LI B R.AFLY 
 
 OF THE 
 
 U N IVLRSITY 
 
 Of ILLINOIS 
 
 6Z1365 
 
 UGSStc 
 
 no.2-U 
 cop-3 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/axiallyexcitedsu07roya 
 
AX!I ALLY EXCITED SURFACE WAVE ANTENNAS 
 
 Contract No AF33(616) 310 
 
 RDO No R- 112-110 SR~6f2 
 
 TECHNICAL REPORT NO 7 
 
 by 
 D E. Royal 
 
 10 October 1955 
 >-it library of t;;e 
 
 K - v i 4 1955 
 
 UNIVERSITY Of ILLINOIS 
 
 ANTENNA SECTION 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA. iLLSNOSS 
 

 CONTENTS 
 
 Acknowledgement 
 
 Abstract 
 
 lo Introduction 
 
 Page 
 
 v n 
 
 Vlll 
 
 1 
 
 2. The Axially Excited Traveling Wave Antenna 3 
 
 3o Radiation Fields 7 
 
 4. Determination of the Longitudinal Phase Constant 16 
 
 t 4.1 Exciting Waveguide 16 
 
 4.2 Radiating Waveguide 23 
 
 5„ Investigation of the Coupling Mechanism 32 
 
 5.1 Effect of V(x) on Radiation Patterns 36 
 
 60 Experimental Results 41 
 
 60 1 Patterns 41 
 
 60 2 Aperture Measurements 57 
 
 7. Conclusions 80 
 
 Bibliography 82 
 Appendices 
 
 A. Calculation of Radiation Fields 84 
 
 A. 1 Equivalence of the Assumed Aperture Distribution to 
 an Array of Electric Dipoles Normal to the Ground 
 
 Plane 84 
 
 A„ 2 Calculation of the Radiation Field from the Aperture 
 
 Field 87 
 
 B. Determination of Phase Constants 89 
 
 B. 1 H z = Modes - Exciting Waveguide 89 
 
 B.2 E z - Modes 90 
 
 Co Investigation of the Coupling Mechanism 92 
 
 111 
 
ILLUSTRATIONS 
 
 Figure 
 
 Number Page 
 
 2.1 The Axially Excited Antenna with Dimensions of the 
 Experimental Model 4 
 
 2.2 Coupling Apertures Used in the Axially Excited Antenna 5 
 
 301 Coordinate System 8 
 
 302 Line Source Pattern Factor versus <fi for Various Values 
 
 of C/V 11 
 
 303 Line Source Pattern Factor versus </> for C/V = 1, L = 27cm, 
 
 e = 78° 13 
 
 3.4 Line Source Pattern Factor for Two Non Uniform Amplitude 
 
 Distributions 14 
 
 4-1 Exciting Waveguide 16 
 
 4.2 Curves of a versus A for Fixed d, b, e B /e A with C/V as 
 Parameter (H z - Mode) 19 
 
 4.3 Curves of a versus A for Fixed d, b } £g/e A with C/V as 
 Parameter (H 2 - Mode) 20 
 
 4.4 Curves of a versus A for Fixed d. b 3 £r/s a with C/V as 
 Parameter (H z = Mode) 21 
 
 4o5 Relative Phase Velocity versus Free Space Wavelength with 
 
 a as Parameter for Guide Pictured (H z ■ Mode) 22 
 
 4 6 Curves of d/A versus b/A for Radiating Waveguide with 
 
 C/V as Parameter 24 
 
 4.7 Curves of d versus A for Radiating Waveguide with C/V 
 
 as Parameter 25 
 
 4 8 Curves of Relative Phase Velocity versus Free Space Wavelength 
 
 for Radiating Waveguide 26 
 
 4.9 Calculated Relative Phase Velocity versus Free Space Wave- 
 length for Radiating and Exciting Waveguides (H z s Mode) 27 
 
 4.10 Curves of d versus A for Fixed b, e B /e A 3 with C/V as 
 Parameter for Radiating Waveguide (H - Mode) 29 
 
ILLUSTRATIONS (Cont. ) 
 
 Figu : e 
 
 Number Page 
 
 4.11 Relative Phase Velocity versus Free Space Wavelength with 
 
 d as Parameter for Radiating Waveguide (H z - Mode) 30 
 
 4 12 Infinite Dielectric Sheet over Conducting Plane, d, versus 
 
 \ with C/V as Parameter and C/V versus X with d as Parameter 31 
 
 5.1 Coupled Transmission Lines 32 
 
 5.2 Behaviour of P 3 for Two Values of a 38 
 
 5.3 Patterns Calculated for Various Values of Pj and 3 2 40 
 
 6,1 End-Fed Antenna 42 
 
 6 2 a-b Patterns for 5/16" Hole Array 43-44 
 
 6 3 a-b Patterns for 5/16" Hole Array 45-46 
 
 6 C 4 a-b Patterns for 1/4" Slot 47.-48 
 
 6.5 a-b Patterns for 1/4" Slot 49-50 
 
 6 6 a b Patterns for End-Fed Antenna 51-52 
 
 6.7 Patterns Calculated from Equation 5.11 56 
 
 6 8 Aperture Phase and Amplitude Distribution Measuring 
 
 Equipment 58 
 
 6.9 Aperture Phase and Amplitude Distributions 60 
 
 6 10 Aperture Phase and Amplitude Distributions 61 
 
 6 11 Aperture Phase and Amplitude Distributions 62 
 
 6 12 Aperture Phase and Amplitude Distributions 63 
 
 6.13 Aperture Phase and Amplitude Distributions 64 
 
 6 14 Aperture Phase and Amplitude Distributions 65 
 
 6 1 r > Aperture Phase and Amplitude Distributions 66 
 
 6 16 Aperture Phase and Amplitude Distributions 67 
 
. V I 
 
 ILLUSTRATIONS (Cont. ) 
 
 Figure 
 
 Number Page 
 
 6" 17 Aperture Phase and Amplitude Distributions 68 
 
 6ol8 Aperture Phase and Amplitude Distributions 69 
 
 6=19 Aperture Phase and Amplitude Distributions 70 
 
 6 20 Aperture Phase and Amplitude Distributions 71 
 
 6»21 Aperture Phase and Amplitude Distributions 72 
 
 6-22 Aperture Phase and Amplitude Distributions 73 
 
 6=23 Aperture Phase and Amplitude Distributions 76 
 
 6o24 Aperture Phase and Amplitude Distributions 77 
 
 6 25 Aperture Phase and Amplitude Distributions 78 
 
 6 26 Magnitudes of the Forward and Backward Traveling Waves 
 
 in the Radiating Waveguide as Calculated from Equation 5.3, 
 
 Together with Their Sum and Difference 79 
 
ACKNOWLEDGEMENT 
 
 The idea of applying axial excitation to surface wave antennas was 
 suggested to the author by his advisor, Professor Victor H. Rumsey in 
 October, 1954= The author takes pleasure in acknowledging the guidance 
 and assistance of Professor Rumsey throughout the course of the investi- 
 gation, and expresses his appreciation of the support of Air Force 
 Contract No. AF33(616) 310. 
 
 v i i 
 
ABSTRACT 
 
 A method of exciting surface wave antennas is described. A surface 
 wave antenna is basically an aperture in a conducting surface which is 
 excited so that the distribution of field in the aperture consists of a 
 traveling wave. The antenna is excited by coupling it along its direc- 
 tion of propagation to an adjacent waveguide through a long slot or 
 series of holes. The adjacent waveguide carries a traveling wave having 
 the same propagation velocity as the desired mode in the antenna. 
 
 In order to test the theory an experimental antenna was built. To 
 observe the action of the coupling mechanism, the antenna was designed 
 so that the rates of change cf propagation velocity with frequency of the 
 exciting waveguide and of the radiating waveguide (i.e., aperture) were 
 large and differed considerably. In a practical application one can, on 
 the contrary, design the antenna for broadband operation. In addition, 
 the propagation velocities of the two guides were equal at one frequency 
 in the range of interest. A propagation velocity very near that of free 
 space was employed so that the antenna produced an end-fire beam. This 
 was obtained by filling the aperture of the antenna and partially filling 
 the exciting waveguide with dielectric. 
 
 The theory employed to estimate the phase velocities in the antenna 
 and in the exciting waveguide is shown to give results in close agreement 
 with measured values. The aperture field of the desired hybrid mode is 
 shown to be equivalent to an array of normal electric dipoles provided the 
 tangential electric field in the aperture is given by E * VU where U is a 
 scalar such that the tangential component of E at the rim vanishes (U = 0), 
 
 v III 
 
I 
 
 The approximate theory developed to describe the action of the coupling 
 mechanism is shown to predict aperture distributions and radiation 
 patterns in agreement with those obtained in practice 
 
 The aperture distributions and radiation patterns were measured for 
 the cases when the antenna was (a) end- fed, (b) axially excited by a 
 long slot, and (c) axially excited by a series of holes. These measure- 
 ments demonstrate that the hole array is capable of producing a pattern 
 which is remarkably free of side lobes (spurious radiation). Patterns 
 produced by the hole array indicate a side lobe level better than 30 db 
 in one plane and about 20 db in the other plane as compared with a 
 typical value of 10 db for the end- fed antenna in both planes. The 
 purity of mode obtained with the hole array is considerably better than 
 that obtained by feeding the antenna at one end. 
 
lo INTRODUCTION 
 
 This thesis is concerned with the investigation of a new excitation 
 mechanism for surface wave antennas, A surface wave antenna is basically 
 an aperture in a conducting surface which is excited so that the distri- 
 bution of field in the aperture consists of a traveling wave. Such 
 antennas are therefore flush mounted and their great utility lies princi- 
 pally in the field of high speed aircraft. Frequently a flush mounted 
 antenna is required to have an end fire beam. This requirement can be 
 met at sufficiently low frequencies by an array of slots appropriately 
 fed, but at the higher frequencies this scheme is very difficult to put 
 into effect. However, at the higher frequencies, slots many wavelengths 
 long become practical „ If they are excited by waves traveling with the 
 free space phase velocity they can be made to produce end- fire beams. To 
 establish this phase velocity the aperture can be wholly or partially 
 filled with a dielectric. In this way it is possible to obtain an end- 
 fire beam over a reasonably wide frequency band 
 
 The methods which have been used in the past 1 to excite flush- 
 mounted dielectric slab antennas have been found unsatisfactory in certain 
 respects . In these methods the antenna is excited over the transverse 
 cross section at one end. The wave which is initiated travels down the 
 slab, radiating as it proceeds Unfortunately there is often a consider- 
 able amount of stray radiation which results in a poor pattern., The stray 
 radiation can be ascribed to "higher order modes" or "direct radiation 
 from the exciting mechanism;" more precisely, it represents the differ- 
 ence between the observed pattern and the theoretical pattern of a 
 traveling wave distribution 
 
 1 
 
2 
 This work is concerned with the excitation of a dielectric slab 
 antenna along its axis of propagation by coupling it to an adjacent 
 waveguide through a long slot or series of holes. The adjacent wave- 
 guide carries a traveling wave having the same propagation velocity as 
 the desired mode in the antenna. It has been shown 18 that selective 
 mode excitation in one waveguide by another is possible when this condi- 
 tion is established between their propagation constants. The length of 
 the coupling section and its shape influence the mode discrimination 
 property and directivity of energy transfer. The present work demon- 
 strates that this method of coupling can be extended to the case of 
 coupling to a dielectric slab antenna. 
 
