2.4.b. A typical spectrum of modes on a polythene tube in free space A (e =£ =y =y =y =1 e „=2.26, and P=r../r =0.5) is shown in figure 2.2. rl r3 rl r2 r3 r2 12 The main features of the mode spectrum are: (i) The HE^^ mode has no lower cutoff frequency. (ii) Unlike the case for the dielectric rod (p=0), the TEQm and TM„ modes do not have the same value of r„/X at cutoff. This is Om I also true for HE. ,. and EH. modes. l,m+l lm. (iii) As P"KL, the phase characteristics of the TE^ and HE^ mode become indistinguishable, as do those of the TM^ and EH]_m m°des, thus 1 CN ^—N rH rH CN CN ft ft •w - c a >< CN rH i-l CN (0 (X + I—1 rH rH rH P. ft - a c M t—1 rH rH rl rH (J ft rH CN *—* ft CN CN c ft —' c rH >-) CN * V ft rH CN c ft ' CN CN ft CN ft CN U 3-CN ft CN ft CN ft CN ft + CN ft CN ft *-> rH CN ft I CN ft CN ft CN ft CN CN ft CN I CN CN CO ft CN CN CN ft 8 providing the physical distinction between HE and EH modes. (iv) As p-KL, the n=0 and n=l modes appear in widely separated clusters, each cluster consisting of four modes (HElm> TEom' ™0m and EHlm^' (v) The HE, and TE. phase characteristics intersect at some lm Om value of . In most cases, for values of r^/A greater than that at the intersection, the differences in the two curves are too small to be seen graphically. However, the degeneracy of the HE^ and TE02 modes ^or san be seen in figure 2.2. Figure 2.2 Mode Spectrum of Polythene Tube, p=0.5 2.2.2 Cutoff Conditions Lossless surface-wave propagation on the dielectric-tube requires that all quantities appearing in equations 2.3 be real and positive. If u ~e 0=u .e then cutoff occurs when h =0 and h =0, or r3 r3 rl rl 3 1 generally, when P-j^O anc^ pll=^° Hence by applying small argument approximations to certain of the Bessel functions in equations 2.4.a-b, the following cutoff conditions are obtained. J0(p22) YQ(P22) £rl P21 JQ(P21) " 2er2:Jl(?21) ^ A TR—-, s 7, ~—: x TM modes £rl p21 Y0(p21> " 2£r2 \<»2J P21 J0(P21) - 2Jl(p21}  p21 Y0(p21) -.2Yl(p2]L) TE modes 2.6.a P22 - 0 J1(P22) J1(P21) Yl(p22) Yr(p21) HE^^ mode HE, modes lm m > 1 ,2.6.b J1(P21) _J1(P22} Yl(p2iy  Yl(p22)J "J1(P21> J1(P22) Y1(P21>  Y1(P22} '21 P21 £rl (e ,+e „) rl r2' EH, modes lm m >. 1 ,2.6.c At cutoff p22 is given by p22=2.. ,x ) Jl Mr2er2 _ yr3er3} .2.7 from which the value of r_/X can be determined. 2 c The variation of r0/X with P for the TE.,, TMni, EH and HE modes on L C Ul U± ii Ll a polythene tube (er2=2.26) in free space is shown in figure 2.3, 11 3. CAVITY-RESONANCE METHOD FOR MEASURING ATTENUATION-3.1 Introduction . The cavity-resonance method appeared to be the one most suitable for directly measuring the small attenuation coefficient of the HE^Q mode on dielectric-tube waveguides. The main advantages of the method are that only a fairly short length of waveguide is needed and the problems of accurate measurement of power levels or substituted attenuation are avoided. The relationship between attenuation coefficient and the Q factor of a cavity formed from a section of the waveguide and two metallic end plates is discussed in the next section. 3.2 Relation Between Attenuation Coefficient and Q Factor Adopting the nomenclature of reference 7, the Q factor of the resonator xs given by ITT wLN /v 11 _j^WL_ = g S ....................... 3.y 2N +NL 2N +2LaN P P g where W = N /v , N=2aN g g g Then . 2av . 