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

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MEASUREMENT OF THE PROPAGATION CHARACTERISTICS OF SHIELDED AND UNSHIELDED DIELECTRIC-TUBE WAVEGUIDES . by IKUFUMI MAKINO B.Sc , Doshisha U n i v e r s i t y , 1967 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 t h i s thesis as conforming to the required standard Research Supervisors • . > «... Members of the Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make 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 s t u d y . I f u r t h e r a g r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s 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 by t he Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Beefarieg.) tBno^/veerC The U n i v e r s i t y o f B r i t i s h C o l u m b i a Vancouve r 8, Canada Date December 1' , 1310 i i A B S T R A C T 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 f i e l d 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 T E Q ^ and T M Q ^ modes on the tube at frequencies close to cutoff are also reported. i i i TABLE OF CONTENTS Page ABSTRACT . . i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS ... . i v LIST OF TABLES . v i LIST OF SYMBOLS v i i ACKNOWLEDGEMENT i v 1. INTRODUCTION 1 : 2. SURFACE-WAVE PROPAGATION ON DIELECTRIC-TUBE WAVEGUIDES 3 2.1 F i e l d Components 3 2.2 Mode Spectrum 6 2.2.1 C h a r a c t e r i s t i c Equations 6 2.2.2 Cutoff Conditions 9 3. CAVITY-RESONANCE METHOD OF MEASURING ATTENUATION , 11 3.1 Introduction 11 3.2 Relation Between Attenuation C o e f f i c i e n t and Q Factor ... 11 3.2.1 Relation Between Attenuation C o e f f i c i e n t and Q. Factor f o r Surface-Wave Resonator 13 3.3 Relation Between Unloaded Q and Loaded Q 16 4. EXPERIMENTAL APPARATUS 18 4.1 Introduction 18 4.2 Surface-Wave Resonator 20 4.3 Mode E x c i t e r s 26 5. RESULTS 29 5.1 Dependence of Cavity Q Factor on Size of Coupling Aperture 29 5.2 Measurement of Guide Wavelength 30 5.3 Measurement of Radial Decay of E l e c t r i c F i e l d 36 5.4 Measurement of Attenuation C o e f f i c i e n t 38 6. CONCLUSIONS 42 REFERENCES 43 i v LIST OF ILLUSTRATIONS Figure . Page 2.1 The D i e l e c t r i c Tube Waveguide 3 2.2 Mode Spectrum of Polythene Tube, p=0.5 8 2.3 Cutoff Conditions; TE Q 1, TM Q 1, EH and HE 1 2 Modes 10 3.1 Characteristics of D i e l e c t r i c 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 Po s i t i o n of Exciter Relative to D i e l e c t r i c Tube 26 4. 7 T E Q 1 Mode Exciter 28 4.8 TM Q 1 Mode Exciter 28 5.1 Measured Dependence of Cavity Q Factor of HE^ Mode on Coupling Aperture Diameter 29 5.2 Measurement of Guide Wavelength of HE^ Mode by F i e l d Perturbing Bead Method 31 5. 3.a Experimental and Theoretical Phase Characteristics of HE.. 1 Mode on Polythene Tube I 33 5.3.b Experimental and Theoretical Phase Characteristics of HE.^ Mode on Shielded and Unshielded Polythene Tube I I 34 5.3. c Experimental and Theoretical Phase Characteristics of HE.. Mode on Polythene Tube I I I 35 5.4. a Radial Decay of E z 3 for HE^ Mode 36 5.4.b Radial Decay of E , for HE-. Mode 37 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 4 1 LIST OF TABLES Table 3.1 Sample Output from QFACTOR . 4.1 Details of Polythene Tubes .. 5.1 Details of Coupling Apertures v i i LIST OF SYMBOLS a., b . = constants A (p..), B (p..) = functions of Bessel functions c. = j b ± / a ± ^ z i ' ^rL' ^ f l i = ^- o n^^ u^ n a^-» r a d i a l , azimuthal components of e l e c t r i c f i e l d , 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 z i r i 61 f i e l d , respectively, in medium i I (p..) = modified Bessel function of the f i r s t kind n * i j J (p..) = Bessel function of the f i r s t kind n * i j 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. i j 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 v i i i 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 V Q , 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 3̂  = coupling coefficient A f = bandwidth e . = relative permittivity of medium i r i 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 f i n a l draft. 