2. THE AXIALLY EXCITED TRAVELING WAVE ANTENNA 
 
 Figure 2.1 is a drawing of the axially excited traveling wave 
 antenna which is investigated in this work. It can be considered to 
 consist of three parts,, a radiating waveguide,, a coupling aperture, and 
 an exciting waveguide* The radiating waveguide consists of a dielectric- 
 filled rectangular channel in a ground plane. The coupling aperture 
 (Fig. 2.2) is formed by a plate interposed between the two waveguides 
 which has an opening or series of openings to allow coupling between the 
 guides. The exciting waveguide is partially filled with dielectric. It 
 is terminated at one end in a resistive absorbing card and at the other 
 end in a taper transition to a standard X-band detector mount, A stand- 
 ard rectangular waveguide would replace the detector if the antenna were 
 used for transmission. 
 
 To explain how the antenna operates, consider the transmitting case, 
 A TE wave from the exciting section strikes the dielectric loaded filter 
 section. The higher modes from the discontinuity are evanescent. The 
 dominant (hybrid) mode for this loaded guide continues on through the 
 section adjacent to the coupling plate. Part of the energy in the wave 
 passes through this plate and excites a mode in the open dielectric slab 
 which is very nearly like that in the exciting waveguide. This mode 
 traveling in the open slab constitutes the desired radiation mode. The 
 energy which is not passed through the coupling plate proceeds into the 
 section containing the resistive card wherein it is dissipated. The 
 resistive load is used in this investigation only for purposes of test- 
 ing. In practice, most of the energy would be radiated and the amount 
 

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 absorbed in the resistor would be negligible. The particular placement 
 
 of the waveguides and coupling plate shown gives a symmetrical excita- 
 tion. 
 
 Realization of the desired beam shape depends upon the generation 
 of the proper mode in the open slab. This in turn depends upon the shape 
 and size of the coupling aperture between the open slab and the exciting 
 waveguide and on the mode in the latter. The coupling per unit length 
 between the two is a maximum when their propagation constants are 
 identical. High efficiencies should be realizable by adjusting the 
 length of the coupling aperture so that nearly all the energy in the 
 incident wave is coupled to the antenna before reaching the resistive 
 load. 
 
 The antenna is analyzed from the viewpoint of two rather loosely 
 coupled waveguides. Because they are both radiating, the first into 
 free space and the second into the first, their propagation constants 
 have real as well as imaginary components. However, in order to obtain 
 approximate estimates of their phase velocities, the exciting waveguide 
 is considered closed and lossless, and the radiating slab is considered 
 to have the same fields as the closed guide in the limiting case for the 
 dimension "a" \,see Fig, 4.1) approaching infinity. There is justification 
 in the literature for these assumptions, and probe measurements of the 
 phase velocity made at the surface of the radiating slab give phase 
 velocities in close agreement with them. 
 
 This approximate analysis points out some of the effects of axial 
 Ltat on upon the antenna's aperture distribution and pattern. 
 
3. RADIATION FIELDS 
 
 The diffraction of waves through a hole in a perfectly conducting 
 plane screen can be expressed in terms of the tangential electric field 
 in the hole by the application of Huygen's principle. One statement of 
 this principle is that if we have an infinite perfectly conducting plane 
 boundary containing a hole, one side of the boundary being source free, 
 the waves diffracted (radiated) through the hole can be considered to be 
 generated by a sheet of magnetic current sources M - E * n situated in 
 the hole provided: (a),, the hole is removed, i.e., the conducting plane 
 is complete, (b), E is the tangential electric field in the hole due to 
 the actual sources (which is zero over the rest of the plane) and, (c), 
 n is the unit normal to the aperture pointing into the source free side 
 of the plane. Proceeding from this result we can remove the conducting 
 plane by observing that the image of M in the plane is in phase with M 
 and in this case coincides with M Therefore M radiating in the presence 
 of the conducting plane can be replaced by 2M radiating in free space. 
 The electric field at a point P on the source-free side is then obtained 
 from the electric vector potential by use of the formula 
 
 E p «Vpx/ fe a , E(0) f_!! ds(0) (3.1) 
 
 where Vpx is the curl taken at the point. P and r po is the distance from P 
 to a point Q in the hole, as in Fig. 3.1 below. 
 
 For pattern purposes it is desired to generate a mode in the antenna 
 which produces an end- fire beam. A mode that is capable of this and 
 
Suitoce ol AntentKj 
 
 Figure 3 1 Coordinate System 
 
 which can exist in a partially dielectric- filled waveguide is described 
 by 
 
 H T V x z f 
 
 (3.2) 
 
 where z is a unit vector normal to the ground plane and 
 
 r - a "J^X* a i i\ K y 
 
 f s A e cos p_ (z + d) cos — 
 
 z b 
 
 (see Section 4). It is assumed that the tangential electric field in 
 the aperture of the ground plane is given by 
 
 jwe E = V x U 
 
 (3.3) 
 
 re U is obtained from Eq. 3.2. The assumed distribution of tangen - 
 'ill fci so obtained is not correct because the type of field represented 
 by Eq 3.2 canii o t be set up in the antenna. It is, however, a good 
 
9 
 approximation. Now the field of an infinitesimal electric dipole with 
 its axis oriented in the z direction is given by 
 
 H = Vx|j (3.4) 
 
 where J is the surface density of dipole moment. It is shown in 
 Appendix C that if tangential E in the aperture has the form given by 
 Eqs. 3.3 and 3.2, then the resulting radiation field is the same as that 
 due to an array of infinitesimal electric dipoles oriented normal to the 
 ground plane in free space provided that the surface density of dipole 
 moment is given by 
 
 J = -2 1 • Vf (3,5) 
 
 where Vf is evaluated at the aperture of the antenna. This equivalence 
 is strictly true when the antenna is mounted in an infinite ground plane 
 and if the form of f is such that the component of E which is tangential 
 to the rim of the hole vanishes there. Neither of these conditions 
 imposes an inordinate restriction upon the use of the equivalent dipole 
 array in obtaining radiation patterns. An infinite ground plane is a 
 useful approximation to the usually large (in wavelengths) ground plane 
 which would be employed in practice. The requirement on f is tantamount 
 to saying that we require the Huygens source in the aperture, n x E(£) to 
 be equivalent to some actual source behind aperture. Rumsey points out 
 that if this boundary condition is violated, the energy density in the 
 aperture must be infinite, i.e., there must be actual sources in the 
 aperture. 
 
10 
 Using the electric dipole array which is equivalent to the assumed 
 
 aperture fields described by Eqs. 3.2 and 3.3, the following expressions 
 
 are obtained for their radiation (far zone) fields: 
 
 E r - fy = E e = C *^ I a (9,0) I b (9,0) sin 9 
 H r = H = 1 E Q H 9 = 
 
 b/2 
 I a (9,0) = C e j^ysinQsin4> CQS ^ dy 
 
 -b/2 
 
 L/2 
 I b (9 f 0)= r e (-Y x *jP.ineco.«, fe (36) 
 
 "^ (See Fig. 3.1) 
 
 The derivation of these expressions is given in Appendix C. The amplitude 
 factor C is independent of r, 9, and <p. 
 
 The expression for Eq involves the two array factors along the x and 
 y axes, I and II, and the unit pattern, sin 9. 
 
 For the narrow width of antenna considered here, the integration with 
 respect to y has negligible effect on the front and back lobes and acts 
 principally to reduce certain side lobes by about 40 percent. The sin 9 
 term also reduces the side lobe level of certain patterns and has little 
 effect on front and back lobes. The combined effect of these two terms 
 j l to reduce the side- lobe level of all the patterns below that corre- 
 sponding to the quantity 1^, which represents the pattern of a line source 
 distribution. 
 
 Sf:vf;ral calculated patterns are shown to bring out the effects of 
 rariationa in the (x) axial distribution. Figure 3.2 shows patterns for 
 
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 various values of 3 x /3 in the range of interest for an end- fire antenna. 
 For this group the attenuation constant is zero. The length L corre- 
 sponds to that of the experimental antenna. It is apparent from Fig. 3.2 
 that for this L a value of P x /3 very near unity must be used if an end- 
 fire beam is desired. 
 
 Figure 3.3 demonstrates the effects of introducing attenuation in 
 the axial distribution. If it is assumed that a traveling wave, initi- 
 ated at one end of the antenna, radiates one-third of its energy by the 
 
 nepers 
 
 time it reaches the other end, a value of a = .0174 is obtained. 
 
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 to the assumption that 90 percent of the original energy is radiated. 
 For both curves P x /(3 is equal to unity. The principal effects of intro- 
 ducing attenuation in the traveling wave are a reduction of the null 
 depth and increase in the peak side lobe level. 
 
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 tions have been considered. The insert in the graph shows the shapes of 
 the distributions. Patterns A and B result from the trapezoidal distri- 
 bution A and the cosinusoidal distribution B, respectively. Both patterns 
 show a reduction in the side-lobe level over that obtained for the uniform 
 amplitude distribution. The main lobe obtained with the cosinusoidal 
 distribution is somewhat broader than that obtained with the uniform 
 distribution but its side lobes are considerably lower. 
 
 The effects of the attenuation constant, relative phase velocity, 
 ind amplitude distribution are, to a first approximation, independent of 
 one another. Therefore, the combination of their effects can be visual 
 
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15 
 
 ized from graphs such as these show. Before proceeding further with the 
 patterns, the fields in the exciting and radiating guides will be con- 
 sidered, with particular emphasis upon their phase velocities. 
 
4. DETERMINATION OF THE LONGITUDINAL PHASE CONSTANT 
 
 4-1 Exciting Waveguide 
 
 Since the antenna is designed to produce an end- fire beam, the 
 radiating and therefore the exciting waveguide must support energy 
 traveling with a phase velocity equal to or slightly less than that of 
 free space. To permit this the exciting waveguide is partially filled 
 with dielectric as shown in Fig. 4.1. 
 
 j. Region 
 
 nNWnvnnN 
 
 £ a Region A 
 
 Figure 4 1 Exciting Waveguide 
 
 It can be shown (see Appendix A) that the magnetic field given by 
 
 H=vx^f 
 
 (4.1) 
 
 where i is a unit vector in the z direction, satisfies Maxwell's equa- 
 tions in the waveguide of Fig. 4.1 if 
 
 t - cos p n d e x 
 
 rcy . 
 
 Bz' 
 
 cos (3. (z - a) sin — in Region A 
 
 A z U 
 
 . i ft % rev 
 
 cos P A .(d - a) e x cos 3 R z sin — - in Region R (4.2) 
 
 Az 
 
 16 
 
17 
 
 with 
 
 P* 2 - 3. 2 - (3 2 + 
 
 re 
 
 i b ~, 
 
 P B 2 -P Bz 2 (4.3) 
 
 and 
 
 £ B ?Az tan \M " a > " £ A P B2 tan 3 Bz d ° U ° 4) 
 
 ^Az' ^Bz» ^x' ^A and ^B are t ^ ie P ro P a g a tion constants in the z and x 
 directions and in Region A and Region B, respectively. The fields defined 
 by Eq. 4,1 are described in this work as H - modes. Under similar 
 conditions a set of modes defined by 
 
 E = V x I g (4.5) 
 
 can exist in the structure of Fig. 4.1. These are the E z = modes. The 
 H = mode was chosen for the investigation because it gives a desirable 
 radiation pattern as has been pointed out in Section 3. It can be made 
 the dominant mode by the proper choice of dimensions. 
 