2N . I = S_ + P_ • 3.2 Q gvp coWL where 3 = to/v =2TT/X P • g For very long resonators, the second term in equation 3.2 can be neglected and the expression for Q becomes <; Mr) •• •••••••••• 3-3 g 12 Then the attenuation coefficient a is given by ,3.4 4 9 In previous experimental investigations of surface waveguides ' , values of a have been obtained by measuring Q and g and using the trans mission-line formula, a = .3.5 2Q which assumes v /v =1 in equation 3.4. This assumption may lead to significant errors. As an example, the factor v /v for the dominant P g HE^ mode on a polythene tube (e^^=2.26) waveguide has been completed and is shown in figure 3.1. Inspection of this figure shows that equation 3.5 is valid for such waveguides when the phase-velocity reduction is very small or very large, but can lead to appreciable errors for intermediate values. In the present investigation, no provision was made for measuring v and hence the ratio v /v together with the term in equation g P g . 3.2 involving end plate losses were evaluated using the theory given below 1.4 v _E v g 1.2 1.0 p=0.1 f\ I0'3 -// / 0,5 ^0.8 ^"""^0.9 0 0.4 0.8 r2/X 1.2 1.6 2.0 Figure 3.1 Characteristics of Dielectric-Tube Waveguide, HE^ Mode 13 3.2.1 Relation Between Attenuation Coefficient and Q Factor for  Surface-Wave Resonator From equation 3.2, a_l ALYI _ 2Np\ * / MA _ .2(VNP2+V\ 2 \,v J\q wWl7 2\.vg/\Q uLO^+W^Wg). / 3.6 where, for the dominant HE^^ mode, W a2a2 verl fAl (p21} 1 4 12 4v„Z„h, \I^(p1;L) W 0 0 1 _ a3a2ffer2 2 4 4V0h2 L h: Ti+&2+(koVi)2] h-^owl^ h2 V B2sA+(k0Z0C2)2sB+4gk0Z0C2SAB W = 3 a2a2TTer3 / Ax (P22 ) 4V0h3 W<P32>, h^ TK+[32+(k0Z0c/]sK+4BkoZ0c3K2 (p32) .3.7 ^V^n /A1(P21)N Pl ST+C?32ST - 4CL3(:° RL II I 1 \ z L> Z0 ^lP N 2a0a0fTR 2 2 m P2 ~zrJ Vc23 [ZT,—: v4c2" Jo J ^0er2yAl(p21)' 0 Bl(p21) J. AB N 2a2a27TRm/'Al(p22)1 p3 Kl(p32), Vr3 u zo K1(p32) .3.8 The functions ST, S,,S . S, S,„ TT , T, and T,r are integrals of functions I A B AB' K' I A K of Bessel functions which are defined and evaluated in reference 7. Table 3.1 shows a sample output from a computer program called QFACTOR which was used to obtain the unloaded Q factor of the surface wave resonator, the factor (v /v ) and the attenuation coefficient of the P 8 .14 dominant mode by solving equation 2.4.b for ^(g), where the. phase coefficient 0 was decided by the number of half wavelengths contained in the length of the cavity. R1=0.012700(M) R2=0.015875(M) Rl/R2=0.8000 ER1=1.000 ER2=2.260 ER3=1.000 T1=0.0 T2=0.0005 T3=0.0 SIGMA=0.3536E 08(MHO/M) LENGTH OF THE RES0N4TOR=1.757(M) L F( HZ) R2/LAMBDA KO/BETA VP/VG QU ALPHA(DB/M) ALPHA(D8/FT) 95 7.8842E 09 4.1748E -01 9.7252E -01 1.0854E 00 1. 0081E 04 7.8611E -02 2.3961E -0 2 96 7.9606E 09 4.2153E -01 9.7171E -01 1.0865E 00 9. 8862E 03 8 .1105E =02 2.472LE -02 97 8 .0368E 09 4.2557E -01 9.7091E -01 1.0875E 00 9. 7002E 03 8.3622E -0 2 2.5488E -02 98 8.1130E 09 4.2960E -01 9.7010E -01 1.08S5E 00 9. 5222E 0 3 8 .6165b -02 2 .6263E -02 99 8.1889E 09 4.3362E -01 9.6930E -01 1.0895E 00 9. 3519E 03 8.8731E -02 2.7045E -0 2 100 8.2648E 09 4 .3764E -01 9.6849E -01 1.090 5E 00 9. 1886E 03 9.1321E -02 2.7835E -02 101 8.3405E 09 4.4165E -01 9.6769E -01 1.0915E 00 9. 0322E 03 9.3934E -0 2 2 .8631E -0 2 102 8.4161E 09 4.4565E -01 9.6689E -01 1.0924E 00 8. 8818E 03 9 .6573E -02 2.94366 -02 103 8 .4916E 09 4.4965E -01 9.6609E -01 1.0933E 00 8.7372E 03 9.9236E -02 3.0247E -0 2 104 8.5669E 09 4 .5364E -01 9.6529E -01 1.0942E 00 8. 5980E 03 1 .0192E -01 3 . 1067E -02 105 8 .6421E 09 4.5762E -01 9.6449E -01 1.0951E 00 8 .4641E 03 1.0463E -01 3 .1893E -0 2 106 8.7172E 09 4 .6160E -01 9.6369E -01 1.0960E 00 8. 3349E 03 L .0737E -01 3 .2727E -02 107 8.7922E 09 4.6557E -01 9.6289E -01 1.0968E 00 8. 