ABSTRACT INVESTIGATION OF OPEN-CAVITY RADIATORS By Min-Houng Hong An open-cavity radiator, or a simplified model of recently developed ”backfire" antennas, is investigated in this research. This antenna consists of a simple, open-ended circular cavity with a primary radiator placed at an appropriate location inside the cavity. The circuit property and the radiation characteristics of this radiator are studied. The waveguide excitation theory is employed to find the field excited in the cavity. The aperture field is then determined by sum- ming the propagating modes at the open end of the cavity. Subsequently, the radiation field is calculated based on the aperture field. The input resistance of the radiator is obtained from the total radiated power carried by the propagating modes and the input current of the primary radiator. Various primary radiators such as a dipole, a dipole array, a transmission line and a circular loop are considered in this study. Min-Houng Hong An eXperimental study has been conducted inparallel with the theoretical analysis and a satisfactory agreement has been obtained between theory and experiment. This study may help clarify the mechanism of radiation of this new radiator and prove useful in its optimum design. INVESTIGATION OF OPEN- CAVITY RADIATORS BY Min-Houng Hong A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1970 64-17% 7-/-70 To my par ents Mr. 8: Mrs. Chien-Chuan Hong ii ACKNOWLEDGMENTS The author wishes to eXpress his sincere appreciation to his major professor, Dr. K. M. Chen, for his guidance and encourage- ment in the course of this research. He also wishes to thank the committee member Dr. D. P. Nyquist for giving the valuable suggestions in this research, and to the other members, Dr. B. Ho and Dr. R. Hamelink, for reading the thesis. Finally, the author owes a special thank to his wife, Ai-Hwei, for the encouragement and understanding that only a wife can give. The research reported in this thesis was supported by Air Force Cambridge Research Laboratories under contract F 196 28- 70-C-OO7Z. iii TABLE OF CONTENTS ACKNOWLEDGMENTS ................... LIST OF FIGURES O O O O O O O O O O O O O O O O O O O O O LISTOFTABLES. ...... INTRODUCTION ..... . . . . . . . ......... . WAVEGUIDE EXCITATION THEORY AND THE RADIATION FIELDS OF THE OPEN-CAVITY RADIATORS 0 O O O O O O O I O O O O O O O O O O O O O O O 1 Geometry and Statement of the Problem . . . . . . 2 Waveguide Excitation Theory . . . . . . . . . . . . 2. 2.1 General Field Expressions in Waveguide. . 2.2.2 LorentzLemma. . . . . . . . . . . . . . . 20 20 2.2.3 Excited Fields. . . . . . . ...... . . . 2. 3 Calculation of Reflection Coefficients rlq and I‘Zq. Z. 4 Input Resistance of the Primary Radiator ..... . 2. 5 Radiation Fields of the Open-Cavity Antenna . . . . Z. 5. 1 Geometry and General EXpressions for the Radiation Fields. . . . . . . . . . . . . Z. 5. 2 Evaluation of F9q(9, 4)) and F¢q(8, ¢) for TEModes.................. 2.. 5. 3 Evaluation of F9q(9, 4)) and F¢q(8, 4)) for TMModeS.................. .4 Modified F9q(9,¢) and F¢q(9,¢). . . . . . . 5 Radiation Fields Due to the Individual Waveguide Modes . . . . . . . . ...... OPEN-CAVITY RADIATORS WITH DIPOLE AND DIPOLEARRAYEXCITERS. . . . . . . . . . . . . . . . Introduction . . . ...... . . . . ........ 3. I 3. Z EXpansion Coefficients and Input Resistance of the Radiator with a Dipole Exciter . . . ........ iv Page iii 17 22. 26 Z7 Z8 32 33 WOOD) O‘U‘Irh 3. 2. 1 Geometry and Trial Antenna Current . . . 3.2.2 Expansion Coefficients. . . . . . . . . . . 3.2.3 InputResistance.............. ExPansion Coefficients and Input Resistance of the Radiator with a Dipole Array Exciter . . . . . 30301 GeometrYoooooooooooooooooo 3.3.2 Expansion Coefficients B and Cq. . . . . q 3.3.3 InputResistance. . . . . . . . . . . . . . EXperimental Setup. . . . . . . . . . . . ..... Comparison between Theory and Experiment . . . conCIuSion o o o o o o o o o o o o o o o o o o o o o OPEN-CAVITY RADIATORS WITH TRANSMISSION LINE EXCITERS O O O O O O O O O O O O O O O O 9 O O O O 4.1 4.2 4 3 4.4 Introduction ....... . . . . . ......... EXpansion Coefficients and Radiation Resistance of an Open-Cavity Radiator with a Transmission LineExciter.................... 4.2.1 Geometry. . . . . . . . . . . . ...... 4.2.2 Expansion Coefficients. . . . . . . . . . . 4.2.3 Radiation Resistance. . . . . . . . . . . . Comparison between Theory and Experiment . . . Conclusion..................... OPEN-CAVITY RADIATORS WITH CIRCULAR LOOP EXCITERS O O O O O O O O O O O O 0 O O O O O O O O 0 O 0 5.1 5.2 mm .0 U14: Introduction..................... Expansion Coefficients and Input Resistance of an Open- Cavity Radiator with a Circular Loop Placed ina Transverse Plane . . . . . . . . . . . 5.2.1Geometry.................. 5.2.2 EXpansion Coefficients. . . . . . . . . . . 5.2.3 InputResistance. . . . . . . . . . . . . . EXpansion Coefficients and Input Resistance of an Open-Cavity Radiation with a Circular Loop Placed ina Longitudinal Plane. . . . . . . . . . . 5.3.1 Geometry and EXpansion Coefficients . . 