 For this structure there is no convenient formula connecting the 
 guide wavelength to the cutoff wavelength as for a homogeneously filled 
 guide. The quantity of primary interest is the relative phase velocity 
 in the guide C/V, or its equivalent ^/^ x ° Equation 4.4 cannot be solved 
 explicitly for C/V but if we let a " ; °° to simulate the radiating condi- 
 tion, a single family of curves can be constructed for a particular 
 e g/ e A from which C/V can be obtained as a function of A for any partic- 
 ular pair of values of b and d. However, in the case of the closed guide 
 the additional parameter a which is involved makes it necessary to con- 
 struct a three dimensional manifold to present the same information. 
 
18 
 
 The cross section of the exciting guide was determined as follows. 
 Polystyrene was used for both the radiating and exciting guides because 
 of its excellent electrical properties at 10,000 mc. This settled the 
 relative dielectric constant at 2.56. The dimension b was arbitrarily 
 chosen to be the same for the exciting guide as for the radiating guide. 
 It remained to determine the dimensions a and d. Equation 4.9 can be 
 solved explicitly for a for assumed values of the other parameters. 
 Putting a = °° to simulate the radiating waveguide, the dimension b was 
 chosen as 1.7 cm for an operating wavelength of about 3 cm. Figures 4.2, 
 4.3, and 4.4 present families of curves of a versus X for several values 
 of the parameter C/V in the range of interest for three different values 
 of d. From each of these a family of curves such as Fig. 4.5 can be 
 constructed. Figure 4.5 is constructed from Fig. 4.3 and depicts the 
 variation of C/V with respect to X for b = 1.7 cm, d - 0.516 cm, 
 E B /e. s 2.56 and for various values of a. A value of a = .7 cm was 
 chosen for the experimental antenna in order to bring out the effect of 
 a change in phase velocity. This puts C/V equal to 1 at A - 3.26 cm and 
 gives a variation in C/V from 1.2 to 0.9 between X = 2.67 cm and 
 X ■ 3. 56 cm. 
 
 A comparison of Figs. 4.2, 4.3, and 4.4 shows how the C/V for this 
 guide is affected by variations in a and d. It is obvious that the more 
 dielectric there is in a particular cross section, the lower is the 
 phase velocity at any given frequency. As would be expected, the range 
 of C/V obtainable at any frequency for given b and a lies between what 
 Can be obtained with an empty guide on the one hand and a completely 
 filled guide "r, the other hand. 
 
19 
 
 3.0 
 X in cm. 
 
 FIGUKE 1 2 CURVES OF a VERSUS X FOR FIXED d b s ^ WIT?! C/V AS 
 PARAMETER H z = MODE A 
 
20 
 
 I.I- 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \c/v 
 
 = 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X/V= 
 
 .9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \c/v= 
 
 1 1 
 
 
 
 
 
 
 
 
 
 i7 
 
 8 
 
 E 
 o 
 
 c 
 ° 7 
 
 6 
 
 5 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 \C/V= 
 
 1.2 
 
 
 \ 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •— — ( 
 
 
 b = ! 7 cm 
 d p 516 
 
 il - 2.56 
 
 e A 
 
 
 
 
 
 
 2 * 
 
 t 
 
 1 
 
 1 
 
 \\\ 
 
 <A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~d* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1 
 
 4 2.6 2 8 3.0 3 2 3.4 
 
 X in cm 
 
 "IGURE H 3 CURV^r OF a VERSUS A FOR FIXED b, d, ^&, WITH C/V AS 
 PARA,;LTfcR ll z MODE e A 
 
21 
 
 6 — 
 
 b = 
 
 : 17 cm 
 
 d = 
 
 6 cm 
 
 _B 
 
 - 2 56 
 
 2.6 
 
 2.8 
 
 3.2. 
 
 24 
 FIGURE 11 CURVES OF a VERSUS A FOR FIXED d, b IB- WITH C/V AS 
 
 3,0 
 X in cm 
 
 PARA .LTER 
 
 H- 
 
 MODE 
 
22 
 
 25 2.fl 3 3 2 34 
 
 »> M i i i 
 FIGURE l| 5 RELATIVE PHASE VELOCITY VE,;5JS FREE SPACE WAVFLENGTH WITH a 
 AS PARAMETER FOR QUIDE PICTURED 
 
 H, ::ODE 
 
23 
 12 Radiating Waveguide 
 
 In order to approximate the fields in the radiating waveguide the 
 dimension a of Fig. 4.1 is set equal to infinity. The requirement that 
 the power flow be in the positive z direction is then imposed. The 
 following equations result: 
 
 -jP x X -(3 A (zd) Tty 
 i ~ e e sin — in Region A 
 
 b 
 
 - j P X Tty 
 
 - e x sec 3 R ,d cos (3 z sin — - in Region B (4 6) 
 
 D 21 D 2 Irk 
 
 e.(3_ tan (3 R d - e R 3 A . (4.7) 
 
 A Bz Bz B Az 
 
 These results are valid provided 
 
 / \ 2 
 
 ■ A 2 ^ P x 2 
 
 H > . (4.8) 
 
 b / 
 
 Equation 4.7 together with Eq. 4.3 can be solved explicitly for d 
 given e B / e A , b and C/V. In Fig. 4.6 is shown a family of curves of d/X 
 as function of b/X for a set of values of C/V for £ B / e A s 2.56. For the 
 present antenna design the same information in the form of Fig. 4.7 was 
 used to obtain the desired curves of C/V versus X for various values of 
 d which are shown in Fig. 4.8. From this figure d can be chosen to give 
 the desired range of C/V. 
 
 A choice of d - 0.775 cm was made. Curves showing the relative 
 phase velocity calculated from this approximate analysis for both the 
 exciting waveguide and the radiating waveguide are shown in Fig. 4.9 for 
 the experimental antenna as built. Also shown is a curve of measured 
 values of C/V obtained from the aperture measurements described in 
 
24 
 
 7 
 
 .6 
 
 w 
 A 
 S 
 
 o 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ! 1 
 
 
 ~ /„ _ o cff 
 
 
 
 
 1 
 
 
 
 c 3 / &A -• <.,ao 
 
 H z ■ MODE 
 
 
 
 
 
 
 
 
 
 
 ^-d-— 
 
 oo 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 1 
 
 
 \ \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 u\ 
 
 \ \ 
 
 
 
 
 
 
 
 i4 
 
 
 
 
 \\ 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^ 
 
 
 
 
 ^T" 
 
 
 
 
 
 
 
 
 
 
 
 
 7T~~~~~ 
 
 
 
 
 
 
 
 
 
 
 
 11 t 
 
 
 
 
 
 
 
 
 
 
 
 \00 
 
 
 
 
 
 
 c 
 
 / v- 0\ 
 
 
 
 
 s£/.</ 
 
 
 
 
 
 
 c 
 
 a .6 i.O u7 
 
 "•3 r »F d/> VERSUS h/A FOR RAD 1 AT INC. WAVEGUITE WITH 
 
25 
 
 i 
 
 1.8 
 1.6 
 
 1.4 
 E 
 
 u 
 
 .£ 1.2 
 1.0 
 
 .8 
 .6 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 b s 
 
 1 - 7 cm 
 
 2.-6 
 MODE 
 
 
 
 
 
 
 
 T 
 
 
 r 
 
 j 
 
 a 
 
 T - 
 
 Q-CO 
 
 
 
 
 
 
 
 J 
 
 J 
 
 >to- 
 
 Hz 
 
 
 
 
 
 
 
 
 -^d-*- 
 
 
 
 
 
 
 
 
 
 /£/V 
 
 = 1.2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 'I.I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 // 
 
 
 ^9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 i 1 
 
 .4 2.6 2.8 
 
 :URt 7 . RVES OF d vERSUS A 
 PARAMETER 
 
 3.C 3.2 3.4 
 
 X '- cm 
 
 FOi? RADIATING WAVEGUIDE WITH C/V AS 
 
26 
 
 ,2 
 
 .1 
 
 C 
 
 
 09 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b = L7 cm 
 H z - MODE 
 
 e 1 s 2 56 
 
 
 
 
 1 
 
 1 
 
 /// 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 —-a-*. 
 
 i ► 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Si 
 
 ^* 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 V V 
 
 \3 
 
 ^ 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 d = K 
 
 
 •OS 
 
 
 .f\ 
 
 
 
 
 i OS. 
 
 u\ 
 
 s^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s\ 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <?4 ^S 28 3.0 3.2 3.4 
 
 FIGURE «4 8 CURVES OF RELATIVE PHASE VELOCITY VERSUS FREE SPACE 
 WAVELENGTH *0R RADIATING WAVEGUIDE 
 
27 
 
 1.2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 >.Ra 
 
 diating 
 
 guide 
 
 
 
 
 
 
 
 
 E* 
 
 iting c 
 
 
 i^^^ 
 
 
 
 
 
 
 
 
 
 I.I 
 
 
 
 
 > V. 
 
 
 =5^0 
 
 
 
 
 
 
 
 
 
 
 
 
 ^"N^O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 > | ^ 
 
 
 
 
 
 
 
 N 
 
 \ 
 
 N^ 
 
 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 Sv O 
 ^v ( 
 
 ) 
 
 
 
 
 
 
 
 
 
 
 
 
 o\ 
 
 
 09 
 
 
 
 
 
 
 
 
 
 
 
 . ( 
 
 N. 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 ]~ 
 
 #s 
 
 ^» 
 
 { 
 
 :% 
 
 
 
 
 
 
 
 
 ^ 
 
 € A 
 
 Ab/ 
 
 /// 
 
 C A 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 ^=2." 
 
 
 
 — *— >. 
 
 h a ^ 
 
 
 
 
 
 6 
 
 
 
 
 € B_- 
 
 >.56 
 
 
 
 
 
 p 
 
 % 
 
 2 
 
 6 
 
 9 
 
 S 
 
 A 
 
 3 
 
 o 
 
 3 
 
 2 
 
 J 
 
 4 
 
 
 FIGURE i+ 9 CALCULATED RELATIVE PHASE VELOCITY VERSUS FRES SPACE WAVE- 
 
 LENGTH FOR RADIATING MID EXCITING WAVEGUIDES 
 
 is- MODE, CIRCLED POINTS ARE MEASURED VALUE? 
 
28 
 Section 6. Considering the approximations involved in the theoretical 
 curves, they are in good agreement with the experimentally obtained 
 values. 
 
 It is apparent that C/V varies rapidly with wavelength. While this 
 effect was purposely sought in this antenna in order to bring out the 
 operation of the coupling mechanism, it is undesirable in practice 
 because of the pattern variation it produces. The frequency sensitivity 
 of C/y is controlled by the width b of the waveguides. To point this 
 out, Fig. 4.10, 4.11, and 4.12 have been constructed. Figures 4.10 and 
 4.11 are to be compared with Figs. 4.7 and 4.8, respectively. The 
 difference in the two sets is caused by doubling the dimension b. The 
 variation of C/V with X for b = 3.4 cm. (Fig. 4.11) is greatly reduced 
 from the variation which is obtained for b = 1.7 cm (Fig. 4.8). As a 
 matter of interest, the curves of Fig. 4.12 are shown. These are for an 
 infinite dielectric sheet, a = b - °°. In this case it is evident that 
 for a given dielectric thickness, d, there is an infinitely large range 
 of wavelengths for which the variation of C/V with wavelength is 
 negligible. 
 