1. INTRODUCTION During the l a s t f o r t y years or so, many inv e s t i g a t o r s have considered the problem of surface-wave propagation along d i e l e c t r i c tubes. In 1932, Zachoval"'" obtained the c h a r a c t e r i s t i c equation for T M Q ^ modes and solved t h i s g r a p h i c a l l y f o r a range of tube parameters. 2 Two years l a t e r , the existence of these modes was v e r i f i e d by L i s k a , whose measurements of guide wavelength showed good agreement with Zachovai's theory. In 1949, Astrahan obtained the c h a r a c t e r i s t i c equations for TE and hybrid modes and measured values of guide wavelength f o r the HE^, T M Q ^ and TE modes which agreed very w e l l with theory. At 4 about the same time, Jakes gave expressions for the attenuation c o e f f i - cients of TM„ and TE„ modes and measured the attenuation of the TM' Om Om 01 and TEQ-^ modes on polystyrene tubes. A technique f o r obtaining the attenuation c o e f f i c i e n t of any mode was outlined by Unger^ i n 1954 using a method s i m i l a r to Jakes', but the analysis was completed only f o r the 6 ^ 1 1 m o c * e o n tubes with small diameter to wavelength r a t i o s . Mallach made a rough estimate of the attenuation of the H E - Q mode by measuring the radius at which the magnitude of the e l e c t r i c f i e l d f e l l to 1/e of i t s value at the tube surface. In 1968 Kharadly and Lewis'' completed a comprehensive study of the pos s i b l e usefulness of the d i e l e c t r i c tube as a low-loss waveguide. They concluded that a moderately thin-walled tube propagating the dominant mode could have propagation c h a r a c t e r i s t i c s g r e a t l y superior to those of conventional m e t a l l i c waveguides at m i l l i m e t e r - wave frequencies. Also, they proposed a method f or 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 i n a layer of low-density, low-loss d i e l e c t r i c of s u f f i c i e n t r a d i a l extent that a ne g l i g i b l e portion of the wave was carried outside t h i s d i e l e c t r i c . So far as i s known, no accurate measurements of the attenua- t i o n c h a r a c t e r i s t i c s of the dominant HE mode on dielectric-tubes have been made. This seems surprising i n view of the fact that t h i s mode i s the one most l i k e l y to be used i n practice. The objectives of the investigation reported here were therefore: (i) to obtain experimental data on the attenuation and phase coe f f i c i e n t s of the HE mode on commercially available polythene tubes from direct measurements, using the cavity-resonance method. ( i i ) to ascertain experimentally the effect of shielding the tube with low-density, low-loss polyfoam. Chapter 2 reviews b r i e f l y some of the features of surface-wave propagation on diele c t r i c - t u b e waveguides. The theory i n this chapter i s drawn from reference 7. In Chapter 3, the theory underlying the cavity- resonance method for measuring the attenuation c o e f f i c i e n t of low-loss waveguides i s discussed. This i s followed i n Chapter 4 by a description of the experimental apparatus used. Experimental results for the propagation ch a r a c t e r i s t i c s of the HE^ mode on-polythene tubes i n a i r and surrounded by a polyfoam sh i e l d are given i n Chapter 5, together with results for the phase c o e f f i c i e n t of the t e Q ^ and TM ^ modes at frequencies close to cutoff. Conclusions drawn from this investigation and suggestions for further work are contained i n Chapter 6. 