5.3.2 InputResistance. . . . . . . . . . . . . . Comparison between» Theory and Experiment . . . Conclusion..................... Page 33 33 37 37 39 39 41 41 44 48 62 62 63 63 63 7O 71 73 77 77 78 78 79 81 82 82 87 88 91 APPENDDL. . ......... ... ......... REFERENCES... ... vi 3.4 3.5 3.6 3.7 3.8 LIST OF FIGURES Geometry of an open-cavity radiator. . . . . . . . . Geometry> and fields of a waveguide excited by Sources Ja O O O O O O O O O O O O O O O O O O O O O 0 Illustration for Lorentz Lemma . . . . . . . . . . . Geometry for calculation of the radiation fields . . . Radiation patterns for (a) TE mode, (b) TE 11 12 mode, (c)TE21mode................. Radiation patterns for (a) TM mode, (b) TM mode, (c) TM12 01 ll mode................. Geometry of the radiator with a dipole exciter . . . Geometry of the radiator with a dipole array exciter Experimental setup for the open-cavity radiator . . Open-cavity radiator with a dipole exciter inside theanechoicchamber. . . . . . . . . . . . . . . . . The eXperimental setup outside the anechoic chamber Radiation patterns of an open-cavity radiator with a dipole exciter (h: 0. 25 X0, 11: 0. 25 x0, L = 0. 8 X0) . Radiation patterns of an open-cavity radiator with a ' ' =.2)\ : .ZX =.)\ d1poleexc1ter (h 0 5 o' 11 O 5 o' L 10 0) Radiation patterns of an open-cavity radiator with a dipole exciter (h: 0. 25 X0, 11: 0. 25 X0, L: 1. 2 X0) . vii Page 18 30 31 38 38 42 43 43 50 51 52 Figure Page 3. 9 Radiation patterns of an open- cavity radiator with a dipole exciter (h=0.05)\ , 1 = 0.25). , L: 1.0). ). . 53 o l o o 3. 10 Radiation patterns of an open-cavity radiator with a dipole exciter (h: 0.15 X0, 1 = 0. 25 x0, L21. 0 X0). . 54 1 3. 11 Radiation patterns of an open-cavity radiator with a dipole exciter (h: 0. 35 x0, 11: 0. 25 X0, L: l. 0 k0). . 55 3. 12 Radiation patterns of an open- cavity radiator with a dipole exciter (h=0.25ko, ll: 0.15X , L=1.O)\O). . 56 o 3. 13 Radiation patterns of an open-cavity radiator with a dipole exciter (h: 0. 25 X0, £1: 0.10 X0, L: 1. 0 X0). . 57 3. 14 Radiation patterns of an open- cavity radiator with a dipole exciter (h: 0. 25 X0, 1 l: 0. 05 X0, L=1. 0 X0) . 58 3. 15 Radiation patterns of an Open- cavity radiator with a dipole array exciter (h: 0. 25 X h = 0.22 X 1:0.25x,d=o.25i, L=008X)....0.... 59 1 o l o’ 3. 16 Radiation patterns of an open-cavity radiator with a dipole array exciter (h = 0.25 X , h1= 0.22 X0, 1:0.25x,d=0.255{,L=190>.).. ..... . 60 1 o l o o 3. 17 Radiation patterns of an open-cavity radiator with a dipole array exciter (h = O. 25 X h _ 0.22 x0, 2 0.,25x d=o.2551, L: 102x1).. 61 1:0'1 o’ 4. 1 Geometry of an open-cavity radiator with a trans- mission line exciter and the equivalent circuit of the transmission line exciter. . . . . . . . . . . . . 64 4. 2 Radiation patterns of the open-cavity radiator with a transmission line exciter at L = 0.8 x0 . . . . . . . 74 4. 3 Radiation patterns of the open-cavity radiator with a transmission line exciter at L = 1.0 KO . . . . . . . 75 4. 4 Radiation patterns of the open-cavity radiator with a transmission line exciter at L = 1.2 X0 . . . . . . . 76 viii 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 Geometry of an open-cavity radiator with a cir- cular loop exciter placed in a transverse plane. . . Geometry of an open- cavity radiator with a cir- cular loop exciter placed in a longitudinal plane . . Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d = 0O 09 x0, L = 0O 8 x0). O O O O O O O O O O O O O O Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (dzooogxonglooxo)00.000.00.000. Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane ((1:0009 x03 L=102x0) o o o o o o o o o o o o o 0 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d:0019x09 Lzoo8xo) o o o. no. on no. u 0 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d=0019x0’ L=100xo) o o o o o o o oo o oo o 0 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d=0019)\o' L=102xo) o 000 o so so... 0 0 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane (d:0006xo.L:008x0)0000000000.... Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane (d=0.06xo’ L=100xo)oooooooooooooo Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane (d=0.06 x03L=102x0) o 00000 000 o 0000 ix Page 78 83 93 94 95 96 97 98 99 100 101 Figure A. 1 A.2 Page The circular cylindrical waveguide. . . . . . . . . 102 Field configurations in a circular waveguide for TEmodes...................... 106 Field configurations in a circular waveguide for TM mOde 8 O O O O O O O O O O O O O O O O O O O O O O 1 l 0 LIST OF TABLES Table Page 3. 1 Experimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Dipole Exciter, h = 0.25 1&0, 11: 0.25 X . . . . . 47 o 3. 2 Experimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Dipole Exciter, h = 0. 32 x , 11: 0.25 x0. . . . . 47 o 3. 3 EXperimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Dipole Array Exciter, h = 0.25 X , £1: 0. 25 x andd=o.25x.....?....f’ ....... ° . 47 l o 4. 1 Experimental and Theoretical Radiation Resistances of an Open-Cavity Radiator with a Transmission Line Exciter, 11: 0.045 1.0, 2b = 0.5 X0 . . . . . . 72 5. 1 Experimental Input Impedance and Theoretical Input Resistance of an Open- Cavity Radiator with a Circular Loop Exciter Placed in a Transverse Plane, d=0.097\.£:0.25k............... 90 o 1 o 5. 2 Experimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Circular Loop Exciter Placed in a Transverse Plane, d=0.19)\o,£l=0.25)\0............... 90 A.1 Valuesofpl'llforTEModes........ 104 O A.2 ValuesofpnlforTMModes............ 108 3 xi CHAPTER 1 INTRODUCTION Antennas employing the "backfire" principle conceived by (1. 