29 
 
 08 
 07 
 0.6 
 
 0.5 
 
 c 
 ■o 
 0.4 
 
 n ~3. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 - 3.1 cm 
 e 3 /e A = 2 56 
 
 
 T 
 
 1 
 
 i 
 
 ■ ■■ ■»- 
 
 0=°° 
 
 
 
 
 
 
 
 
 
 
 —^12 
 
 y/^ 
 
 
 
 --b* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I.I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 JP 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.95 
 
 U.O 
 
 0.2 
 0.1 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ■«- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 Fl 
 
 .4 Z6 2.8 3.0 3.2 3.4 
 
 X :n cm 
 3URE 4- 10 CurtVES OF d VERSUS A FOR FIXED b, e B /e A , WITH C/V AS 
 PARAMETER FOR RAD 1 A iNG WAVEoUIDE 
 
 l' z = MODE 
 
30 
 
 1.3 
 1.2 
 
 I.I 
 
 1.0 
 
 > 
 
 o 
 .9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 C 
 
 .550 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .442 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .338 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 d=0 
 
 .226 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h = ^ LI rm 
 
 
 
 
 t 
 
 -O 
 
 1 
 
 XV 
 
 
 
 
 
 a=°° 
 
 
 
 
 
 
 
 
 - B / - A 
 
 
 
 
 
 h~-c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 .4 2.6 2.8 3.0 3.2 3.4 
 
 X in cm 
 
 figui'f >| ii relative phase velocity versus free space wavelength 
 
 it:: 'Ara:ieter fcr radiating waveguide h z o mode 
 
31 
 
5. INVESTIGATION OF THE COUPLING MECHANISM 
 
 The coupling aperture available in the axially excited antenna 
 extends the length of the radiating slab, which is about ten wavelengths. 
 The coupling per wavelength therefore can be kept reasonably low. Thus 
 the coupling mechanism can be investigated on a loose coupling basis to 
 get an idea of the factors involved. In the simplest approximation the 
 structure is considered as two coupled transmission lines as shown in 
 Fig. 5. 1. The exciting waveguide represented by line 2 is terminated in 
 
 Line I 
 
 fv(x) 
 
 -20' 
 
 L/2 
 
 Line 2 
 
 20 
 
 *.i 
 
 •20 
 
 L/2 
 
 Figure 5 I Coupled Transmission Lines 
 its characteristic impedance at both ends. The radiating guide, line 1, 
 is terminated in an impedance Z\ at each end. The coupling extends from 
 -L/2 to + L/2 and is zero elsewhere. Let it be assumed that the effect 
 of the exciting line is to impress a shunt current of strength I'(x) per 
 unit length upon the excited line. We are interested in the voltage V(x) 
 of the excited line at any point along the coupling interval due to the 
 Integrated effect of I'(x) over this interval, It is shown in Appendix C 
 that these considerations lead to the following expressions for the 
 
 32 
 
33 
 voltage on line 1: 
 
 pX 
 
 V(x)--^ZosinhYi(|rx)+ZicoshYi( L *)} \ {ZosinhYi^J+ZicoshYi <\»B.)}l' (Z)dZ 
 Di z 2 J L 2 2 
 
 "2 
 
 1/2 
 +^Z sinhYi( L bc)+Z,coshY 1 (^x)}\ {ZosiiihYi(M)+Z a coshYi(k-£)}l'(£)d2 
 
 x 
 
 (5.1) 
 where Yi is the propagation constant of line 1 and 
 
 Di = -{(Zo 2 + Zi 2 ) sinh YiL + 2ZiZ cosh YiU . 
 
 A reasonable assumption for the impressed current I'(x) is that it 
 varies as I e where Y2 is the propagation constant of the exciting 
 line (2) and I is constant. By allowing Y2 to be complex, we can take 
 partly into account the reaction of the excited line upon the exciting 
 line. Io is proportional to the coupling coefficient times the strength 
 of the field in the exciting line. For this form of I ' (x) „ V(x) for 
 L/2 1 x £ L/2 becomes 
 
 V(xH ^ 2 J e Y2X y 
 
 Di(Y2 ~Yi ) 
 
 «=» Y Y T 
 
 +e [-,-Y2sinh b cosh [( Yi + Y2)= + b] +\ iCOsh b sinh [( Yi^Ys^+b]] 
 
 -Yix 
 
 L-,-Y2smh b cosn LI Yi J "Y2^ + bj +\ - x cosh b sinh K Yi + Ys J^bJJ 
 
 z ^ 
 
 +e [-Y 2 sinh b cosh[( Yz + Yi)^ + b] +YiCOsh b sinh[( Y2 + Yi)^ + b]]> 
 
 2 j 
 
 Yi f Y2 
 
 (5.2) 
 
34 
 
 V(*>-— <!« 
 
 — +Z Z 1 L+(Zo 2 +Z 1 2 )x 
 Yi 
 
 sinhYiL+ 
 
 Z Zi(i *2jO+(Z , +Zi , >1- 
 
 Yi 2] 
 
 cosh YiL 
 
 -e 
 
 Yix 
 
 (Zo 2 - Zl 2 )uMi 
 
 2 Yi 
 
 Yi ■ Ya 
 
 (5.2) 
 
 o 2 2 
 
 where d = Z - Zi and b = tanh 
 
 -i ZY 
 
 Z 
 
 The representation of V(x) as a sum of traveling waves is convenient 
 in determining its resulting radiation patterns . It is apparent that in 
 the case of arbitrary Zi and Yi / Y2 there will be three traveling waves 
 set up in the excited line. Corresponding to these waves, three patterns 
 of the line source type are to be expected as components of the radiation 
 field of the antenna. The resulting radiation pattern is seen to be an 
 involved function of the propagation constants of the two lines and of the 
 relative (toZo) value of the terminating impedances of the excited line. 
 
 If Zx = 0, then Eq. 5.2 becomes 
 
 I0Z0Y1 
 
 V(x)=- 3— " 3" ' * , -<|e ' z "sinhYiL + e * sinh(~Yi + Y2)*?~e * sinh(Yi + Y2)^ 
 
 (Y2 "Yi )sinhYiU 
 
 V(x) 
 
 I X 
 
 0^0 
 
 2sinhYiL 2 
 
 Yi t Yi 
 
 Yix I -Y'lX 
 
 re Lx sinhYiL-^coshYiLJe 
 
 Yi = Y2 
 
 (5.3) 
 
 Tins expression for V(x) is of interest in that a dipole surface density 
 obtained from it vanishes at the ends of the aperture. 
 
35 
 
 Now suppose that Zi - Z . Then Eq. 5,2 bee 
 
 omes 
 
 ioZoJ - Y2 x (-Yi+Ya^YiX , > (-Yi-Ya^x 
 V(x)-— s 2-r<;2Yie ~(Yi + Y 2 )e 2 -(Yi-YaJe 
 
 2(Y2 -Yi ) 
 
 Yi f Y: 
 
 t -v-ix -YiL+YiX 
 V(x)'i2±2J-(x+Lfl/2 Yl )e Yl +S 
 
 2 2 2y± 
 
 Yi = Yj 
 
 (5.4) 
 
 An unusual feature of this result is the appearance of a " reflected" 
 wave even though the terminating impedances are assumed equal to the 
 characteristic impedance of the line. At this point it might be 
 pertinent to inquire as to what should the termination be to eliminate 
 a backward traveling wave. Inspection of Eq. 5.2 reveals that the 
 required value of Z x is a function of frequency. In particular, its 
 value is given by 
 
 Zi 
 
 Zc 
 
 — coth (~Y2+Yi) £ + 
 
 2y 2 2 
 
 Yi (Y1-Y2) , 2 . v i 
 
 — + — : — 1 — coth (»Y2 + Yi) \t 
 
 Y2 4y 2 2 
 
 Yi / Yi 
 
 h. 
 
 z 
 
 YiL 
 
 \ 2 
 
 1 + 
 
 YiL 
 
 Yi = Y: 
 
 (5.5) 
 
 Since the value of PL employed in this antenna is about 60 the normal- 
 ized impedance which would be required to eliminate a reflected wave in 
 it is not far from unity when Yi is near Y2< 
 
36 
 5.1 Effect of V(x) on Radiation Patterns 
 
 The radiation patterns which would be obtained from an aperture 
 
 distribution with a surface density of dipole moment which varies as 
 
 V(x) can be obtained quite simply, for the long narrow radiating slab, 
 
 from the expression 
 
 L/2 
 
 V(x) e JP sin9cos ^ x dx . (5.6) 
 
 L/2 
 
 This excludes the effect of the factor I a (9,0) sin 9 in Eq. 3.6. The 
 form of V(x) which has been obtained for Yi i Y2 is 
 
 ™ Y V ™ Y Y 
 
 V(x) % gi(Zo,Z 1 ,Yi,Y2,L) e ' + g 2 (Z 0l Z 1 , Yi , Y2, L) e 
 
 + g 3 (Z ,Z lf Yi,Y2,L) e 1 . (5.7) 
 
 The pattern which results from this is 
 
 sinh(-Y2^j(3')L sinh(-Yi + jP')L' Binh(Yi+j£')£ 
 
 P * Si — J ^TTi + gs — ; r^TTi + gs — : -— j (5.8) 
 
 where |3' = 3 sin 9 cos 4>. 
 
 The manner in which the three line source patterns combine depends on 
 
 the relative magnitudes and phases of the factors g^ . 
 
 In the case of Yi " Y2 it was shown that V(x) assumes the form 
 
 V(x) % hi(Zo,Zi,Yi,U e" YlX + h 2 (Z ,Z 1( Yi,L) e YlX + h 3 (Z ,Z 1 , Yi.Dxe" YlX . 
 
 (5.9) 
 
37 
 It is of interest to examine the effects of the third term of Eq„ 5=9 on 
 the pattern. The pattern due to this term alone is 
 
 r 2 (viB')x LcoshC-Yi+jP'jL 2 sinh(- Yl+ jP')L 
 
 P 8 * \ xe (Y * jP)x dx- _ — I 2 (5.10) 
 
 .J, <-Y.+jP'> <.. Yl+ j0') 2 
 
 It is easily shown that if the factor (~Yi + j3') vanishes, the limiting 
 value of this expression is 0. Equation 5-10 has been graphed in Fig. 5.2 
 for typical values of $ x = 3 ~ 2. 18 rad/cm and L = 27 cm, and for two 
 values of ol 1} 0, and .0485 nepers/cm. It is seen that this term contrib- 
 utes to the forward beam of the pattern. 
 
 The first and second terms of Eq. 5.9 produce traveling wave line 
 source patterns with main lobes which are in the forward and backward 
 directions, respectively. 
 