3 2. SURFACE-WAVE PROPAGATION ON DIELECTRIC TUBE WAVEGUIDES 2.1 F i e l d Components The tube configuration of i n t e r e s t i s shown i n f i g u r e 2.1. It consists of two c o a x i a l d i e l e c t r i c regions of i n f i n i t e length and r e l a t i v e p e r m i t t i v i t i e s e , and e „ embedded i n a t h i r d i n f i n i t e d i e l e c t r i c v r l r2 of r e l a t i v e p e r m i t t i v i t y e r3> where and e r 2 > e r l e _ > e -r2 r3 _ .2.1 In a l l cases, i t w i l l be assumed that the r e l a t i v e permeability of the region, u ., i s unity. Propagation i s assumed in'the z - d i r e c t i o n , r i ' with t-9-z dependence of the form exp j(wt-n0.-6z) i n the l o s s l e s s case. "r3 Figure 2.1 The Dielectric-Tube Waveguide 4 Under these conditions, omitting the factor exp j(wt-n6-$z), the f i e l d components are given by E z l = a l V h l r ) R ' n y r i k n z n E r l = J hT a l W > + \ \ I n V ) 1 h, r 01 2_ "1 n ^ l h l r H z l = b l V h l r > H r l 1 n 1 n e , k — a i W ) + ^ b i V h i r ) h l r Z0 e k H e i = J h f i~ a i i ^ + f ~ h W > 1 0 1 r E z 2 = a2 J Our) H Y (h.r) n Z n 2 « a 2 A n ( h 2 r ) o i • ' nu _ k Z E r 2 = ll a2 V h 2 r ) " [l b2 VV> 2 h2 r • - n 6 * /v \ i y r 2 k 0 Z 0 . .» . • '82 " " TT a2 V h 2 r ) + J h ~ b2 V h 2 r ) 2 ^ ^ H z 2 = b2 J (h r) + r 1 Y ( h . r ) n 2 b 2 n 2 = b 2 B n(h 2r) \ 2 = ?7 a 2V h 2 r ) - j f b 2 VV» h2 r Z Q 2 £ k H62 = "J zfhT a2 A n ( h 2 r ) " X " b2 B n ( h 2 r ) 0 2 h2 r 0 < r < ,2.2.a r l ± r ± r2 ,2.2.b Jz3 r3 a 3 K n(h 3r) j K ' (h„r) +' n y r 3 9 ° ° b„ K (h,r) h „ 3 n i ,2 J n 3 3 h« r n^ v tu \ - y r 3 K0 Z0 , ' , . a_ K (h r) - 3 b 0 K_(h,r) z3 .2 3 n v 3 r b.K (h„r) 3 n 3 3 n 3 H = -r3 a3 K n ( h 3 r ) + J IT b3 K n ( h 3 r ) h 3 r Z Q 3 H 63 J 1^a3K>3r)+7|--b3Kn(h3r) 3 0 h^ r r 2 < r < «> .2.2.c where, from the wave equation 2 2 2 h- = 3 - u , * e , k n 1 " r l r l 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 l i s t of symbols. Upon setting n=0 (no 6-variation), equations 2.2 separate into two sets corresponding to the circularly symmetric modes designated T M Q ^ 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 ; i f the TM component i s dominant, nm i t i s termed EH . The nature of TE or TM dominance and the significance nm of the subscript m in the mode designation i s discussed f u l l y in reference 7. 6 2.2 Mode Spectrum 2.2.1 Characteristic Equations By matching the axial and tangential f i e l d components in media 1 and 2 at r=r^ and those in media 2 and 3 at *=*2> e l 8 n t homogeneous equations in eight unknowns, a.̂ ,.b̂ , i=l-4, are obtained. These may be solved to give the following characteristic equations for the hybrid modes: « r i / £ r 2 A n ( p 2 2 ) , £ r 3 K n ( p 3 2 ¥ r 2 B n ( P 2 2 ) + ^ _ n _ ^ _ \P 22 An ( p22 ) P32 Kn ( p32 ) A P22 Bn ( p22 ) P32 Kn ( p32 ) y r 3 K n ( p 3 2 ) 32yJ ,2.4.a and ^ £r2 An ( p21 ) , W n ^ ' lP 2l A n ( p21 ) P l l I n ( p l l ) ^ f y r 2 B n ( p 2 1 ) + ' p21Bn ( p21 ) Pll I„<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 u r^- The character- i s t i c 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 i n free space A (e =£ =y =y =y =1 e „=2.26, and P=r../r =0.5) is shown in figure 2.2. r l r3 r l r2 r3 r2 1 2 The main features of the mode spectrum are: (i) The HE^^ mode has no lower cutoff frequency. ( i i ) Unlike the case for the dielectric rod (p=0), the T E Q m and TM„ modes do not have the same value of r„/X