2) have been the subjects of extensive experimental (3) EhreSpeck studies. More recently, Ehrespeck has developed a ”short- backfire” antenna which consists of a simple open-ended circular cavity with a dipole exciter placed at an appropriate location inside the cavity and a small reflecting plate placed in the open end. A gain of 15 dB above isotropic, with side lobes of at least -2.0 dB and a back lobe lower than -30 dB was achieved with this configuration. In Spite of its simple geometrical structure, this radiator has a comparable performance as a more sophisticated reflecter-type antenna. Although this antenna has been studied experimentally, very little theoretical work has been conducted. (4) Chen, Nyquist and Lin have developed an approximate calculation of the radiation fields of a short ”backfire" antenna based upon the assumption that the aperture field is distributed approximately cosinusoidally in both horizontal and vertical planes as evidenced by a near-zone measurement. They conclude that a short "backfire" antenna is essentially a circular aperture antenna with the dipole functioning merely as an exciter for the aperture field. Zucker(5) has theoretically studied a long "backfire" antenna and has provided some useful information for design. It is apparent that more extensive theoretical and eXperi- mental studies are needed to understand the basic operational principles of this radiator. It is also anticipated that if an adequate theory is developed, it will not only lead to an understanding of the basic principles of this antenna but perhaps may also lead to a better design for the backfire antenna or the development of a new class of open-cavity radiator type antennas. It is for these reasons that the present investigation was made. The model of the radiator for this study is similar to that of a short ”backfire" antenna except that the reflecting plate at the antenna aperture is ignored and the rim length of the antenna is increased. The reasons for adopting this model are for theoretical simplicity and for the eXperimental fact that a short ”backfire" antenna radiates the same if the reflecting plate is removed while the antenna rim is increased. In this investigation, both theoretical and experimental studies have been conducted to find the radiation and circuit pro- perties of an open-cavity radiator excited by various primary exciters. For the primary exciters, a dipole, a dipole array, a transmission line and a circular loop have been considered. The current distribution on the primary exciter is assumed. The wave- guide excitation theory is employed to find the eXpansion coefficients of the normal modes excited in the cavity. The reflection coefficients of the normal modes at the open end of the cavity are calculated approximately. The aperture field is obtained by summing the pro- pagating modes at the open end of the cavity; the radiation field is then calculated based on the aperture field. The input resistance is obtained by calculating the total radiated power carried by the propagating modes. The effects of the cavity dimensions and the geometries and dimensions of the exciters on the radiation characteristics of the radiator are studied. Theory has been confirmed by experiment. The present investigation should prove useful in the understanding of the basic operational principles of a ”backfire antenna" and its design. CHAPTER 2 WAVEGUIDE EXCITATION THEORY AND THE RADIATION FIELDS OF THE OPEN-CAVITY RADIATORS 2.1 Geometfl and Statement of the Problem The geometry of an open- cavity radiator is as shown in Fig. 2.1. This antenna consists of a simple, open-ended circular cylinder with a primary radiator placed at an appropriate location inside the cylinder. The configuration of the primary radiator can be of various shapes. Various current sources 3a with the frequency w are assumed to be on the primary radiator. The circular cylinder is assumed to be perfectly conducting with a radius of A and a length of L = 11+ 1 2. This cylinder is short-circuited by a perfect plane reflector at z : -£1 and the other end is open at z : £2. The center of the primary radiator is located at z = 0. Inside the cavity, or the open waveguide, cylindrical coordinates (r', ¢', z) are adopted to express the waveguide fields. Outside the cavity, a new coordi- nate system is used to eXpress the radiation fields. ’ Y __| I A I l I I , .3; i 4- a I I I | I J z=-1l z=0 2:12 Y Fig. 2.1 Geometry of an open-cavity radiator 1 I I l I I V,S III 4+4+ I I 4+»+ E1,H1 ——-.- I : ———o- £22,142 : 3'. I -——-’Z E1.Hl ‘— II a : -.-—— E2,H2 l I I l f f ._ S _ 2-21 0 2—22 Region I: z < 21 Region II: 2 > 22 Source Region: z1 _<_ z E 22 —-> Fig. 2. 2 Geometry and fields of a waveguide excited by source Ja Fig. 2. 3 Illustration for Lorentz Lemma 2. 2 Waveguide Excitation Theory ' In a circular cylindrical waveguide with a radius A which is relatively large compared with the wavelength, several waveguide modes can propagate along the waveguide. With a known current source located in the waveguide, the EM field can be expanded in the normal waveguide modes and the expansion coefficients of all modes excited by the source (propagating and evanescent) are deter- mined by the waveguide excitation theory. Since the cylindrical waveguide used in this study has a finite length, the reflections due to the discontinuities at both ends also need to be considered and evaluated. 