2103E 03 1.1013E -01 3.3569E -0 2 108 8.8671E 09 4.6953E -01 9.6210E -01 1.0977E 00 8. 0898E 03 1 .1292E -01 3.4419E -02 109 8 .9418E 09 4.7349E -01 . 9.6131E -01 1.0985E 00 7.9735E 03 1.1574E -01 3.5277E -0 2 1 10 9.0164E 09 .4.7744E -01 9.6052E -01 1.0993E 00 7. 8611E 03 I.1857E -01 3.614^E -02 111 9.0909E 09 . 4.8138E -01 9.5973E -01 1.1001E 00 7. 7521E 03 1.2144E -01 3.7016E -0 2 112 9.1653E 09 4.8532E -01 9.5895E -01 1. 1009E 00 7. 6466E 03 1 .2433E -01 3.7897E -02 113 9.2396E 09 • 4.8926E -01 9.5816E =01 1.1017E 00 7. 5443E 0.3 1.2725E -01 3.8787E -0 2 114 9.3138E 09 .4 .9318E -01 9.5738E -01 1. 1025E 00 •7. 4451 E 03 1 .3020E -01 3.9685E -02 115 9 .3878E 09 • 4.9710E -01 9.5660E -01 / 1.1033E 00 7. 3489E 03 1.3317E -01 4.0591E -0 2 116 9.4617E 09 • 5.0102E -01 9.5582E -01 1.1041E 00 7. 2553E 03 1 .3617E -01 4. 1506E -02 117 9.5356E 09 5.0493E -01 9.5505E -01 . 1. 1048E 00 7. 1645E 03 1.3920E. -01 4.2429E -0 2 118 9.6093E 09 . 5 .0883E -01 9.5427E -01 1. 1056E 00 7. 0761E 03 .1 .4226E -01 4.3360E -02 119 9 .6829E 09 5.1273E -01 9.5350E -01 1.1063E 00 6. 9902E 03 1.4534E -01 4.4301E -0 2 120 9.7564E 09 5 .1662E -01 9.5273E -01 1.1071E 00 6. 9065E 03 1 .4846E -01 4.5250E -02 . Ln Table 3.1 Sample Output from QFACTOR 16 3.3 Relation Petween Unloaded Q and Loaded"Q In practice the loaded Q factor, Q , of a cavity resonator is given by ,3.9 where f is the resonant frequency and Af is the bandwidth at the half-power points of the transmission character istic. To determine the bandwidth Af, either the amplitude or the phase of the transmission characteristic of the reso nator can be used (figure 3.2"^). The unloaded Q can be obtained by measuring the loaded Q and the coupling coefficients of the cavity in put and output apertures. . In the case--where there are two coupling apertures, the unloaded Q .is given by % = V1+fW .....3.10 In the case where the output coupling $2 is negligibly small, equation 3.10 becomes Qu - Q^a+B-p ...3.11 To obtain the coupling coeffi cient 6^, it is necessary to measure the amplitude rv 1,0 \ 0.707 S i l -"^ —X . 1 r 1 \. f-f phase 90° 45° r Af 0 J f-f -* 45° X. r 90°' Figure 3.2 Transmission Characteristics of a Resonant Cavity overcoupled undercoupled /critically coupled R=0 R=°° Figure 3.3 Variation of Input Impedance of a Resonant Cavity 17 input impedance of the resonator at resonance. If the normalized resistive component R, which is equal to the coupling coefficient $ • is ' found to be greater than unity, the cavity is overcoupled. If R is found to be less than unity, the cavity is undercoupled and if R is found to be unity the cavity is critically coupled. 18 4. EXPERIMENTAL APPARATUS 4.1 Introduction Although dielectric-tube waveguides would be most advantageously used at millimeter-wave frequencies, it was more convenient to conduct the present investigation at X-band frequencies. 'This placed less stringent tolerance requirements on the dimensions of the tube, making it possible to use commercially available tubes. The general layout of the microwave apparatus is shown in figure 4.1. To improve the frequency stability of the X-13 klystron, the latter was water cooled and a klystron synchronizer (FEL Model 136-AF) was used. For measurement of the Q factor of the surface-wave resonator, it was necessary to measure the bandwidth Af of the resonator Q curve accurately. This was facilitated by use of a beat-frequency technique which made it possible to measure frequencies in the X-band range with an error of not more than ± 50 KHz. By comparison, the