at cutoff. This i s Om I also true for HE. ,. and EH. modes. l,m+l lm. ( i i i ) As P"KL, the phase characteristics of the TE^ and HE^ mode become indistinguishable, as do those of the TM^ and E H ] _ m m°des, thus 1 CN ^ — N r H r H CN CN ft ft • w - c a >< CN r H i-l CN (0 ( X + I—1 r H r H r H P . ft - a c M t—1 r H r H r l r H (J ft r H CN *—* ft CN CN c ft — ' c r H >-) CN * V ft r H CN c ft ' CN CN ft CN ft CN U 3- CN ft CN ft CN ft CN ft + CN ft CN ft *-> r H CN ft I CN ft CN ft CN ft CN CN ft CN I C N CN CO ft CN CN CN ft 8 providing the physical d i s t i n c t i o n between HE and EH modes. (iv) As p-KL, the n=0 and n=l modes appear i n widely separated cl u s t e r s , each cluster consisting of four modes ( H E l m> T E o m ' ™0m a n d E H l m ^ ' (v) The HE, and TE. phase char a c t e r i s t i c s intersect at some lm Om value of . In most cases, for values of r^/A greater than that at the inter s e c t i o n , the differences i n the two curves are too small to be seen graphically. However, the degeneracy of the HE^ a n d T E02 m o d e s ^ o r san be seen i n 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 a l l 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 r l r l 3 1 generally, when P-j^O a n c^ 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. J 0 ( p 2 2 ) Y Q(P 2 2) £ r l P21 JQ (P21 ) " 2 e r 2 : J l ( ? 2 1 ) ^ A TR—-, s 7, ~—: x TM modes £ r l p21 Y0 ( p21> " 2 £ r 2 \<»2J P21 J 0 ( P 2 1 ) - 2 J l ( p 2 1 } p21 Y 0(p 2 1) -.2Y l (p 2 ] L) TE modes 2.6.a P 2 2 - 0 J 1 ( P 2 2 ) J 1 ( P 2 1 ) Y l ( p 2 2 ) Y r ( p 2 1 ) HE^^ mode HE, modes lm m > 1 ,2.6.b J 1 ( P 2 1 ) _ J1 ( P22 } Y l ( p 2 i y Y l ( p 2 2 ) J "J1(P21> J 1 ( P 2 2 ) Y1 ( P21> Y1 (P22 } '21 P21 £ r l (e ,+e „) r l r2' EH, modes lm m >. 1 ,2.6.c At cutoff p 2 2 is given by p22=2.. , x ) Jl M r 2 e r 2 _ y r 3 e r 3 } .2.7 from which the value of r_/X can be determined. 2 c The variation of r 0/X with P for the TE.,, TMni, EH and HE modes on L C Ul U± i i 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 s u i t a b l e for d i r e c t l y measuring the small attenuation c o e f f i c i e n t of the H E ^ Q mode on d i e l e c t r i c - t u b e waveguides. The main advantages of the method are that only a f a i r l y short length of waveguide i s needed and the problems of accurate measurement of power l e v e l s or substituted attenuation are avoided. The r e l a t i o n s h i p between attenuation c o e f f i c i e n t and the Q f a c t o r of a c a v i t y formed from a s e c t i o n of the waveguide and two m e t a l l i c end plates i s discussed i n the next sec t i o n . 3.2 Relation Between Attenuation C o e f f i c i e n t and Q Factor Adopting the nomenclature of reference 7, the Q f a c t o r of the resonator xs given by I T T w L N / v 1 1 _j^WL_ = g S ....................... 3.1 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 gv p coWL where 3 = to/v =2TT/X P • g For very long resonators, the second term i n equation 3.2 can be neglected and the expression f o r Q becomes <; M r ) •• •••••••••• 3-3 g 12 Then the attenuation c o e f f i c i e n t a i s given by ,3.4 4 9 In previous experimental i n v e s t i g a t i o n s of surface waveguides ' , values of a have been obtained by measuring Q and g and using the trans- m i s s i o n - l i n e formula, a = .3.5 2Q which assumes v /v =1 i n equation 3.4. This assumption may lead to s i g n i f i c a n t e r r o r s . As an example, the f a c t o r v /v f o r the dominant P g H E ^ mode on a polythene tube (e^^=2.26) waveguide has been completed and i s shown i n fi g u r e 3.1. Inspection of t h i s f i g u r e shows that equation 3.5 i s v a l i d f o r such waveguides when the phase-velocity reduction i s very small or very l a r g e , but can lead to appreciable errors f o r intermediate values. In the present i n v e s t i g a t i o n , no p r o v i s i o n was made for measuring v and hence the r a t i o v /v together with the term i n equation g P g . 