 In order to apply these results to the experimental antenna it is 
 necessary to have values for Zt/Z , Yij> an d Y2- The approximation that 
 Z; » is employed. Applying Eq. 5.6 to Eq. 5.3, we obtain for this 
 
 case: 
 
 IoZoYxL Jsinh(~Y2 + jP')L> sinh(~Yi + Y 2 )| sinh( Y i+j P' )^ 
 Y2 2 ~Yi 2 [ (-Y2 + j(3')5- sinh Yi L (Yi+j3') L 
 
 sinh(Yi+Y 2 )^ sinh( -Yi+j 3' )jj 
 sinh YiL (-Yi + j3')L 
 
 Yi / Y 2 (5.11) 
 
38 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 I 
 I 
 
 1 
 
 O 
 
 in 
 
 d 
 
 CM 
 
 O ii 
 
 CO 
 
 CM 
 
 w ■ 
 
 $ ' 
 
 LL. 
 
 o 
 
 CO 
 LU 
 
 _i 
 
 S | 
 
 oc 
 o 
 u. 
 
 n 
 o. 
 
 u_ 
 o 
 
 OS 
 
 s 
 8 I 
 
 LU 
 00 
 
 Csl 
 
 •u 
 
 OS 
 
 •=) 
 
 -•) 
 
 L'_ 
 
 
 
 
 
 
 
 
 
 
 
 II 
 II 
 j| 
 
 1 1 
 
 
 
 
 
 
 
 
 
 
 
 V 
 y 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 \\ 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 it 
 (i 
 
 \ \ 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 it 
 ( i 
 \\ 
 
 ) } 
 
 
 
 
 
 
 
 
 
 in 
 
 CD 
 
 o o 
 
 ll" II 
 
 cj a 
 ■ i 
 
 i 
 ! i 
 03 < 
 
 
 fi 
 
 
 
 
 
 
 
 
 
 
 ft 
 (i 
 
 \\ 
 
 J 3 
 ft 
 
 
 
 
 
 
 
 
 
 
 Ys 
 
 ft 
 \\ 
 
 V \ 
 
 
 
 
 
 
 
 
 
 
 
 ft 
 \\ 
 \\ 
 
 ft 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v » 
 
 { 1 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 f € 
 
 \ \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 \ 
 
 
 
 
 
 
 
 
 00^ 
 
 -^-T*" 
 
 ^i --- 
 
 " „ 
 
 — 
 
 
 
 
 
 
 
 
 
 f i 
 1 i 
 V \ 
 
 s < 
 
 
 
 
 
 
 
 
 
 
 
 Sv 
 
 1 — 
 
 **. „^ 
 
 
 ID <fr 
 
 l £ dl 
 
 oj O 
 
 s 
 
39 
 
 P - IoZo L 
 
 sinh (Yt + j(3')L sinh(^Yi + jP')^ 
 — ^ coth YiL ^ 
 
 sinh YiL ( Yl+ j P ')L 
 
 (-Yi+jP'jL 
 
 cosh(- Yi + j3')^ sinh(-Yi + j3') L " 
 L (-Yi+jP')£ j(- Yl+ jP')Ly 
 
 Yi = Y2 
 
 (5.11) 
 
 Equations 5=11 predict a rather complicated pattern behaviour. However, 
 a study of them allows one to make general predictions concerning the 
 patterns. Equations 5.11 have been plotted in Fig. 5.3 for values of the 
 parameters representative of those of the experimental antenna. The 
 parameters chosen are listed in the graph. 
 
 The behaviour of these patterns will be compared to the experimental 
 results in the next section. 
 
40 
 
6. EXPERIMENTAL RESULTS 
 
 6.1 Patterns 
 
 Patterns have been taken of the axial ly excited traveling wave 
 antenna of dimensions as given in Fig, 2.1 using different coupling 
 plates. Three representative plate constructions are shown in Fig. 2.2. 
 The best patterns were obtained with the aperture comprising a single 
 row of closely spaced 5/16" diameter holes with centers on the longi- 
 tudinal axis of the antenna. Patterns for this aperture are shown in 
 Figs. 6.2a, 6.2b, 6.3a, and 6.3b. Figure 6.2a shows patterns for Eq(4>) 
 and E^ifi) (see coordinate system, Fig. 3.1) taken for 6 = 78°. Fig. 
 6.2b shows patterns of Eq(0) taken for 4> = 0°. Figures 6.2a and 6.2b 
 were obtained with the resistive termination on the exciting waveguide. 
 Figures 6.3a and 6.3b present corresponding patterns with the resistive 
 termination replaced by a short circuit. The patterns obtained with 
 the aperture consisting of a double row of 3/16" diameter holes (Fig. 2.2) 
 are quite similar to these. Figures 6.4a, 6.4b, 6.5a. and 6.5b present 
 the same information for a 1/4" width longitudinal slot which yielded 
 the best of the patterns obtained for slot apertures. These coupling 
 apertures all extended the length of the radiating waveguide. 
 
 For purposes of comparison, the patterns obtained with the radiating 
 waveguide fed at one end are shown in Figures 6.6a and 6.6b. For this 
 case the exciting waveguide has been removed and one end of the radiating 
 waveguide has been replaced by a tapered transition to X-band waveguide. 
 The structure is sketched in Fig. 6.1. When the antenna of Fig. 6.1 is 
 
 41 
 
42 
 
 TZZZZZZZZZZZZZZ 
 
 *s 
 
 Figure 6 I End-Fed Antenna 
 
 excited by the lowest order transverse electric mode polarized as shown, 
 the desired hybrid mode is excited in the radiating waveguide. 
 
 Returning to the original antenna, an estimate of the attenuation 
 constant a 2 of the exciting waveguide can be obtained by comparing the 
 patterns obtained with the resistive absorber to those obtained with 
 short circuit. These patterns were obtained using the antenna for 
 reception. By reciprocity, the same patterns would be obtained if the 
 antenna were used for transmission. For the transmitting case we can 
 write 
 
 Efficiency = 1 - ^ = 1 - jpl (6.1) 
 
 Ei I Pi 
 
 and 
 
 h. . P J. . e °* L (6.2) 
 
 E2 P2 
 
 where 
 
 Ei Strength of the incident wave in the exciting guide, measured at the 
 
 start of the coupling section. 
 Eg Strength of the reflected wave in the exciting guide with short 
 
 ' if' uit., measured at the end of the coupling section. 
 P I' Magnitudes of the front and back lobes, respectively, obtained 
 with the short circuit in place. 
 
43 
 
 3 i+O cm 
 
 E {<P) FOR 6 - 78° 
 
 l (0) FOR 
 
 . 7fjO 
 
 FIGURE 6 2a PATTERNS FOR 5/16" HOLE ARRAY 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
44 
 
 2 99 cm 
 
 3 273 cm 
 
 3 358 cm 
 
 g 3 40 cm 
 
 f 3 46 cm 
 
 E (0) FOR = 0° 
 
 FIGURE 6 2b PATTERNS FOR 5/16" HOLE ARRAY 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
45 
 
 3 40 cm 
 
 E e (0) FOu 9 - 78° 
 
 E (0) FOR 9 
 
 - 78< 
 
 FIGURE 6 3a PATTERNS FOR 5/) 6" HOLE ARRAY 
 
 EXCITING WAVEGUIDE TERMINATED IN SHORT CIRCUIT 
 
46 
 
 CI 2 99 cm 
 
 I 3. 123 cm 
 
 C 3.273 cm 
 c 
 
 ^ 3 358 cm 
 
 ,o cm 
 
 3 16 cm 
 
 E (e) FOR <£ = 0° 
 FIGURE 6 3b PATTERNS FOR 5/16" HOLE ARRAY- 
 EXCITING WAVEGUIDE TERMINATED IN SHORT CIRCUIT 
 
47 
 
 2.99 cm 
 
 3 273 cm 
 
 3 40 cm 
 
 E d {4>) FOR £ - 78° 
 FIGURE 6 4a PATTERNS FOR l/4 n SLOT 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 E (0) FOR 6 - 78 c 
 
48 
 
 3 2 99 cm 
 
 u b 3 123 
 
 cm 
 
 p 3 273 cm 
 
 $ 3 358 
 
 cm 
 
 3 40 cm 
 
 f f 3 % cm 
 
 Ip{6) FOR 0° 
 FIGURE 6.4b PATTERNS FOR 1/^" SLOT 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
49 
 
 3 40 cm 
 
 E e ($) FOR 6 - 78° 
 FIGURE 6 5a PATTERNS FOR 1/4" SLOT 
 
 EXCITING WAVEGUIDE TERMINATED IN SHORT CIRCUIT 
 
 E^(0) FOR 6 78 ( 
 
50 
 
 Q 2 99 cm 
 
 b 3 123 
 
 cm 
 
 C 3 273 cm 
 
 3 358 cm 
 
 3 40 cm 
 
 3 46 cm 
 
 E e {6) FOR 4> - 0° 
 FIGURE 6 5b PATTERNS FOR \/'V SLOT 
 
 EXCITING VVAVEGUIL - TEF&HnATED IN SHORT CIRCUIT 
 
51 
 
 3 46 cm 
 
 b 3 !23 
 (9Q * _\ 
 
 r.m 
 
 Eq(-/0 FOR 6 ■ 78° 
 
 ■e 
 
 FIGURE 6 6a PATTERNS FOR END-FED ANTENNA 
 
 E (0) FOR 9 78° 
 
52 
 
 3 40 cm 
 
 3 46 cm 
 
 E e {6) PATTERNS <fi = 0° 
 FIGURE 6.6b PATTERNS FOR END-FED ANTENNA 
 
53 
 
 Table 6,1 has been prepared for the 5/16" hole array coupling plate from 
 
 the patterns of Figs. 6.3 by use of Eqs. 6.1 and 6.2. This formula for 
 
 Wavelength 
 
 3, 
 
 ,123 
 
 cm 
 
 3. 
 
 273 
 
 cm 
 
 3. 
 
 358 
 
 cm 
 
 3. 
 
 40 
 
 cm 
 
 3, 
 
 46 
 
 cm 
 
 Pi 
 P 2 
 
 
 Efficiency 
 
 1.64 
 
 
 
 .63 
 
 2.10 
 
 
 
 .77? 
 
 2.28 
 
 
 
 .81 
 
 2.20 
 
 
 
 .79 
 
 2.00 
 
 
 
 .75?? 
 
 
 Table 
 
 6. 
 
 1 
 
 0185 
 
 0278? 
 
 0303 
 
 0292 
 
 0256?? 
 
 the efficiency (Eq. 6.1) is valid only if no back lobe occurs with the 
 resistive termination. In practice a small back lobe is usually obtained 
 (see Figs. 6.3). This introduces some error in Table 6.1 as indicated. 
 
 Table 6.2 below has been prepared from Figs. 6.5 for the antenna 
 using the 1/4" coupling slot. Tables 6.1 and 6.2 reveal that the two 
 waveguides are coupled much closer by the hole array than by the slot. 
 
 Wavelength 
 
 Pi. 
 P 2 
 
 Efficiency 
 
 <x 2 
 
 .40?? 
 
 .009?? 
 
 60?? 
 
 .017?? 
 
 .47? 
 
 .012? 
 
 .35? 
 
 .0078? 
 
 48?? 
 
 .012?? 
 
 .29?? 
 
 0067?? 
 
 2.99 cm 1.29 
 
 3.123 cm 1.59 
 
 3.273 cm 1.38 
 
 3.358 cm 1.24 
 
 3.40 cm 1.39 
 
 3 46 cm 1.19 
 
 Table 6„2 
 
 Since the aperture field is approximately equivalent to an array of 
 non»»J ell <hpoles, we would expect to see little energy radiated 
 in the tec to i about 6 ■ 0° However, the Bg(6) patterns of Fig. 6.4b for 
 
54 
 the slot show that considerable energy is radiated in that region. 
 Figure 6.4a indicates that from 3.358 cm to 3.46 cm an undesired mode 
 tends progressively to swamp out the desired mode. Tie patterns for 
 Eq(<£) show the presence of a mode having a high phase velocity and a 
 larger magnitude than that of the desired mode. Also, although E^ for 
 the latter mode is zero, Fig. 6.4a shows that this component is quite 
 large. Both of these effects can be ascribed to the ridged waveguide 
 mode. (Incidentally, it is found that while the width of the slot is 
 not critical, its placement with respect to the axial center line of 
 the two waveguides is quite critical. Noticeable assymetries in the 
 patterns of Eq(<£) develop when the center line of the slot is displaced 
 from the center line of the guides by as little as .001".) 
 