ordinary reaction type of cavity frequency meter has a typical accuracy of ± 1MHz in the same frequency range. Details of the components of the surface-wave resonator are given in the following sections. power supply-klystron synchronizer water isolator cooled JHF multiplier mixer \psc J x3-.8-4.2GHz •a digital counter 0.1 msec/cm attenuators coarse fine *y>A— frequency meter saw/cw modulation slotted section tapered field perturbing waveguide bead VL surface-wave resonator chart recorder Figure 4.1 Layout of Apparatus 20 4.2 Surface-Wave Resonator The surface-wave, resonator, shown in figures 4.2-4.5, consisted of a length of dielectric tube [1] approximately 1.78m long bounded at both ends by flat, circular, aluminum plates, 0.61m in diameter and 1;2 cm in thickness, mounted at right angles to the waveguide. Since it was desirable to use as long and as straight a tube as possible in order to obtain accurate measurements of the attenuation coefficient, it was necessary to devise some method of adequately supporting and tensioning the tube. This was achieved by passing the ends through holes in the end plates of the resonator and radially gripping the•tube walls between these plates and close fitting, circular, short-circuiting plugs, [2 and 3], inside the tube. Leakage of energy outside the resonator through the dielectric-filled, annular apertures thus formed in the end plates was prevented by the use of annular short-circuiting plungers [4] at each end of the dielectric tube. The end plates of the resonator were kept parallel and in alignment by four tie rods [5]. Alignment of the end plates was carried out using a laser in a manner similar to that used for aligning optical cavities. Table 4.1 shows details of the polythene tubes,used in the investigation. The other end plate of the resonator had a number of holes [7], 0.13 cm in diameter, lying along a radius of the plate, through which was inserted a small wire probe sensitive to the longitudinal component of the electric field within the resonator. By moving the probe from one sampling hole to another the radial field decay could be investigated. Normally, all the holes in the end plate, except the one containing the probe, were closed by tightly fitting aluminum plugs. * The numbers given in the text correspond to those appearing in figures 4.3-5. 21 For the measurement of the radial decay of the radial tube r1(cm) r2(cm) P=r1/r2 component of the electric field inside the resonator, I 0,953 1.270 . 0.750 another probe, mounted on a modified slotted-line car II 1.270 1.588 0.800 • riage, was moved radially III 1;588 1.905 0.833 across some cross-sectional plane inside the resonator. Table 4.1 Details of Polythene Tubes Figure 4.2 General View of Surface-Wave Resonator Figure 4.3 Surface-Wave Resonator scale 1/10 25 26 4.3 Mode Exciters Excitation of the dominant HE^ mode on the dielectric tube was achieved by means of a small circular aperture [6] fed by a circular waveguide, which also formed one of the tube tensioning plugs [3] mentioned in section 4.2. Excitation of the TE^ mode was achieved by replacing the annular short-circuiting plunger [4] at the input end of the resonator by two polystyrene-filled rectangular waveguides of transverse dimensions 0.8 cm by 1.3 cm (figure 4.7), which butted up against the exposed end of polythene tube III. The waveguides were excited 180° (figure 4,6.b) out of phase by using the set— . . up shown in figure 4.6.a. The dimensions of the tapered wave-quides were such as to equalize the phase