3.2 i n v o l v i n g 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 r 2/X 1.2 1.6 2.0 Figure 3.1 C h a r a c t e r i s t i c s of D i e l e c t r i c - T u b e Waveguide, H E ^ Mode 13 3.2.1 Relation Between Attenuation Coefficient and Q Factor for Surface-Wave Resonator From equation 3.2, a _ l A L Y I _ 2Np\ * / M A _ . 2 ( V N P 2 + V \ 2 \ , v J\q wWl7 2 \ . v g / \ Q u L O ^ + W ^ W g ) . / 3.6 where, for the dominant HE^^ mode, W a2a2 v e r l fAl (p21} 1 4 1 2 4v„Z„h, \I^(p 1 ; L) W 0 0 1 _ a3 a2 f f er2 2 4 4V0h2 L h: T i + & 2 + ( k o V i ) 2 ] h-^owl^ h2 V B 2 s A + ( k 0 Z 0 C 2 ) 2 s B + 4 g k 0 Z 0 C 2 S A B W = 3 a 2 a 2 T T e r 3 / A x ( P 2 2 ) 4V0h3 W<P32>, h^ T K + [ 3 2 + ( k 0 Z 0 c / ] s K + 4 B k o Z 0 c 3 K 2 (p 3 2) .3.7 ^ V ^ n /A1(P21)N P l S T + C ? 3 2 S T - 4 C L 3 ( : ° R L I I I 1 \ z L> Z0 ^ l P N 2a 0a 0 fTR 2 2 m P2 ~ z r J V c 2 3 [ Z T , — : v 4 c 2 " Jo J ^ 0 e r 2 y A l ( p 2 1 ) ' 0 B l ( p 2 1 ) J. AB N 2 a 2 a 2 7 T R m / ' A l ( p 2 2 ) 1 p3 K l ( p 3 2 ) , Vr3 u zo K 1 ( p 3 2 ) .3.8 The functions S T, S,,S . S, S,„ T T , T, and T,r are integrals of functions I A B AB' K' I A K of Bessel functions which are defined and evaluated i n 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 c o e f f i c i e n t of the P 8 .14 dominant mode by sol v i n g equation 2.4.b for ^ ( g ) , where the. phase c o e f f i c i e n t 0 was decided by the number of h a l f wavelengths contained i n the length of the c a v i t y . 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( H Z ) R 2 / L A M B D A KO/BETA V P / V G QU A L P H A ( D B / M ) A L P H A ( D 8 / F T ) 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 p r a c t i c e the loaded Q f a c t o r , Q , of a cavity resonator i s given by ,3.9 where f i s the resonant frequency and Af i s the bandwidth at the half-power points of the transmission character- i s t i c . To determine the bandwidth Af, e i t h e r the amplitude or the phase of the transmission c h a r a c t e r i s t i c of the reso- nator can be used (figure 3.2"^). The unloaded Q can be obtained by measuring the loaded Q and the coupling c o e f f i c i e n t s of the ca v i t y i n - put and output apertures. . In the case-- where there are two coupling apertures, the unloaded Q .is given by % = V 1 + f W .....3.10 In the case where the output coupling $2 i s n e g l i g i b l y small, equation 3.10 becomes Q u - Q^a+B-p ...3.11 To obtain the coupling c o e f f i - c i e n t 6^, i t i s 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 C h a r a c t e r i s t i c s of a Resonant Cavity overcoupled undercoupled / c r i t i c a l l y coupled R=0 R=°° Figure 3.3 V a r i a t i o n 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 i s undercoupled and i f R i s found to be unity the cavity i s c r i t i c a l l y coupled. 18 4. EXPERIMENTAL APPARATUS 4.1 Introduction Although d i e l e c t r i c - t u b e waveguides would be most advantageously used at millimeter-wave frequencies, i t was more convenient to conduct the present i n v e s t i g a t i o n at X-band frequencies. ' T h i s placed le s s stringent tolerance requirements on the dimensions of the tube, making i t po s s i b l e to use commercially a v a i l a b l e tubes. The general layout of the microwave apparatus i s shown i n f i g u r e 4.1. To improve the frequency s t a b i l i t y of the X-13 klystron, the l a t t e r was water