 The use of the hole array cuts out the ridged waveguide mode. 
 Comparing Fig. 6.2a to 6.4a and Fig. 6.2b to 6.4b shows the marked 
 improvement effected by replacing the slot with the array. The Eq(0) 
 patterns indicate the predominance of the desired mode; the E^(0) 
 patterns show a large reduction in the magnitude of E^; and the Eq(0) 
 patterns reveal a large reduction in side lobe radiation. 
 
 The general behaviour of the patterns obtained using the hole array 
 follows qualitatively that predicted by Eqs 5.11. The behaviour of 
 Eqs. 5.11 is such as to indicate, for the magnitudes and variation with 
 A. of Yi and Y2 representative of this antenna, that the pattern be- 
 haviour would be as follows: In the neighborhood of X = 3 cm the Eg(0) 
 pattern comprises two main lobes, one at = 0°, the other at = 180°, 
 the back lobe being some 65 percent of the forward lobe. A small side 
 
55 
 lobe radiation is also indicated. As X increases, the back lobe reduces 
 in size; for X up to about 3=25 cm, the main lobes lie on the axis: for 
 X > 3.25 cm, the forward beam begins to split into two conical beams 
 corresponding to the two y involved. As X is increased, the size of the 
 back lobe tends to increase, A study of the patterns of Fig. 6.2a and 
 6,2b shows that, in the main, this behaviour is obtained in practice. 
 
 The patterns of Fig, 5.3 are replotted in polar form in Fig. 6.7. 
 These patterns were calculated from Eq. 5.11 for several combinations of 
 the parameters in the range of interest. Curve A gives the pattern 
 which might be expected at the short wavelength end of the antennas 
 operating range. It shows a higher back lobe and lower side lobes than 
 are observed experimentally. The large back lobe seen in Curve A is due 
 in part to the assumption employed in its calculation that a 2 - a«. Had 
 ot 2 been assumed less than tt lt which would be in accord with experimental 
 evidence (see Table 6.1 and Fig. 6.12), the front to back lobe ratio 
 given for (3,, = (3 2 by Eqs. 5.11 would have been larger. Because it has 
 been assumed in curve A that C/V 1, this curve would be expected to 
 show a lower side lobe level than the experimental pattern taken at 
 2 99 cm, for at this wavelength the C/V of the antenna is about 1.1. 
 
 Curve B shows a type of pattern to be expected in the middle of the 
 operating range of the antenna. 
 
 Curve C or D predicts the pattern at the long wavelength end of the 
 operating range, the principal difference in the two being in the shape 
 '< f the back Lobe. The estimates of the phase velocities for the two 
 'Jes obtained iri Section 4 indicate that P< (3 2 a t this wavelength, 
 
56 
 
 3i/P 
 
 3 2 /3 
 
 2 9 cm 
 
 pjp = .99 (3 2 /p = 95 X = 3 17 
 
 Pi/P 
 
 97 p 2 /p = 93 X - 3.46 
 
 Pi/0 - 93 (3 2 /|3 - ,S7 X 
 
 FIGURE 6 7 PATTERNS CALCULATED FROM EQUATIONS 5 II FOR VALUES 0! 
 PARAMETERS AS LISTED 
 
 FOR ALL CURVES a 1 > ou - 048 = 78° L = 27 
 
57 
 Pattern D corresponds to this situation. However, pattern C more nearly 
 resembles the experimental pattern obtained at 3.46 cm than does pattern 
 D, This would seem to indicate that in practice (3* > (3 2 at this 
 wavelength. As in pattern A, the back lobes of patterns B, C, and D 
 would be smaller had ot 2 been assumed less than <t x . 
 
 In view of the approximations employed in the theory, it is felt 
 that a reasonable agreement between theory and experiment is indicated. 
 In the end-fed antenna, while there is no possibility of producing a 
 ridged-waveguide mode, the lobe structure of its patterns deviates 
 considerably from that to be expected of the desired mode, The patterns 
 obtained with the axially excited antenna using the hole array seem to 
 show a predominance of the desired mode. 
 6 2 Aperture Measurements 
 
 A series of probe measurements has been made of the phase and 
 amplitude distribution of the field at the surface of the antenna. The 
 equipment used is sketched in Fig. 6.8. The traverse was built in such 
 a way that the probe was moved along a straight line parallel to the 
 axis of the antenna with its tip touching the surface. The electric 
 field picked up by the probe was compared by a null technique to the 
 reference signal which was adjustable in phase and magnitude. The 
 traverse mechanism was placed behind blocks of absorbing material to mimi- 
 mize reflections. Most of the measurements were taken with a coaxial probe be- 
 <t its small size and good sensitivity to the desired mode in the an- 
 a The probe was made up from a 10" length of rigid wall 50 .0. cable which 
 bad «n outside diameter of less than 4mm, Some measurements were taken witha 
 
58 
 
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59 
 probe made of a 12" length of X-band waveguide tapered to about 1/8" * 
 
 9" at its tip. The flexible cable connecting the probe to the magic 
 T was a 3' length of 50 fl cable, selected from among several tested for 
 minimum variation in transmission with flexing, and was arranged in a 
 long loop to minimize its flexing with probe travel- 
 
 In all the graphs the phase readings were taken at the positions 
 where the amplitude distribution was a maximum or a minimum The abso- 
 lute level of the amplitude distributions in the graphs is arbitrary. 
 
 Figures 6.9 and 6. 10 give the amplitude and phase distribution of 
 the aperture field of the axially excited antenna using the 5/16" hole 
 array coupling plate at a wavelength of 3.123 cm with the exciting 
 waveguide terminated in an absorber and in a short circuit, respectively. 
 They correspond to the patterns of the same wavelength given in Figs. 
 6.2 and 6.3. Figures 6.11 and 6.12 give the same information for the 
 1/4" coupling slot and correspond to patterns given in Figs. 6.4 and 
 6.5. For purposes of comparisons, Fig. 6.13 shows the aperture field 
 of the end-fed antenna taken at the same wavelength. The best pattern 
 the end- fed antenna produced was obtained at X r 3.273 cm. Figure 6.14 
 shows the aperture field of this antenna obtained at that wavelength. 
 All of these distributions were measured with the coaxial probe, 
 
 Since the antenna produces its best patterns when using the hole 
 array, a series of aperture measurements was obtained of this arrange 
 rnent at different wavelengths and is shown in Figs. 6.15 through 6.22. 
 
 [napection of Eqa. 5.3 shows that the aperture distribution they 
 l"' ,Jlf ' ' '■ '" reasonable agreement with that obtained when using the 
 
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 Distance along Antenna in cm 
 
 FIGURE 6 
 
 9 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 
 5/16" HOLE ARRAY COUPLING PLATE 
 
 
 EXCITIIiG WAVEGUIDE TERMINATED IN ABSORBER 
 
 k = 3 123 
 
 cm k/K A = 1 07 COAXIAL PROBE 
 
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 Distance along Antenna in cm 
 
 FIGUkE 6 10 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 5/16" HOLE ARRAY COUPLING PLATE 
 EXCITING WAVEGUIDE TERM! ATED IN SHORT CIRCUIT 
 
 3 123 /// x | 07 COAXIAL PROBE 
 
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 Distance along Antenna in cm 
 
 IGURE 6 II APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 1/4" COUPLING SLOT 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 A = 3 123 cm X/\ x 1 07 COAXIAL PROBE 
 
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 FIGURE 6 12 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 1/4' COUPLING SLOT 
 EXCITING V'AVEGUIDE TERMINATED IN SHORT CIRCUIT 
 
 3 l23 cm \/\ x 1 07 COAXIAL PROBE 
 
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 Distance along Antenna in cm 
 
 -I8URE 6 13 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 END- FED ANTENNA 
 
 V = 3 123 cm A/X x 1 07 COAXIAL PROBE 
 
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 Distance along Antenna in cm 
 
 I6URE 6 14 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 END-FED ANTENNA 
 
 3 273 cm A/A x 1 038 COAXIAL PROBE 
 
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 FI8URE 6 !5 
 
 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 
 5/ «8 HOLE ARRAY COUPLING PLATE 
 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 
 X - 2 7Q cm X/\ x - 1 15 COAXIAL PROBE 
 
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 FIG'iRE 6 «6 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 5/ '6 HOLE ARRAY COUPLING PLATE 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 2 9i* cm A/* x 1 !I2 COAXIAL PROBE 
 
68 
 
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 Distance along Antenna in cm 
 
 FIGURE 6 17 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 5/16" HOLE ARRAY COUPLING PLATE 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 X = 2 99 cm X/X x - 1 098 COAXIAL PROBE 
 
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 FIGURE 6 18 APER 
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 3 32 36 40 44 4 
 Distance along Antenna in cm 
 
 TURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 HOLE ARRAY COUPLING PLATE 
 TING WAVEGUIDE TERMINATED IN ABSORBER 
 3 273 cm X/A x -■ 1 022 COAXIAL PROBE 
 
 3 
 
70 
 
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 Distance along Antenna in cm 
 
 FIGURE 6 !9 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 5/16^ HOLE ARK/' COUPLING PLATE 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 A = 3 358 cm \/X x - 988 COAXIAL PROBE 
 
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 Distance aiong Antenna in cm 
 
 FIGURE 6 20 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 5/16" HOLE ARRAY COUPLING PLATE 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 3 40 cm A// x 976 COAXIAL PROBE 
 
72 
 
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 FIGURE 6 21 
 
 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 
 
 5/16" HOLE ARRAY COUPLING PLATE 
 
 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 
 A - 3 46 cm 
 
 \/\ x 0, 925 COAXIAL PROBE 
 
73: 
 
 Amplitude in db 
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 8 32 36 40 44 48 
 Distonce along Antenna in cm 
 
 JRE PHASE VIP AMPLITUDE DISTRIBUTIONS 
 HOLE ARRAY COUPLING PLATE 
 ING WAVEGUIDE TERMINATED IN ABSORBER 
 
 A// y dl2 COAXIAL PROBE 
 
74 
 hole array, as the following shows, Eqs. 5.3 state that for Yi f Y2 
 
 wher< 
 
 Assume: 
 
 V(x) Z Fi + F 2 + B = R + B 
 
 Fi = -e sinh(Yi + Y2)jr 
 
 it " Y2X L 
 
 r 2 ~ e sinh Yi L 
 
 V Y 
 
 B " e sinh(-Yi + Y2)^ 
 
 X - 3.25 cm 
 
 oti ^ .048 (Estimated from Fig 4.9) 
 
 a 2 = 028 (From Table 6,1) 
 
 Pi - 1.86 (From Fig. 4=9) 
 
 2 1,91 (From Fig. 4.9) 
 
 Curves of |R| and |B| are plotted in Fig. 6.26 for the values of the 
 parameters indicated in the foregoing. Curves of |R| + |B| and lR| - |B| 
 which define the envelope of the standing wave pattern produced by the 
 combination of R and B are also plotted in the same figure. The latter 
 two curves are plotted in decibels to allow them to be readily compared 
 with the measured amplitude distributions obtained for the hole array 
 (Figs. 6.9, and 6.15 through 6, 22). Although the calculated distribution 
 supposedly portrays the situation at X - 3.25 cm. it actually resembles 
 the distribution measured at X = 3.123 cm more nearly than it does that 
 measured at 3.273 cm. The calculation predicts a higher standing wave 
 ratio than is obtained in practice but this would be expected since it 
 
75 
 is based on the approximation that there is a total short circuit at the 
 ends of the radiating guide. 
 