2. 2.1 General Field Expressions in a’Waveguide Fig. 2. 2 illustrates a waveguide of finite length in which a current source 3; is located in the region between z1 and 22' The total volume of the source region is V. S is defined as the total closed boundary surface of V and So is the total surface of conducting wall in V. The fields excited by the source may be expressed as an infinite Fourier series in the orthogonal normal waveguide modes as follows: E1=El++fl‘=zAE++zBE' (2.1a) q qq q qq Z —> +J§qz where E - = (e + e ) e (2. 2a) q q- zq —> -9 -> +jfi7 Hi = (h +h )6 q (2.2b) q " q zq In eqs. (2.1)and (2. 2), q is a general summation index and implies a summation over all possible TE and TM modes, and the time dependence factor of ejwt has been suppressed. The Bq, e , ezq’ hq and hzCl are the propagation constant, transverse and 7.- components of E fields, transverse and z- components of H fields for the qth waveguide mode, respectively. The super "+" and "-" indices represent the waves in positive and negative 2 directions, respectively. The unknown constants Aq, Bq, Cq and Dq are the eXpansion coefficients which are to be evaluated later. We define reflection coefficients rm and I‘Zq as, Aq = — 2.3 I“lq B ( a) Cl Dq = —— 2.3b I-‘2q Cq I I Substituting eqs. (2. 2) and (2. 3) into eq. (2.1), the following are obtained quz _. _. -quz + I‘lq(eq+ e )e I z —> —> Jfiq7 —> —> "jflq/ . HI = §Bq[(-hq+ hzq)e + rlq(hq+ hzq)e ] z < zl (2. 5) .. _. I -3qu Jfiqz . E2 = 2:1 Cq[(eq+ ezq)e + F2 (e - ezq)e ] z > 22 (2.6) -> —> —o -JBqZ —> —> Jflqz H2 = 221 Cq[(hq+ hzq)e - I‘quq- hzq)e ] z > 22 (2.7) 2. 2. Z Lorentz Lemma Consider a volume region V bounded by a closed surface S as in Fig. 2. 3. Let a current source-3a in V produce fields Ea, Ha, while a second source Tb produces fields Eb, Hb. The Lorentz Lemma states(6), ' o —> —> - —> —> d = ' —> . -> - -> . —> Z ‘3 V (EaXHb be Ha) v ‘8 (Eb Ja Ea Jb)dv ( .8) v v With divergence theorem eq. (2. 8) leads to EHEHAd-ETEId 29 §s(ax b- bx a)°n 8-Sv(b a-a b)V (') where S is the total surface enclosing V. 2. 2. 3 Excited Fields In Fig. 2.2, let E and E be the E , and 11’ and 11’ be the 1 a l 2 2 Ha. These fields are produced by the current source Ta defined in the Lorentz Lemma. The E and H are assumed to be b b -> —> - —> jfiqz E : E : e - e 2.108.) b q ( q zq) ( -> -—> - -> -> quz H = H = (-h + h )e (2. 10b) 10 Both Eb and Hb are fields in a source-free region correspond- ing to Tb: 0. Using Lorentz Lemma in our problem, V is assumed to be a region between 21 and zz, S is the total surface enclosing V and So is the total area of the conducting wall in region V. Eq. (2. 9) can now be rewritten as follows: §(+ax-H--—E XHa.) : SE--Tadv s q q vq S(E xH--—E-xH a)-nds+§ (E aXfiq-I- E-xH)-hds a C1 C1 q a 5 8-08 0 = X’E-o-Idv (2.11) q a The surface S-S consists of two cross-sectional surfaces 0 Since the boundary condition on the conducting wall 1 2' -> A —> .. S0 is fix Ea : an = 0, the first term of eq. (2.11) vanishes because -(fixE )°H]ds=0 (2.12) Based on the power orthogonality property of the normal waveguide modes, —> -> A ‘8‘ EixHi-ndszo nylm, (2.13) CS m n eq. (2. 11) leads to 5 (E xH-- E-xH)°f1ds : (E xH---E-x—H) fids a q q a a q a 3-8 z o 1 +5 (7 xH--E-xH)°1Aids a q a 22 —> ->_ ->_ —> A -> -... -*- —> A =5 (E xH - E xH ). (-z)ds +5 (E xH - E xH ). (z)ds a q q a a q q a 21 z2 :S[ZB (Fl-+1" H+)xH' -E ’sz (H ”+r Hp +-)] (-2)ds prplppqqppplpp 1 +5[23cp (E +1“ E )xH -E "'prC (H ++r2pHp '-)] (2) ds 2 p pr q q p P 2 = (-B E xH"—r BH+XH’+BH'XH'+B r H'xH+).2ds z q q q lq q q q q q q Cl lq q 1 ' —>+—->.. —>_ —>_ —>_. —->+ —>_ —>_ A +5 (CExH +Cl" xH -CExH -C xH)°zds q q q q Zq q q q q q q Zq q q = 3 ( Jrx-H-) zds+C ‘8 (E+xH--E-x—H+) zds q lq q q (12 q q q q where‘s ds and‘s ds are the surface integrals over the cross sec- Z2 . l tions at 21 and 22’ By substituting eq. (2. 2) into the above equation, we get +C - :8: x-H+H -H-; xH+H - ds q.1[(q zq)(qzq)(q zq)( ”2 Z 2 (Mr 261' ("'B’)“d = - ex '23 qlq <15 q q 01' —> A '->.. —> 2(qu “.ch (exh)~zds= E -J dv (2.14) 1 c.s q q 12 where 3 ds is the surface integral over the cross-section of the c. s. waveguide and is independent of 2. Similarly, if E; and H; are chosen as the source-free region fields Eb and Hb corresponding to II b: 0, we obtain ' —> —> A '-+ —> 2(CI‘ -B)‘S (e xh)-zds=§E+-Jdv (2.15) qu q CS q q Vq a If Mq and Nq are defined as SE-orl’dv q a. M = v (2.16a) q 25 (3x3)-2ds C.S q q N: _ _. _. A (2.16b) q 25 (e xh )- zds C's. q q then I“ B - C = M lq q q q r C - B e N Zq q q q F2 M +N or B = rqrq lq (2.17a) q lq Zq' I‘l N +M c = rqrq lq (2.17b) q lq Zq The eXpansion coefficients for the EM field excited in the waveguide by a primary source are therefore determined in terms of the source current distribution Ta and the reflection coefficients I‘lq, FZq' 13 2. 3 Calculation of Reflection Coefficients qu and I‘Zq In section 2. 2.1 and in Fig. 2.1, I‘lq and I‘Zq have been defined as the reflection coefficients of qth waveguide mode due to the discontinuities at z = ‘11 and z : 22, respectively. At z = -11, the waveguide is short-circuited by a good conductor. The E field in the region I of z < 0 is —> —> —> quZ —> -> -quZ EzEB e-e e +F e+e e <0 2.4 1 q q[(q Zq) 1q Based on the boundary condition, anI: 0, over the con- ducting wall, the transverse component of E1 at z = ~21 vanishes. That is -10 1 35 1 E(z:-£):ZB(ee +I‘ ee :0 2.18) It 1 q q q lq q I ( The surface integral of the scalar product of Elt and an arbitrary mode Hp over the short-circuited wall S1 is also zero. .8 E (z:-£ )-e ds=5 23B (e +I‘ e )e 0e ds:0 8 1t 1 p q q lq q p s l 1 -j(3 I 1(3 1 .