cooled and a klystron synchronizer (FEL Model 136-AF) was used. For measurement of the Q f a c t o r of the surface-wave resonator, i t was necessary to measure the bandwidth Af of the resonator Q curve accurately. This was f a c i l i t a t e d by use of a beat-frequency technique which made i t po s s i b l e to measure frequencies i n the X-band range with an error of not more than ± 50 KHz. By comparison, the ordinary r e a c t i o n type of ca v i t y frequency meter has a t y p i c a l accuracy of ± 1MHz i n the same frequency range. D e t a i l s of the components of the surface-wave resonator are given i n the following sections. power supply- klystron synchronizer water isolator cooled JHF multiplier mixer \psc J x3- .8-4.2GHz •a d i g i t a l counter 0 . 1 msec/cm attenuators coarse fine *y>A— frequency meter saw/cw modulation slotted section tapered f i e l d perturbing waveguide bead V L 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 f l a t , circular, aluminum plates, 0.61m in diameter and 1;2 cm in thickness, mounted at right angles to the waveguide. Since i t was desirable to use as long and as straight a tube as possible in order to obtain accurate measurements of the attenuation coefficient, i t 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 f i t t i n g , circular, short-circuiting plugs, [2 and 3], inside the tube. Leakage of energy outside the resonator through the d i e l e c t r i c - f i l l e d , 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 t i e 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 f i e l d within the resonator. By moving the probe from one sampling hole to another the radial f i e l d decay could be investigated. Normally, a l l the holes in the end plate, except the one containing the probe, were closed by tightly f i t t i n g 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 r 1(cm) r 2(cm) P=r 1/r 2 component of the electric f i e l d 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 Ex c i t e r s E x c i t a t i o n of the dominant H E ^ mode on the d i e l e c t r i c tube was achieved by means of a small c i r c u l a r aperture [6] fed by a c i r c u l a r waveguide, which also formed one of the tube tensioning plugs [3] mentioned i n s e c t i o n 4.2. E x c i t a t i o n of the T E ^ mode was achieved by re p l a c i n g the annular s h o r t - c i r c u i t i n g plunger [4] at the input end of the resonator by two p o l y s t y r e n e - f i l l e d 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° (figu r e 4,6.b) out of phase by using the set— . . up shown i n figu r e 4.6.a. The dimensions of the tapered wave- quides were such as to equalize the phase JO 10 v e l o c i t i e s of the TE mode of the e x c i t e r and the TE° mode of the surface waveguide. This arrangement could also be used to excite the dominant H E ^ mode by feeding both d i e l e c - t r i c waveguides i n phase. An a l t e r n a t i v e method of e x c i t i n g the tapered waveguides - c a v i t y end plate / / s ; ; / ; ;i phase s h i f t e r . / / / ; ? / s A 7 d i e l e c t r i c tube (a) (b) (c) Figure 4.6 Diagram of TE Mode E x c i t e r and the P o s i t i o n of E x c i t e r R e l a t i v e to D i e l e c t r i c Tube T E ^ mode, that due to 27 3 Astrahan , i s shown in Figure 2 . 