JO 10 velocities of the TE mode of the exciter and the TE° mode of the surface waveguide. This arrangement could also be used to excite the dominant HE^ mode by feeding both dielec tric waveguides in phase. An alternative method of exciting the tapered waveguides -cavity end plate / / s ; ; / ; ;i phase shifter. / / / ; ? / s A 7 dielectric tube (a) (b) (c) Figure 4.6 Diagram of TE Mode Exciter and the Position of Exciter Relative to Dielectric Tube TE^ mode, that due to 27 3 Astrahan , is shown in Figure 2.6.C. This was tried in the present investi-r-gation, but proved to be unsatisfactory, since it excited both the HE.^ and TEQ^ modes simultaneously. For excitation of the TMQ^ mode, the circular waveguide and aperture were replaced by a section of coaxial line, having a tapered inner conductor. This is shown in figure 4.8. 29 RESULTS 5.1 Dependence of Cavity Q Factor on Size of Coupling Aperture Figure 5.1 shows the dependence of the cavity Q factor of the HE^ mode on coupling aperture size for •'• tube II at a frequency of 8.328 GHz. For aper ture sizes of less than about 5 mm, both the loaded Q factor and the unloaded Q factor became virtually constant and the coupling coefficient 3^ was smaller than 0.01. Hence the amount of cavity loading for this range of aperture sizes was negligible. Table 5.1 shows the actual size of aperture used with each particular size of tube. In all cases, the apertures were small enough to ensure that the errors in the measurements were small. tube diameter of aperture (mm) I 7.1 II 6.4 III 6.4 Table 5.1 Details of Coupling. Apertures 7000 6000 Q 5000 4000 • X 5 Diameter (mm) 10 Figure 5.1 Measured Dependence of Cavity Q Factor of HE^ Mode on Coupling Aperture Diameter, tube II, f=8.328GKz O experimental points for unloaded Q factor A experimental points for loaded Q factor 30 5.2 Measurement of Guide Wavelength Measurement of the wavelength* along the surface of the dielectric-tube mounted inside the resonator was carried out using a perturbation . . method similar to that described by Barlow and Karbowiak"''''". Essentially the method involved the determination of the number of half wavelengths contained in the length of the resonator when the latter was resonant at a known frequency. This was achieved with the aid of a small aluminum bead supported in close proximity to the dielectric waveguide by a cotton thread stretched transversely between tiro parallel nylon running cords mounted longitudinally outside the resonator and diametrically opposite one another. By simultaneous axial movement of the running cords the small bead was made to traverse the length of the resonator, remaining throughout at approximately the same distance from the dielectric-waveguide. While no appreciable disturbance of the field was produced by the cotton thread, some energy was scattered by the bead except when-it was situated at a node of the electric field. Thus the output of the probe connected to the resonator exhibited successive variations as the bead was moved along the dielectric waveguide, and it was only necessary to count the number of oscillations in the probe output in traversing the length of the resonator. The number of maxima corresponded to the number of nodes in the longitudinal field distribution of the reso nator and the wavelength was therefore determined. Thus, the accuracy of the method was dependent on the precision with which the length of the resonator could be measured. In the present investigation, this was achieved to an accuracy better than ± 1 mm, leading to an error in the measurement of guide wavelength of not more than 1 part in 1780. Figure 5.2 shows a typical probe output, obtained when the field perturbing 3 pu •U P O <u fi o r-l P. i • ! i •U- i-.Li,.l ..... r. ~ .1. 