__ _. or EB(e ql+I‘ e ql)§ e-e ds:0 (2.19) q q 1q q P 51 Due to the orthogonality of the waveguide modes, eq. (2.19) becomes -Jfi £1 Jfipll B(e 9 +1“ e )3 3.3ds=o (2.20) q 1p S p p 1 Since B and ‘8 H . H ds are both non-zero constants, it leads to C1 8 P 1 -jfi 1 jfi 1 e P l + P e p l : 0 14 where p represents an arbitrary mode. Thus, the reflection coefficient due to the short-circuit discontinuity at z : -fl (referred toz:0)is ..‘Z l Jfiql :- 22 qu e (.1) The calculation of I‘ the reflection coefficient due to the 2q’ open-end discontinuity at z = [2, is much more complicated than I‘lq. The diffraction effects which are due fundamentally to the fact that the sources are distributed over an Open surface, can cause the regenerations of other waveguide modes. This coupling phenomenon which can be handled by a ray-optical theory”) is extremely complicated and not appropriate for the present analysis. Fortunately, the eXperiment and also the ray-optical theory indicate that whenever the dimensions of the aperture are not small compared with the wavelength, which happen to be our case, the diffraction effect is insignificant and the major portion of the (8). aperture field is due to the field from the waveguide An accurate reflection coefficient at the open-end can only be determined by extremely complicated methods(7' 9). Since the reflection coefficient at a large aperture is usually small, a simple method based on trans- mission line principle will be employed to calculate the reflection coefficient. Assume that the reflection coefficient for the qth mode at the open-end, z : 22, is k2q and it is defined as the ratio of (eq)r/ (eq)i or the ratio of the transverse components of the 15 reflected and incident electric field vectors. When extrapolated 8 to the plane of the open-end, qu can be interpreted as, I ) Q - Z 0 q k = (2,22) 2 + Z q 120 q where LO and Zq are the field impedances of Space and qth mode, respectively. The transverse component of electric field in the region II of z > 0 is _. _. -quz _, quz E = 2) C e e + 1" e e 2. 23 At 2 2 [2 or the open end, The incident wave : E quq (r,¢ )e _. mgr, The reflected wave = E C I“ e (r, q 2 —> q 2 E C I‘ e r e = E C k e r, e 59 1 -J'I3 1 q 2 q Z-* or EC I‘ e -k e e r :0 (2.24) If :p is the transverse component of the electric field vector of an arbitrary p th mode, the surface integration of the product of I; and eq. (2. 24) over the waveguide open-end SZ should also be P zero, that is . jfi 1 fifl f. XCF e -k e e -e ds:0 I; [q q( Zq Zq ) q] p 2 it” as! .. .. or ZC(I‘ qu-k e qz)S‘e oeds:0 q q Zq Zq p p s 2 jfi l -jfi I . 2 —> -> C(I‘ e p -k e p )3 e .e ds:0 (2.25) p Zp 2p S p p 2 Eq. (2. 25) has been derived with the help of the orthogonality property of the waveguide modes. Since C and S‘ :p - Hp ds are 3 2 non-zero constants, eq. (2. 25) leads to .19 l :13 l q 2 q 2 _ I‘qu - que _ 0 or -jZB 2 Q - Z -j28 l q 2 o q q 2 r = k e = —— e (2.26) 2 2 Z q q 40+ q 1" is the reflection coefficient of the qth mode at z:0 due to the Zq open-end at z = 12. 2. 4 I_nput Resistance of the Primary Radiator The input resistance Rin of the primary radiator is defined at the terminals of the primary radiator and is equal to the total real power radiated divided by one half of the square of the input current of the primary radiator. The real power radiated from the exciter can be obtained by calculating the real part of the complex Poynting vector of the propagating modes. That is ' -> 4* A P = l Re‘S (E xH )- zds for prOpagating modes 2 2 2 c. s. only (2. 27) Substituting eqs. (2.6) and (2.7) into eq. (2.27), and taking ad- vantage of waveguide orthogonality, P is obtained as l7 1 ' 4+ —>_ —->+ -+_ >5: Pz—Re 2C E + E x C H 2 S [q q< q rzq qn [3 q.< q. I Z ' *+ "’- ""+ "’-* A =—zc Res E+ E xH+I‘H -ds ' ' jZB z -j2I32 , I Z 2 q * q 1 -> --> =- E 'C | Re(1- |I‘ 1+1“ 8 - I" e )5 '-—-(e -e )ds 2 q‘ q 2g 2g 2q c s Zq q q where q are for those propagating modes only. Because only the propagating modes have been considered, the wave impedance and I; J'Zfiqz 12an ,. ’are real, and Re[l"2qe -, (I‘qu ) ] = 0, therefore 1 Z —> -+ P :— Z (1 - II‘ ()5 (e 'e )ds (2.28) 2 q Z 2q c.s. q q The input resistance of the primary radiator, Rin’ is then defined as 2 - e -e s. q Zq 2q c.s. q q 2 R : P : 1 ' * >I< 1“ 11 11 OO 00 (2.29) The reactive component of the input impedance was attempted with an induced EMF method without much success. The reason is that there are infinite number of higher order, cut-off modes present on the exciter surface and it is hard to obtain a sensible and accurate reactance. 2. 5 Radiation Fields of the Open-Cavity Antenna The radiation fields of an open-cavity radiator are calculated based on the aperture fields at the open-end of the cavity. 2. 5.1 Geometry and General EXpressions for the Radiation Fields Figure 2.4 shows the geometry of the problem. The x'-y' plane is the aperture plane and s' is the surface which forms the aperture. The radiation fields are maintained by the aperture 18 fields Ea and H3. Spherical coordinates (r, 9, ¢) are adopted to respresent the radiation fields, while the aperture fields are ex- pressed in terms of cylindrical coordinates (r', ¢',1 2). P(r, 9, 4)) is an arbitrary observation point in the radiation zone and P'(r', ¢', 1 Z) is a point on the aperture. The distance between P and P' is R = '3'. - if" and the radiation zone approximation for R is -> A r - r '. r --- for phase terms R : ( (2. 