6 . C . This was tried in the present investi-r- gation, but proved to be unsatisfactory, since i t excited both the H E . ^ and T E Q ^ modes simultaneously. For excitation of the T M Q ^ 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 virtua l l y constant and the coupling coefficient 3̂  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 a l l 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 d i e l e c t r i c - tube mounted i n s i d e the resonator was c a r r i e d out using a perturbation . . method s i m i l a r to that described by Barlow and Karbowiak"''''". E s s e n t i a l l y the method involved the determination of the number of h a l f wavelengths contained i n the length of the resonator when the l a t t e r was resonant at a known frequency. This was achieved with the aid of a small aluminum bead supported i n close proximity to the d i e l e c t r i c waveguide by a cotton thread stretched transversely between tiro p a r a l l e l nylon running cords mounted l o n g i t u d i n a l l y outside the resonator and d i a m e t r i c a l l y opposite one another. By simultaneous a x i a l 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 d i e l e c t r i c - waveguide. While no appreciable disturbance of the f i e l d was produced by the cotton thread, some energy was scattered by the bead except when- i t was s i t u a t e d at a node of the e l e c t r i c f i e l d . Thus the output of the probe connected to the resonator exhibited successive v a r i a t i o n s as the bead was moved along the d i e l e c t r i c waveguide, and i t was only necessary to count the number of o s c i l l a t i o n s i n the probe output i n t r a v e r s i n g the length of the resonator. The number of maxima corresponded to the number of nodes i n the l o n g i t u d i n a l f i e l d d i s t r i b u t i o n of the reso- nator and the wavelength was therefore determined. Thus, the accuracy of the method was dependent on the p r e c i s i o n with which the length of the resonator could be measured. In the present i n v e s t i g a t i o n , t h i s was achieved to an accuracy better than ± 1 mm, leading to an error i n the measurement of guide wavelength of not more than 1 part i n 1780. Figure 5.2 shows a t y p i c a l probe output, obtained when the f i e l d perturbing 3 pu •U P O <u fi o r-l P. i • ! i •U- i-.Li,.l ..... r . ~ .1. 4..,.. • i f t - ! ' 0.5 L(m) 1.0 1.5 1.757 Figure 5.2 Measurement of Guide Wavelength of HE Mode by F i e l d Perturbing Bead Method, Tube I I I , f=8.323 GHz, 4=101 3 2 bead was moved along the length of the resonator. From this, i t was deduced that there were 1 0 1 half wavelengths in the length of the resonator at a frequency of 8 . 3 2 3 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 , I I and I I I . The experimental, results agree well with particular theoretical curves computed for values of relative permittivity'in the range 2 . 2 6 to 2 . 3 1 . (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 T M Q ^ modes, the latter were individually excited on tube I I I , 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 I I I found before, £ ^ = 2 . 2 6 . These results are shox-m in figure 5 . 3 . c Figure 5 . 3.b shows the effect of surrounding tube I I by a low- density polyfoam shield of cross-sectional dimensions 5 0 cm by 5 0 cm. As can be seen, the dispersion characteristics of the shielded tube are l i t t l e 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 . 0 4 1 . 33 34 X X 0.95 0.9 0.4 0.5 r 2/X 0.6 Figure 5.3.b Experimental and T h e o r e t i c a l Phase C h a r a c t e r i s t i c s of he-Q Mode on Shielded and Unshielded Polythene Tube I I ( i ) unshielded tube O experimental points t h e o r e t i c a l curve f o r e „=2.28 ( i i ) shielded tube • experimental points t h e o r e t i c a l curve f o r e =2.28, e =1.041 35 r 0/A =0.5285 2 c vj\ =0.7380 2 c 1.0 0.95 0.9 Figure 5.3.c Experimental and T h e o r e t i c a l Phase C h a r a c t e r i s t i c s of EE , T E Q 1 and TM Q 1 Modes on Polythene Tube I I I (i) HE 1 ] L mode. O experimental points t h e