 The phase velocities read off the phase graphs of Figs. 6.15 through 
 6.22 have been plotted in Fig. 4.9. The "measured phase velocity" is 
 obtained by fitting a measured phase distribution (see Fig. 6.15 to 
 6.22) to a straight line. This measured phase velocity should fall be- 
 tween the velocities of F x and F 2 . It is seen from Fig. 4.9 that the 
 measured values fall closer to F 2 than Fi and this is to be expected 
 since the amplitude of F 2 is, on the average, greater than Fi . 
 
 In Figs. 6.23, 6.24, and 6.25, aperture distributions measured at 
 3.123 cm with the tapered waveguide probe are shown for the 5/16" hole 
 array, 1/4" slot, and the end- fed slab. Figure 6.23 shows that the 
 waveguide, probe picks up a field component present with the hole array 
 which goes undetected by the coaxial probe. Its high relative phase 
 velocity (1.3 from the phase graph) prevents its being detected in the 
 patterns. In the cases of the slot and the end- fed antenna,, the 
 patterns obtained with both probes are about the same. All the aperture 
 distributions for the end fed antenna show a rather large change in 
 amplitude close to the feed end,, This may be due to the presence of 
 high order modes from the discontinuity at the end. 
 
76 
 
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 Distance along Antenna in cm 
 
 3 
 
 FIGURE 6 23 APERTURE PHASE AND AMPLITUDE DISTRIBUTIONS 
 5/l6 ;t HOLE ARRAY COUPLING PLATE 
 EXCITING WAVEGUIDE TERMINATED IN ABSORBER 
 \ ■■ 3 123 cm WAVEGUIDE PROBE 
 
77 
 
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 3 32 36 40 44 48 
 Distance along Antenna in cm 
 
 E PHASE AND AMPUTUDE DISTRIBUTIONS 
 UPLIHG SLOT 
 
 G WAVEGUIDE TERMINATED IN ABSORBER 
 23 cm WAVEGUIDE PRODE 
 
78 
 
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 FIGURE 6 25 APERTL 
 END-FE 
 
 K - 3- 
 
 3 32 36 40 44 4 
 Distance along Antenna in cm 
 
 re phase and amplitude distributions 
 :d antenna 
 
 123 cm WAVEGUIDE PROBE 
 
 3 
 
79 
 
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 i Distribution in db 
 
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 -12 
 IRE 6 26 
 
 -8-4 4 8 12 
 Distance along Antenna in cm 
 
 MAGNITUDES OF THE FORWARD AND BACKWARD TkAVELING WA^S 
 IN THE RADIATING WAVEGUIDE AS CALCULATED FROM EQUATION 
 5 3, TOGETHER WITH THEIR SUM AND DIFFERENCE THE FOL- 
 LOWING PARAI^TER VALUES ARE EMPLOYED: 
 
 0485 fl, 1.86 
 
 0278 |3 2 - 1.91 
 
7. CONCLUSIONS 
 
 This work has demonstrated that it is feasible to excite a traveling 
 wave dielectric slab antenna by coupling it along its direction of 
 propagation to an adjacent waveguide. A higher degree of mode purity has 
 been achieved by this means than by exciting the antenna at one end. The 
 theory employed to estimate the phase velocities in the radiating and 
 exciting guides has been shown to give results in close agreement with 
 measured values. The aperture field of the desired hybrid mode has been 
 shown to be equivalent to an array of normal electric dipoles provided the 
 surface density of the dipole distribution is related in a certain simple 
 fashion to the Hertzian potential describing the hybrid field. The theory 
 developed to describe the action of the coupling mechanism has been shown 
 to be able to predict aperture distributions and radiation patterns in 
 agreement with those obtained in practice. 
 
 Several criteria for the practical design of a wide bandwidth end- 
 fire antenna employing axial excitation have made themselves apparent 
 during the course of the investigation. To obtain an end-fire beam over 
 a wide range of frequencies, the relative phase velocities of both the 
 exciting and radiating guides must be maintained very near unity over the 
 band, and this can be accomplished by proper dimensioning of the guides. 
 The coupling theory indicates that a large attenuation constant in the 
 radiating guide, a small attenuation constant in the exciting guide, 
 together with equal or very nearly equal phase constants are required to 
 obtain patterns having large front to back lobe ratios when the radiating 
 
 80 
 
81 
 
 guide is abruptly terminated. The magnitude of the coupling between the 
 guides maximizes under these same conditions. Most of these principles 
 are, perhaps, intuitively obvious. This work shows that they lead to a 
 method of exciting surface wave antennas which has considerable practical 
 value. 
 
BIBLIOGRAPHY 
 
 1. Rotman, Walter The Channel Guide Antenna, Air Force Cambridge 
 Research Laboratories, Cambridge, Massachusetts, January, 1950. 
 
 2. Rotman, Walter Metal Clad Progressive Phase Dielectric Antenna, 
 Air Force Cambridge Research Laboratories, Cambridge, Massachusetts, 
 January, 1951. 
 
 3. Rumsey, V. H. "Traveling Wave Slot Antennas," Journal of Applied 
 Physics, Vol, 24, No. 11, 1358-1365, November, 1953. 
 
 4. Walter, Carlton H, "End-Fire Slot Antenna," Ohio State University 
 Research Foundation, Columbus, Ohio, Technical Report No. 486-12, 
 prepared under Contract No. AF 18(600)-85 with Air Research and 
 Development Command, Wright Air Development Center, Wright-Patterson 
 Air Force Base, Ohio. 
 
 5. Pincherle, L. "Electromagnetic Waves in Metal Tubes Filled 
 Longitudinally with Two Dielectrics," Physical Review, 66, September, 
 1944, p. 118. 
 
 6. Rumsey, V. H. TE and TM Traveling Wave Slots, Paper presented at 
 URSI IRE Conference, San Diego, California, April, 1950. 
 
 7. Mueller, G. "Dielectric Antenna," Annual Report (October, 1949) and 
 Quarterly Progress Reports Nos. 1-7 (October, 1948 to July, 1950) 
 Contract No. W36-039-ac~38168, Ohio State University Research 
 Foundation, Columbus, Ohio. 
 
 8. Hansen, W. W. Radiation Electromagnetic Waveguide , U.S. Patent 
 2,402,622, 26 November 1940. 
 
 9. Cullen, A. L. "Channel Section Waveguide Radiation," Phil. Mag., 
 Vol. U0, April, 1949, p. 417. 
 
 10. Hines, N. and Walter, C. W„ The Long Slot Antenna, Paper presented 
 at Second Annual Technical Conference, IRE, Dayton, Ohio, 5 May 1950, 
 
 11 
 
 Hansen, W. W. , Seeley, S. :, and Pollard, E. C. Notes on Microwaves, 
 Radiation Laboratory, Massachusetts Institute of Technology, 
 Cambridge, Massachusetts. 
 
 12. Final Engineering Report 400-11, Traveling Wave Slot Antenna, 
 
 Antenna Laboratory, The Ohio State University Research Foundation; 
 prepared under Contract AF 33(038)^9236 with Air Materiel Command^ 
 Wright-Patterson Air Force Base, Ohio, 6 January to November 14, 1951 
 
 82 
 
83 
 BIBLIOGRAPHY (Cont. ) 
 
 13. Van Bladel, J. and Higgins, T. J. "Cutoff Frequency in Two Dielec- 
 tric Layered Rectangular Waveguides," Journal App. Phys., Vol. 22, 
 March, 1951, pp. 324-334. 
 
 14. Montgomery, C. G. , Dicke, R. H. ', and Purcell, E. M. Principles of 
 Microwave Circuits, McGraw Hill Book Co., Inc., New York, 1948, 
 pp. 385=389. 
 
 15. Booker, H. G. "Girders and Trenches as End-Fire Antennas," Report 
 N30, THE Journal, Swanage, England, 1941. 
 
 16. Cullen, R. L. "On The Channel Section Waveguide Radiator," Phil. 
 Mag., Vol. UO, April, 1949. pp. 417-428. 
 
 17. Hines, J. N. , Rumsey, V. H. , and Walter, C. H. "Traveling Wave 
 Slot Antennas," Proceedings of IRE, Vol. hi, No. 11, November, 1953. 
 
 18. Miller, S. E. "Coupled Wave Theory and Waveguide Applications," 
 Bell System Technical Journal, May, 1954. 
 
 19. Rumsey, V. H. Huygen's Principle For Electromagnetic Waves, Report 
 by The Ohio State University Research Foundation, Columbus 10 , Ohio, 
 Contract DA 36-039 sc-5506, Signal Corps Project 22-122 B-0 
 (C-036401.1), Department of Army Project 3-99-05-022. 
 
APPENDIX A 
 CALCULATION OF RADIATION FIELDS 
 
 A I Equivalence of the Assumed Aperture Distribution To an Array of 
 Electric Dipoles Normal to the Ground Plane 
 
 It is assumed that the tangential electric field in the aperture 
 
 (see Fig. 3.1) satisfies 
 
 jweE a - V U(x,y) (A-l) 
 
 where 
 
 U - Ae =YxX cos ^ • (A-2) 
 
 b 
 
 The subscript "a" denotes points in the aperture. By Eq. 3.1, the 
 electric vector potential resulting from this aperture distribution is 
 
 J-fCCixL dt d .. T J_Tr i i,i I (l s^Ids. (A-3) 
 
 2n ) \ ^ r jwe2rc J J a r 
 
 Now 
 
 zxVJ = -VxzU 
 
 d a 
 
 and 
 
 0(V x b) = V x (0b) - (V0) x b 
 
 where b is any vector and <f> is any scalar. 
 
 Setting 
 
 b = z U and - siEl (A- 4) 
 
 we obtain 
 
 E - - t-V C C [V x (1 U ^-) - V a (fi^I) x £ U] ds 
 
 j(i)e2tc J ) r r 
 
 84 
 
85 
 
 Consider 
 
 V x (z U ^^) ds 
 
 - t x V (U a^ll) ds. (A-5) 
 
 Nov 
 
 nxV0ds= \ d | 
 
 5 
 
 where s is an unclosed surface bounded by the closed curve, C; the unit 
 vector n is normal to s and directed positively outwards, and the vector 
 j_ is tangential to C. Using this relation gives 
 
 - 2 x v a (U ^^) ds 
 -a r 
 
 u^ai 
 
 (A-6) 
 
 uy 
 
 T 
 
 ~r 
 
 Figure A I 
 
 The surface 'a" is, as before, the aperture of the antenna and C is its 
 rim Now let us impose the condition that tangential E at the rim 
 vanish. Since E : V U therefore U = on e will insure that the rim 
 condition! are satisfied. Thus F reduces to 
 
 j(ije2rc 
 
 ?a 
 
 g x 7, U ds 
 
 (A-7) 
 
86 
 and the field radiated from the aperture is given by 
 
 L, = V x F = - t-V 2 x C Cv kl?I x^Uds (A-8) 
 
 ^ jwe2n J j " | r | 
 
 a 
 
 upon replacing V Q by -V. 
 