30) r --- for amplitude terms Y P(r.6.¢) Fig. 2.4 Geometry for calculation of the radiation fields 19 The radiation fields at P(I") maintained by the aperture fields Ea and Ha are given by(lo) r jko e'Jkor r r E = - Z 0 411 r (L4) + £0 N9) ( . 31a) r jko ekaOr r r = - 2 E4) 4" r (L9 1; N¢) ( .3lb) where .. A —>r -> ' A —> —> Jk r '.r N (r):5 ana(r')e 0 ds' (2.32a) s' _. A _ jk r'» r ->r —> r. —> —> O L (r) = 5 - an (r') e ds' (2. 32b) 5' a —> —> I. —> h —> I = 1 1 1 1 Z Ea(r ) r Ear(r ) + 4) Ea¢(r ) ( . 32c) -> 4' A a. A. ->' 2 32d Ha(r )— rHaru ) +4» Ha¢(r) <. ) Since the aperture fields Ea and H3 may consist of all possible TE and TM modes, all exicted modes should be considered. In section 2. 3, the diffraction and coupling effects at the aperture have been neglected, therefore the superposition method will be employed to obtain the total radiation fields by summing up the radiation fields maintained by the aperture fields of all excited modes. The unit normal vector 91 on the aperture is :2', therefore eqs. (2. 32) for the qth mode yield the following: 20 —> r—b \' A A -> A —-> ‘jkor'.1‘. : _ 1 1 I l a l Lq (r) .281 z x[r Ear(r ) +4) Ea¢(r )] eA ds . A ... A _. jk r'r'.r : ‘8 [r'Ea¢(l") - ¢'Ear(r')] e 0 ds' (2. 33a) 3' _> r_’ A 1 A jk r' r Nq(r):8 zx-Z—-[sz (r')]e 0 ds' \ l A A S . lq ,. A jk r'r'or : _ __ '1 HI 1 *1 O 1 Z 5' Z [r Ear(r ) +4) Ea¢(r )] e ds ( . 33b) 5 q A A The expressions of unit vectors r' and 4)‘ in terms of spherical coordinates (r, 0, 4)) are r' = 1’: sine cos(4)-4)') + 8 cost) cos(4)-4)') - 4)sin (4)-4)') (2. 34a) 4)’ = r sinB sin (4)-4)') + 0 cosG sin (4)-4)') + 4) cos (4)-4)') (2. 34b) Therefore N14?) : -S Z—Z.l— [E (F'){frsin0 cos (4)-4)') + 8 cosecos (4)-4)') q s' q ar .. $sin(4>-4)')} + Ea¢(?') {r sine sin (4)-4>') +0 c050 sin(4)-4)') .\ jk r'sinB cos (4)-4)') +4>cos(4)-4)')I] e 0 ds' (2.35a) Eqr(;) = 5‘ (13”.; Sing COS(¢—¢I) +8 €059 COS(¢-¢')-$ Sin(4)-4)')} [E s' a - Ear(;')flr sine sin(4)-4)') +9 cos 0 sin(4)-4)') +4) cos(4)-4)') }] jk r'sine cos(4)—4)') o e ds' (2. 35b) r r r r ' The N N , L and L are then determined eq’ ¢q Gq ¢q ’ 21 NGZG") : .. cose 3;'-Z-1:1-[Ear(?t)cos(¢-¢')+Ea (F')sin(¢_¢l)]. jk r'sinO cos(4)-4)') e 0 ds' (2.36a) r—’ _ - _!-__ “7| _ l _ 9| ' .. I N¢q(r)-— L thra¢cos<¢ 4») Ear“ )sm<¢ 4» >1 jk r'sin9cos(4)-4)') e 0 ds' (2.3613) L93?) = cost) 5) Ea¢(?')eos(¢-¢') -Ear(?')sin(¢-¢')]. s jk r'sinG cos(4)-4)') e 0 ds' (2.36c) L¢;(?) = - 584 Ear(?')cos(¢-¢')+Ea¢(?'>sin(¢-¢')] - jk'r'sinG cos(4)-4)') e 0 ds' (2. 36d) Therefore the radiation fields maintained by the qth mode of the aperture field are I . -jk r " r -> Jko e O ' go —> _ __ I I _’ jk r'sinB cos(4)-4)') E (r')sin(4)-4)')] e ° ds' (2,372.) 214) ..‘k E r -> -J 0 e J Or ' E}; 9 E —>' ' E —>' ¢q(r) — 4” r 581(Zq +cos )[ a‘4)(r )cos(4)-4) )- ar(r )- jk r'sinOcos(4)-4)') sin(4)-4)')] e 0 ds' (2.37b) From eq. (2. 23), the transverse component of E2 at z : £2 is -J'I3 1 1'13 1 —> _ - q 2 q 2 —¢ E2t(z_£2) _ 2 Cq(e + 1"qu )eq (2.38) q 22 Then E and E of the qth mode can be found as ar a4) -J°(3 ’2 J'fi I Ear(r"¢') : Cq(e q 2+rzqe q Z)erq(r|' 43') (2. 39a) Ea¢(r"¢" - cq(e q +r2qe q )e¢q-')+e q(r'. ¢')sin(¢-¢')] ¢ °ejzr'COS(¢-¢')r'd¢'dr' (2.413.) a ‘17 F¢q(e. 4n = 50 L} e¢q )81n(¢-1 .err'COS(¢-¢')r'd¢'dr' (2.411)) 2 = k sine (2.4lc) o 2. 5. 2 Evaluation of F q(8, (b) and F 9 q(9, 4)) for TE Modes 4) The Appendix gives the normal TE mode eXpressions. With- out losing the generality, we choose the er and e as follows: Jn(kc r') erq(r',¢’) : kn r' sinncb' (2.42a) Cq e¢q(r',¢') = Jn'(kcqr')cosn¢' (2.4Zb) 23 The Bessel function satisfies the following recurrence relations . _ £3 _ , _ 1 _ _ :11 Jm(z)_ z Jm(z) Jm+l(/.) _ Z[Jm-l(z) Jm+l(z)] — z Jm(z)+Jm_l(z) (2.43a) m l ‘2' Jm‘z =zlJm+1‘z)+Jm-1‘Z)] (2'43“ Therefore the e and e can be stated as rq ¢q l I l . I I : _ ' Z erq(r, 4) ) 2 [Jn-l(kcqr) + Jn+l(kcqr)] 8111 no ( . 44a) 1 I I __ ' _ ' I e¢q(r ,4) ) —. 2. [Jn_l(kcqr) Jn+l(kcqr)] cos n4; (2.44b) and erqcos(¢>-¢') + e¢qsin(¢-¢') _ -1. ' ' I I I - I ' _ Z[ Jn_l(kcqr){51nn¢ cos(¢-¢ )+cosn¢ 51n(¢-¢ )} + Jn+l(kcqr) {sin n¢'cos(¢-¢')- cosn¢'sin(cb-¢')}] _ l ' - _ I ' - I_ Z — Z [In-1(kcqr)51n{(n 1M) +¢}+_Jn+l(kcqr)81n{(n+lM) ¢}] ( .45) With the help of the Bessel-Fourier series, we have jzr'cos(¢-¢') 0° n e : J (zr') + E Zj J (zr')cosn(¢-¢') (2.46) 0 11:1 n ‘Tr Also ‘8 sinmxcosnxdx : 0 -1r TI' 11' S‘ sinmxsinnxdx=§ cosmxcosnxdx: O formfn -1r -17 TI' .TI’ 5‘ sinzmx dx = 5 coszmxdx : 1r -TI' -1T By substituting eqs. (2.45), (2.46) and the above eqs. into Feq, it becomes 24 A 11' 1 ' . ' I ' I Feq : E S, 3:11}: Jn-1(k<:qr )SmHn-IW W}+Jn+l(kcqr )SmunHM) -¢}] ° 00 [J (zr')+ E ijJ (zr')cosm(¢-¢')] or'dcp'dr' o m:1 m A : fijn'lsinndDS [Jn_1(kcqr')Jn_l(zr')- J o (kcqr' ”n+1 (kcqr')] n+1 .r'dr' (2.47) The Lommel integral formula gives x d S; xJn(ax)Jn(§x)dx = (12ng [Jn(ox)'§'; Jn(BX)-Jn(5X)a; Jn(0X)] (Z. 48) Eq. (2.48) and the recurrence relations are used to lead to ,A I I I I ‘80 Jn-l(kcqr )Jn_l(zr )r dr A : —-—-— k ' - ' k kg 22 [Jn-l( ch)ZJn-1(ZA) Jn-l(ZA)kchn-l( CqA)] Cq A 2n _ k2 zz [kchn(kch){z—A Jn(zA)-Jn+l(zA)}-zJn(zA)- cq 2n {k A Cq Jn(quA)-Jn (quA)}] (2.49) +1 ,A I I I I I ‘80 Jn+l(kcqr )Jn+l(zr )r dr A ___ __ k I - ' k k2 _ zZ [Jn+l( ch)z Jn+l(ZA) Jn+l(ZA)kchn+l( ch)] cq A . ' = —2——3[ Jn+1 (quA)z Jn(zA)-Jn+l (zA)quJn(quA)] (Z. 50) k -z Cq With eqs. (2. 