o r e t i c a l curve f o r e r2 =2.26 • ( i i ) T E Q 1 mode ' O experimental points t h e o r e t i c a l curve f o r e 0=2.26 r2 ( i i i ) TM Q 1 mode • experimental points t h e o r e t i c a l curve f o r 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 f i e l d 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, i t 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 c o e f f i - 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 r 2/X 0.6 Figure 5.5.b Experimental and T h e o r e t i c a l Attenuation Cha Tub (i) ( i i ) a r a c t e r i s t i c s of HE Mode on Shielded and Unshielded Polythene e n unshielded tube - O experimental points t h e o r e t i c a l curve f o r e r 2=2.28, tan6 2=0.00068 shielded tube • experimental points t h e o r e t i c a l curve f o r e tan63=0.00007 _ 2.28, e =1.041, r2 r3 .  6 . CONCLUSIONS Accurate measurements of the attenuation c o e f f i c i e n t of the H E ^ mode on c e r t a i n d i e l e c t r i c - t u b e waveguides have been made using a cavity-reso- nance method. The r e s u l t s obtained confirm previous t h e o r e t i c a l p r e d i c - tions although there i s an element of uncertainty concerning the exact d i e l e c t r i c properties of the commercial grade polythene tubes used. The measurements also y i e l d e d the phase c o e f f i c i e n t of the mode which was confirmed by measurement of r a d i a l decay of the e l e c t r i c f i e l d outside the tube. Enclosing the d i e l e c t r i c tube i n a low-density, low-loss poly- foam s h i e l d r e s u l t e d i n only a s l i g h t degradation of the attenuation c h a r a c t e r i s t i c s of the waveguide. Areas i n which future t h e o r e t i c a l and experimental work on d i e l e c t r i c tube waveguides might be c a r r i e d out include the following: ( i ) E f f e c t s of d i s c o n t i n u i t i e s and bends on the propagation c h a r a c t e r i s t i c s of the H E ^ mode ( i i ) Measurement of v of the HE mode g 11 ( i i i ) Coupling between waveguides (iv) Development of e f f i c i e n t mode e x c i t e r s and f i l t e r s (v) Extension of a l l these topics to millimeter-wave frequencies 43 REFERENCES 1. Zachoval, L. , "'Elektromagnetische Wellen an d i e l e k t r i s c h e n Rohren",- Ceska Akademic Ved a Umeni Praze B u l l e t i n I n t e r n a t i o n a l , 1932, Vo l . 33, P. 136. 2. L i s k a , J . , "Elektromagneticke Vlyn na d i e l e k t r i c y c h T r u b i c i c h " , Casopis pro Pestovani Maternatiky a Fysiky, 1934, V o l . 63, P. 97. 3. Astrahan, M.M., " D i e l e c t r i c Tube Waveguides", Ph.D. D i s s e r t a t i o n , Northwestern U n i v e r s i t y , I l l i n o i s , 1949. 4. Jakes, W.C.,"Attenuation arid Radiation C h a r a c t e r i s t i c s of D i e l e c t r i c Tube Waveguides", Ph.D. D i s s e r t a t i o n , Northwestern U n i v e r s i t y , I l l i n o i s , 1949. 5. Unger, H., "Di'elektrische Rohre als W e l l e n l e i t e r " , Archiv der Ele k t r i s c h e n Ubertragung, 1954, Vo l . 8, P. 241. 6. Mallach, "Untersuchungen an d i e l e k t r i s c h e n W e l l e n l e i t e r n i n Stab- und Rohrform", Fernraeldetech Z., 1955, Vo l . 8, No. 1, P. 8. 7. Kharadly, M.M.Z., and Lewis, J.E., "Properties of Di e l e c t r i c - T u b e 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", E l e c t r o n i c s L e t t e r s , 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", V o l . 3, P. 470, Polytechnic Press, 1963. 11. Barlow, H.M., and Karbowiak, A.E., "An Inves t i g a t i o n of the Character- i s t i c s of C y l i n d r i c a l Surface Waves", Proc. IEE, 1953, Vo l . 100, Pt. I l l , P. 321.

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