4..,.. • i ft-!' 0.5 L(m) 1.0 1.5 1.757 Figure 5.2 Measurement of Guide Wavelength of HE Mode by Field Perturbing Bead Method, Tube III, f=8.323 GHz, 4=101 32 bead was moved along the length of the resonator. From this, it was deduced that there were 101 half wavelengths in the length of the resonator at a frequency of 8.323 GHz. The measured and theoretically predicted variation of the guide wavelength of the HE^ mode with ^A is shown in figures 5.3.a-c for tubes I, II and III. The experimental, results agree well with particular theoretical curves computed for values of relative permittivity'in the range 2.26 to 2.31. (The exact dielectric properties of the commercially availabl polythene tubes used were not known.) As a check on the cutoff frequencies of the higher order TEQ-^ and TMQ^ modes, the latter were individually excited on tube III, using the appropriate mode exciter and the variation of guide wavelength with r^/X at frequencies close to cutoff was measured. Good agreement between experiment and theory was obtained using the particular value of relative permittivity for tube III found before, £^=2.26. These results are shox-m in figure 5.3.c Figure 5.3.b shows the effect of surrounding tube II by a low-density polyfoam shield of cross-sectional dimensions 50 cm by 50 cm. As can be seen, the dispersion characteristics of the shielded tube are little different from those of the unscreened tube. The experimental results agreed most closely with the theoretical ones when the relative permittivity of the polyfoam shield was assumed to be 1.041. 33 34 X X 0.95 0.9 0.4 0.5 r2/X 0.6 Figure 5.3.b Experimental and Theoretical Phase Characteristics of he-Q Mode on Shielded and Unshielded Polythene Tube II (i) unshielded tube O experimental points theoretical curve for e „=2.28 (ii) shielded tube • experimental points theoretical curve for e =2.28, e =1.041 35 r0/A =0.5285 2 c vj\ =0.7380 2 c 1.0 0.95 0.9 Figure 5.3.c Experimental and Theoretical Phase Characteristics of EE , TEQ1 and TMQ1 Modes on Polythene Tube III (i) HE1]L mode. O experimental points theoretical curve for er2=2.26 •(ii) TEQ1 mode ' O experimental points theoretical curve for e 0=2.26 r2 (iii) TMQ1 mode • experimental points theoretical curve for e „=2.26 • r2 36 5.3 Measurement of Radial Decay of Electric Field As a check on the dispersion characteristics of the HE^ mode on the-unscreened tube, the radial decay of the longitudinal and radial components of the electric field was measured when the resonator was resonant in the HE, , .. „~ mode at a frequency of 8.328 GHz. The results obtained for tube II are plotted in figures 5.4.a-b together with the theoretical curves computed from-equations 2.2.c for e =2.28. 37 experimental points theoretical curve for e =2.28 . 38 •5.4 Measurement of Attenuation Coefficient The cavity-resonance method was used to"measure the relatively small attenuation coefficient of the polythene-tube waveguide. The eval uation of the attenuation coefficient from the measured Q factor of the reso nator was carried out using equation 3.6. The measured and theoretically predicted variation of the atten uation coefficient with r^/\ of the HE^ mode is shown in figures 5.5.a-c for the same polythene tubes used previously. Assuming that the tubes had the values of relative permittivity found in section 5.3, it was found that the experimental points agreed with the theoretical curves for loss tangents in the range 0.00058 to 0.00085. Figure 5.5.b shows the effect of surrounding tube II by the polyfoam shield. It can be seen that the attenuation coeffi cient of the shielded tube is only slightly higher than that of the unshielded tube. The experimental results agree best with the theoretical ones when the loss tangent is taken to be 0.00007. 