49) and (Z. 50), FGq becomes 25 .n-l ZnTr sing Feqm'd’)‘J k [3 sins J cq o n((3015; sin G)Jn(quA) (2. 51) In calculating F (1(9, 4)), the same procedure is followed as for the <1> f case 0 Feq . 1 e¢qcos(¢-¢'>- erqsmww) =§[Jn_l(kcqr')cosr} - Jn+1(kcqr')‘30S “n+1 mud)“ F¢q zéSéAS-Ejn_l(kcqr')cos{(n-l)¢'+¢}-Jn+l(kcqr')cosf(n+l)¢'-¢l] . a) m [J0(zr') + milzj Jm(zr')cos m(I>-¢')] r'do'dr' n-l 'A F¢q = j ncosncp ‘So [Jn_l(kcqr')Jn_1(zr')+Jn+l(qur')Jn+l(zr')] r'drI (2. 52) The integration of eq. (2. 52) will be carried out differently in order to take advantage of J' (k A) : O. n cq .A I r I I I ‘80 Jn-l(kcqr )Jn-1(1.r )r dr A n I :37 [kchn{Jg+a Jn') (2. 64a) I ._ n ° e¢q(r', ¢ ) — kcqr, Jn(kcqr')51n(n¢') (2. 64b) the modified radiation fields are calculated to be n l ZnfioAsinecosmcb) F (9,4)) :j ' 2 J (p Asin8)J '(k A) (2.65) Gq kcq' Bozsinze n o n cq .n-l 2n1'rsinfl F¢q(9. 4°) - J quBOSine Jn(quA)Jn([30Asin9) = O (2. 66) 2. 5. 5 Radiation Fields Due to the Individual Waveguide Modes -% The radiation field Er in eqs. (2. 40) can be rearranged as follows: . -jk r . . 3k 0 'J5 1 J5 I 1‘ _ o e q 2 q 2 E9(r.9.¢) — 4" r )(31 Cq(e +1“qu )qu(9,¢)(2.67a) . -jk r . Jk 0 ~15 I 36 l r _ o e q 2 q 2 E¢(r.e.¢)- 4" r 2 che +r2qe )I¢q(6,¢)<2.67b) go where 16q(9, 4)) = (1 +2— cos 0)F8q(9’ (ID) (2.67c) q 60 _ I¢q(9, 4)) = (E— + cos 8) F¢q(6, 4)) (2. 67d) q 29 16q and I¢q are defined as the qth waveguide mode radiation pattern functions, since they describe the 9 and 4) dependence of the radiation field Er. Figures (2. 5) and (Z. 6) show some of the radiation patterns in the E-plane (c): = ) and H-plane (4): O) for TE and TM modes, NI=I respectively. These patterns are calculated from the pattern functions I and I , for the case of radius A equal to A or one 9Q ¢>q 0 free- space wavelength. The solid line represents the E-plane pattern while the dotted line indicates the H-plane pattern. In Fig. 2. 5(c), the H-plane radiation pattern of the TE21 mode is same as that of the E-plane. In Fig. 2.6, the E-plane and H- plane radiation patterns for the TM01 mode are identical, and the H-plane field patterns for both TMll and TM12 modes are zero. (b) TEIZ H I I _ 556%) HEN My? WAG ,’ Zl )4 II - \ $13)” ' - ) adia ionptterns for (a) TMOI mode, (b) TM-ll mode, CHAPTER 3 OPEN-CAVITY RADIATORS WITH DIPOLE AND DIPOLE ARRAY EXCITERS 3. 1 Introduction In this chapter, the radiation and circuit properties of an open-cavity radiator with a dipole or a dipole array exciter are studied. Since the eXpansion coefficients of waveguide modes are evaluated based on a given current distribution on the antenna, the antenna currents in a dipole and a dipole array are determined first. The zeroth-order currents for a dipole or for the dipole elements in an array are determined by solving Hallen's integral equations(l 1), The total field excited in the cavity due to a dipole array is obtained by summing up the fields excited by each array element. Theoretical and eXperimental results on the radiation pattern and the input resistance are obtained and compared. The effects of the location of the primary exciter and of rim length of the cavity on the radiation pattern and the input resistance are studied. 32 33 3. 2 Expansion Coefficients and Input Resistance of the Radiator with a Dipole Exciter ‘3. 2.1 Geometry and Trial Antenna Current The geometry of the radiator with a dipole exciter is shown in Fig. 3. l. A thin dipole of length 2h is center-driven and located at the origin inside the open-cavity radiator. The current density on this dipole can be mathematically eXpressed as I —> i A Ja(X.y. 2) = y 51:60). a (xm (z)sinao(h- )y)) for -h 5 v 5 h (3.1) where Io is the input current and Bo is the wave number in free — Space. The circular cylinder is the same as that defined in Chapter 2. This cylinder is shorted at z : -l and has an open end at z : I l 2' 3. 2. 2 Expansion Coefficients From Chapter 2, the expressions for the eXpansion coefficients for the qth waveguide mode in the open-cavity radiator are r M +N = Zq q q q I‘qu‘Zq-l I‘N+M C- qu q " -1 q rlqrzq IOE'JJ' dv ._,+._, 5 q a Eq Jadv — v _ I; where M q 23 (zxK)-2ds q ZS ( xK)-2ds C S q q C S q q The numerators of Mq and Nq for the case of a dipole exciter can be found as 34 ‘-> -> «(D -> —> jfi Z -—> ‘XiE ~J dsz 3 (e -e )e q oJ (x,y,z)dzds q a q zq a c s - i A 5 ' B h d 3 2 _ sinisoh 5c e -y (X)81n o( -|Y]) S ( ° ) .-’+ "’ 1 (I) -> —> ..jB Z _, SVE -J dVZS S (e +e )e q ~J (x,y,z)dzds q a q zq a c.s. -oo I o " -’ A . sinp hi eq.y6(x)smfio(h-|y()ds (3.3) o c.s. —> -> —> —-> A —> —» Since (e xh)og = ex(-1- £xe)-z = —1—-(e-e) q q q Zq q Zq q q ‘ _. A I Z ‘8 eq- y5(x)sin Bo(h-(y()ds q q Zsinfl h —> -> ' o S (e . e )ds c 5. Let us define INq and ID as follows: ' —> A . INq = ‘8 eq- y5(x) Sin BO(h-'y|)ds (3. 5a) c. s. 1 = S (E! . 3 )ds (3. 5b) Dq C. s. q q The cylindrical coordinates (r, q). z) and the rectangular coordinates (x, y, 2) have relations of A Y A . A rsmct +¢cos¢ X rcosq), y:rsin4> Substituting the above relations into eq. (3. 