 Consider an array of electric dipoles normal to and disposed over 
 the surface "a," i.e., the aperture of the antenna. Let the ground 
 plane be removed so that the dipole array is in free space. The array 
 distribution is described by 
 
 zj(x\"y) (A-9) 
 
 where J is the surface density of dipole moment and x, y are the x and y 
 coordinates in the aperture. The magnetic vector potential due to this 
 
 source is 
 
 A ■ \ Ci 3(x,y) s^L ds . (A-10) 
 
 ) ) 4n:r 
 
 The field in space due to this array is 
 
 — — VxVxU 
 
 j(jje4n; ) ) r 
 
 F^ -■ — i^ V x V x \ { \z J(x,y) ^^ds (A-ll) 
 
 which can be transformed to 
 
 E. = -l-Vx^Vx £j(x,y) ^ds 
 a jwe4Tt J ) r 
 
 -1- 7i^ [J(x,y) e-^ Vxt^V (t25l) x i J(x,y)] ds 
 
 jwe4rc 
 
 i n rr. L-jPr 
 
 j Cjje27i 
 
 V x \ \ V ft 
 
 x z % J(x,y) ds . (A-12) 
 
87 
 Inspection of Eqs. A-8 and A- 12 shows that 
 
 Ew " Ed 
 
 J(x,y) - -2 U . (A-13) 
 
 A 2 Calculation of the Radiation Field from the Aperture Field 
 
 The magnetic vector potential due to the assumed aperture distri- 
 bution is, therefore, 
 
 b/2 L/2 
 
 Y x x rty e -j(3r 
 •L?2 
 
 A = z A z = -±_ \ J Cl e x cos -^ *^- dxdy (A- 14) 
 
 -b/2 
 
 Applying the usual approximations to this integral we obtain for the far 
 
 field 
 
 b/2 L/2 
 
 \-C^ C .jPy.ine.i n * co . 2L dy C .(-VJP «ne cos *)x dx 
 
 b/2 - L/2 
 
 This can be written as 
 
 A z = C ^ ^T^- M '<# M e .*>- 
 
 z 4Ttr a D 
 
 Picking out only the terms of order 1/r in the expression 
 
 H ■ V x A 
 results in 
 
 (A- 15) 
 
 H e » H r - ty - -g& - -sin 9 -£ . (A 16) 
 
 Again retaining only terms of order 1/r we have 
 
 H^ = 3 J£l. c -JP r I a <9,0) I b (9,0) sin 9 . (A-17) 
 
 ^ 4ftr 
 
 The electric finld is obtained as 
 
 K r K A - E e » n ty - J -^- £^- I a (9,0) I b (9,0)sin 9. 
 
 (A 18) 
 
88 
 Lumping all constants together we have 
 
 E e = C sl^L l a (e,0) I b (0,0) sin 9 . (A-19) 
 
APPENDIX B 
 DETERMINATION OF PHASE CONSTANTS 
 
 B I H 7 = Modes. Exciting Waveguide 
 
 ■z 
 
 Assume 
 
 H ■ V x z f (B-l) 
 
 where, in a homogeneous region, f is a solution of the scalar wave 
 
 equation 
 
 V 2 f + (3 2 f = (B-2) 
 
 2 2 
 
 with (3 = w |ie . The scalar f can be written as 
 
 f - e =J,3xX F(z,y) (B-3) 
 
 where 3 X is the phase constant in the x direction. Using Eq. B-3, 
 
 Eq, B 2 becomes 
 
 3 2 F 3 2 F 2 2 , 
 
 5-a- + s -r - (-P + P x )F . (B-4) 
 
 Oy dz x 
 
 A function F which satisfies Eq. B-4 and which yields, through Eq. B~l 
 
 and the relation 
 
 jweE = V x H r 
 
 electric and magnetic fields which satisfy the boundary and interface 
 
 conditions imposed upon them by the guide of Fig. 4.1 is given by 
 
 F " Ai cos 3. (z - a) sin — — in Region A 
 
 fifty 
 "b" 
 
 nn;y 
 b" 
 
 Aj cos p_ z sin — — in Region B 
 
 provided that 
 
 n ■ l,2,3,iii (B-5) 
 
 Ai . cos ^ d (B6) 
 
 A y cos p. (d a ) 
 
 89 
 
P B 2 - (3 Bz 2 (B-8) 
 
 90 
 and 
 
 £ b ^Az tan Paz < d a ) " £ a 3 a .. tan 3 Bz d * ( B ~ 7 > 
 
 Equation B-7 together with the general relation 
 
 3a 2 - Pa 2 = 3 2 + — 
 A Az x I k 
 
 constitute the solution for the propagation constants of the H = 
 modes in the waveguide of Fig. 4.4° The lowest order mode corresponding 
 to n - 1 is the desired mode in this investigation. 
 B.2 E z = Modes 
 
 Since it is known that E_ - modes can exist in the configuration 
 of Fig. 4.1, they were investigated by a procedure similar to the fore- 
 going so that waveguide dimensions could be chosen which would suppress 
 them. A function F for these modes is given by 
 
 nrcy 
 F = Ai sin p A (z-a) cos — — in Region A 
 
 nrty . 
 - A 2 sin p R z cos — - — in Region B 
 
 provided that 
 
 and 
 
 n = 0,1,2,3,-4 (B-9) 
 
 Ai sin (3 D d . ,„. 
 
 — = — -&* — - (B-10) 
 
 A 2 sin (3 (d-a) 
 
 Pbz tan P A z ^ d - a) = ^z tan PBz d (B" 11 ) 
 
 The phase constants for these modes can be obtained from Eqs . B-ll and 
 B-8. 
 
 Table B-l lists cut-off wavelengths calculated for the dimensions 
 employed in the experimental antenna. 
 
91 
 
 Mode Waveguide Cut-Off Wavelength 
 
 Radiating 
 
 Exciting 
 
 Radiating 
 
 Exciting 
 
 Exciting 
 
 Exciting 
 
 Table B-l 
 
 H z = 0, 
 
 n = 1 
 
 H z - 0, 
 
 n = 1 
 
 H z = 0, 
 
 n = 2 
 
 H z - 0, 
 
 n - 2 
 
 E z = 0, 
 
 n = 
 
 E z ■ 0, 
 
 n = 1 
 
 4. 
 
 28 
 
 cm 
 
 4. 
 
 66 
 
 cm 
 
 2. 
 
 48 
 
 cm 
 
 2. 
 
 41 
 
 cm 
 
 2. 
 
 2 
 
 cm 
 
 2, 
 
 05 
 
 cm 
 
APPENDIX C 
 INVESTIGATION OF THE COUPLING MECHANISM 
 
 We obtain the contribution to the voltage and current at any point 
 -L/2 £ x £ L/2 on a transmission line due to an impressed shunt current 
 generator having infinite impedance and unit magnitude placed at a 
 point -L/2 £ 5 £ L/2 on the line. The transmission line of character- 
 istic impedance Z is terminated in impedances Z^. and Z 2 at -L/2 and 
 +L/2, respectively. Then the total voltage and current at x due to an 
 arbitrary impressed linear current density !'(£,) is obtained by an 
 integration. 
 
 -L/2 
 
 We can write, assuming a unit current generator at <5, 
 
 I(x,£) ■ P ± cosh y(x + L/2) + Qi sinh y(x + L/2) 
 V(x,£) ■ -Z [Pi sinh y(x + L/2) + Qi cosh y(x + L/2)] 
 
 < £ 
 
 (C-l) 
 
 I(x,£) = P 2 cosh yC-x + L/2) + Q 2 sinh y(-x + L/2) 
 
 x > a 
 
 V(x,£) ■ Z [P 2 sinh y(-x + L/2) *■ Q 2 cosh y(-x + L/2)3. (C-2) 
 
 92 
 
93 
 At x - 5 we have 
 
 V(Z + 0,£) - V(5 - 0,8.) = 
 
 IU + 0,£) - 1(5 - 0,£) - -1 . (C-3) 
 
 At x = -L/2 the voltage -current ratio is 
 
 -z =* -z. "3i 
 
 Zl Zo p, 
 
 and at x = L/2 the voltage-current ratio is 
 
 Q 2 
 
 Z2 = Zo — 
 
 "2 
 
 Hence Eqs . C-l and C-2 can be rewritten as 
 
 I(x,£) - P[Z cosh y(x + L/2) + Zi sinh y(x + L/2)] 
 
 x < I 
 V(x,£) = -Z P[Z sinh y(x + L/2) + Z t cosh y(x + L/2)] (C-4) 
 
 I(x,£) = Q[Z cosh y(L/2 - x) + Z 2 sinh y(L/2 - x)] 
 
 x > £ 
 V(x,£) = Z Q[Z sinh Y (L/2 - x) + Z 2 cosh y(L/2 - x)] (C-5) 
 
 where P B Pi/Z and Q ' P 2 /Z . 
 
 At x - £ we have 
 
 -ZoP[Z sinh Y(^ + L/2)+Z lC osh yU + L/2)] » Z Q[Z sinh Y(L/2-£)+Z 2 cosh y(L/2-£)] 
 
 (06) 
 
 Let 
 
 -DiP ■ Zo sinh y(L/2--£) + Z 2 cosh y(L/2 £) (C-7) 
 
 and 
 
 dO ■ Zo sinh y(L/2+£) + Zr cosh y(L/2 + £). (C-8) 
 
 At x £ we also have 
 
 QCZoCosh Y(L/2-5)+Z a «inh Y(L/2-£)]-P[Z cosh Y(L/2+£)+Z*inhY(L/2+£)3— 1. 
 
 (C 9) 
 
94 
 Therefore 
 
 [Zosinh Y(L/2+£)+Z lC osh Y (L/2+£)] r 
 
 ' " ! [Zocosh Y(L/2-g)+Z 2 sinh y(L/2-£)] 
 
 Di 
 
 [Zpsinh Y (L/2-g)+Z 2 cosh y(L/ 2-g)] . L , T ,« BV „ • L „ ,« .m , 
 
 + [Zocosh Y(L/2+£)+Zismh y(L/2+£)] =-1. 
 
 Di 
 
 (C-10) 
 
 Solving for Di we obtain 
 
 -Di = (Z 2 + ZiZ 2 ) sinh yL + Zo(Zi + Z 2 ) cosh yL . (C-ll) 
 The constants being evaluated, we can write, finally, 
 
 I(x,£h--^[Z sinhY(L/2-£)+Z 2 coshY(L/2~£)] [Z coshY(L/2+x)+Z lS inhY(L/2+x)] 
 Di 
 
 V(x,£)=— [Z sinhY(L/2-£)+Z 2 coshY(L/2-£)] [Z sinhY(L/2+x)+Z lC oshY(L/2+x)] 
 Di 
 
 x < I (C-12) 
 
 I(x 8 £)=-L[Z sinhY(L/2^)+Z 1 coshY(L/2^)] [ZoCoshY(L/2-x)+Z 2 sinh Y (L/2-x)] 
 Di 
 
 V(x,g)=— [ZosinhYa^+fiJ+ZiCoshyt^+fi)] [Z sinhY(L/2"x)+Z 2 coshy(L/2-x)] 
 Di 
 
 x > I . (C-13) 
 
 The total voltage at a point x on the line due to an impressed 
 current density I'(S) is, therefore, 
 
 X 
 
 V( x )= C — [Z sinhY(L/2+£)+Z lC oshY(L/2+£)] [Z sinhY(L/2-x)+Z 2 cosh Y (L/2-x)] 
 
 -1/2 I'(£)d£ 
 
 1/2 
 + C ~[ZosinhY(L/2-5)+Z 2 coshy(L/2-.£).] [Z sinhy(L/2+x)+Z 1 coshY(L/2+x)] 
 
 I'(S)dg 
 
 (OH) 
 
 A similar expression results for the current I(x). Equation 5.1 of the 
 text follows from Eq„ C-14.