40 0.4 0.3 a (dB/m) 0.2 0.1 ^^^^^ 0.4 0.5 r2/X 0.6 Figure 5.5.b Experimental and Theoretical Attenuation Cha Tub (i) (ii) Characteristics of HE Mode on Shielded and Unshielded Polythene Tube n unshielded tube -O experimental points theoretical curve for er2=2.28, tan62=0.00068 shielded tube • experimental points theoretical curve for e tan63=0.00007 _ 2.28, e =1.041, r2 r3 . 6. CONCLUSIONS Accurate measurements of the attenuation coefficient of the HE^ mode on certain dielectric-tube waveguides have been made using a cavity-reso nance method. The results obtained confirm previous theoretical predic tions although there is an element of uncertainty concerning the exact dielectric properties of the commercial grade polythene tubes used. The measurements also yielded the phase coefficient of the mode which was confirmed by measurement of radial decay of the electric field outside the tube. Enclosing the dielectric tube in a low-density, low-loss poly-foam shield resulted in only a slight degradation of the attenuation characteristics of the waveguide. Areas in which future theoretical and experimental work on dielectric tube waveguides might be carried out include the following: (i) Effects of discontinuities and bends on the propagation characteristics of the HE^ mode (ii) Measurement of v of the HE mode g 11 (iii) Coupling between waveguides (iv) Development of efficient mode exciters and filters (v) Extension of all these topics to millimeter-wave frequencies 43 REFERENCES 1. Zachoval, L. , "'Elektromagnetische Wellen an dielektrischen Rohren",-Ceska Akademic Ved a Umeni Praze Bulletin International, 1932, Vol. 33, P. 136. 2. Liska, J., "Elektromagneticke Vlyn na dielektricych Trubicich", Casopis pro Pestovani Maternatiky a Fysiky, 1934, Vol. 63, P. 97. 3. Astrahan, M.M., "Dielectric Tube Waveguides", Ph.D. Dissertation, Northwestern University, Illinois, 1949. 4. Jakes, W.C.,"Attenuation arid Radiation Characteristics of Dielectric Tube Waveguides", Ph.D. Dissertation, Northwestern University, Illinois, 1949. 5. Unger, H., "Di'elektrische Rohre als Wellenleiter", Archiv der Elektrischen Ubertragung, 1954, Vol. 8, P. 241. 6. Mallach, "Untersuchungen an dielektrischen Wellenleitern in Stab- und Rohrform", Fernraeldetech Z., 1955, Vol. 8, No. 1, P. 8. 7. Kharadly, M.M.Z., and Lewis, J.E., "Properties of Dielectric-Tube Waveguides", Proc. IEE, 1969, Vol. 116, No. 2, P. 214-224. 8. Bourk, T.R., Kharadly, M.M.Z., and Lewis, J.E., "Measurement of Wave-quide Attenuation by Resonance Methods", Electronics Letters, 1968, Vol. 4, No. 13, P. 267-8. 9. Schiebe, E.H., King, B.G., Van Zeeland, D.L., "Loss Measurement of Surface Wave Transmission Lines", Journal of Applied Physics, 1954, Vol. 25, P. 790. 10. Sucher, M., and Fox, J., "Handbook of Microwave Measurement", Vol. 3, P. 470, Polytechnic Press, 1963. 11. Barlow, H.M., and Karbowiak, A.E., "An Investigation of the Character istics of Cylindrical Surface Waves", Proc. IEE, 1953, Vol. 100, Pt. Ill, P. 321. 

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