5a), we have , h .17 1qu 5 .3. :q(r,¢). (Ir sin¢ + 3) cos ¢)6(r cos ¢)sin [30(h- ( rsin¢> |)r d¢dr “IT 35 In the integration w. r.t. <1), 5 (r cos 4)) can be eXpressed as HI)— 6(r cos¢) : 6(coscb) (3.6) _, A eq . (r sincb +3) cos¢)6 (cos 4))sin [30(h + r sin¢)d¢ dr 0 -TT h 'Tr—b A A +5 .3 eq-(r sinct>+c1>cosd>)6(cos¢>)sinf‘3o(h-I'Si11)ddr o o h 1T 1T :5“ [erq(¢:7) - erq(¢=-§)] sinBo(h-r)dr (3.7) o where erq is the r-component of the electric field of the qth wave- guide mode. IDq can be expressed as I S'A .n[(e )Z+( )2] d4) dr (3 8) = e r . Dq 0 3n rq ¢q Therefore, the eXpansion coefficients Bq and Cq can be written as 1 z I r +1 B = 0 q Nq zq (3 9) 2 ' h -1 . q smfio IDq qurzq 1 z I F +1 (3 — 0 q Nq lq (3 10) ‘ 2 ' h -1 ’ q sm £30 IDq F qu 2q Up to this point, the eXpansion coefficients are completely deter- mined in terms of integrations w. r. t. r and (p. The determination of INq and IDq for all the TE and TM modes can be made by substituting the mode field distributions into eqs. (3. 7)and (3. 8). (i) TE Modes: The transverse electric fields for the qth TE waveguide mode are 36 n Jn(kcqr) . erq = k r 51n(n¢) (3.11a) cq e4“1 = J;(kcqr) cos(n¢) (3.11b) Therefore I and I can be obtained as Nq Dq h J(k r) IN = 5' kn nr Cq [sin(-r-lz-T-E)- sin(--rlZlT-)] sin Bo(h-r)dr q o cq 2nsin(—) h J nc(k qr) : S——————-— sin [30(h-r)dr (3.12a) kcq o and J2 A 51’ n2 Jn(kcqr) 2 2 2 LDq=S S [ 2 2 sin (n<1>)+JI'1 (kcqr)cos (n¢)]rd¢dr o -11’ kcq r n an(kcqr) 2 an [ 2 r +rJ;1 (kcqr)]dr (3.12b) 0 (ii) TM Modes: The transverse components of the electric field for the qth TM mode are _ I ’ erq- Jn(kcqr)sm(n¢) (3.13a) n Jn‘kc r) e = R q cos(n¢) (3.13b) ¢q Cq r so that h INq: SI J;1(kcqr)[ sin(¥) - sink-{121)} sin [30(h-r)dr ,h = 2 sin(£2T-I)5 Jr'l(kcqr)sin [30(h-r)dr (3.14a) 37 and 2 I d 3. +rJn (kcqr)] r ( 14b) The integrations for INq and IDq are carried out numerically by a CDC 6500 computer. 3. 2. 3 Input Resistance From Chapter 2, input resistance has been defined as Z )9) R. = 2 z (l-II‘Z () e .6 ds (3.15) q q q c.s. q q Equation (3.15) is summing up only the propagating modes. Since the input current is real and I is defined in the previous Dq section, eq. (3.15) can be rewritten as Icqlz 2 Zq (l - iPqu )IDq R. = ..1— 2 1n 2 q 0 (3.16) I 3. 3 Expansion Coefficients and Input Resistance of the Radiator with a Dipole Array Exciter As the extension, an open-cavity radiator with a dipole array exciter will be considered in this section. The currents in the driven element and the parasitic elements are determined first by solving Hallen's integral equations. The superposition principle is then employed to calculate the eXpansion coefficients due‘to individual antenna elements. After some phase 38 Fig. 3. 1 Geometry of the radiator with a dipole exciter. I z=-1 2:0 2:21 2:2 Fig. 3. 2 Geometry of the radiator with a dipole array exciter. 39 modifications, those expansion coefficients are summed up to yield the total eXpansion coefficients which are then used to find the input resistance. 3. 3.1 Geometry Figure 3. 2 shows a dipole array with n parasitic elements placed inside the circular cavity. The driven element is the zeroth element of the array and has a length of ZhO. The input current to this driven array dipole is IO with a frequency of (L). The n parasitic elements are arranged along the z-axis and symmetric to the x-z plane. For the ith element, Ii’ hi and di are the input current, the antenna half-length and the distance between this element and the driven element, reSpectively. The input current I1 is obtained by (12) solving the Hallen's integral equations for the array and taking into account of a ground plane placed at a distance of [I from the driven element. 1 3. 3. 2 Expansion Coefficients Bq and Cq Let us define Biq and Ciq as the expansion coefficients of the qth waveguide mode excited by ith parasitic element. Bq and Cq are the total eXpansion coefficients of the qth mode excited by all the elements of the dipole array. If we use a new coordinate system (xi, yi, zi) w1th xizx, yi=y and z=zi+di for the 1th element, we can find the Biq and Ciq by the same procedure as for single dipole case as discussed in Section 3. 2. The electric field due to the ith element, from eq. (2.4), is E. = E B. [(8 -e )e +1“ (9. +e )e 1'] Z. <0 11 q iq q Zq 1C1 C1 zq 1 (3.17) where rlq is the reflection coefficient of the qth mode due to the short-circuit at 21: - (di+I1). It follows that -jZfi I . -j2{3 (d.+£ ) -J'ZF5 d. I. :_e q11:_e C11 12F q1 3.18 lq lqe ( ) By substituting eq. (3. 18) and zi: z-di into eq. (3.17), we obtain E = - 3- i1 iBiqe ][(eq ezq)e +I‘1q(eq+ezq)e ] ( l9) Similary, Ei2 is j[3 d. -jB z jB z E =ZC e C11[(‘<‘§+?§)e q+r (2’55 )8 q] (3,20) 12 q iq q zq Zq q zq Summing up all the fields due to all the dipole elements, the total E field in the waveguide is n -jI3 d. jf3 2 1'5 2 E=EZBe q1[(7§-2~' )eq+1"(;;+e> )e q] 1 ._ lq q zq lq q zq 1—0 q —> —> jg Z -> -> -jfiqz = z: B [(e -e )e ‘1 +1“ (e +e )e ] (3.21a) q q q Zq lq q zq -15 z 313 z and E = 23 C [(3 +2.7 )e q +r (73' -3 )e q] (3.21b) 2 q q q zq Zq q zq n -jp di Where B = E B, e q (3. 22a) <1 ._ lq. . 1—0 n jfi di and C = )3 c, e q (3.226) q 120 1q are the total eXpansion coefficients for the qth mode excited by the 41 dipole array. It is noted that (10 in eqs. (3. 22) is zero. 3. 3. 3 Input Resistance Equation (3.16) is also valid for the dipole array case except that Cq is the total eXpansion coefficient which has been found in eq. (3. 22b). INq remains the same as the case of a single dipole because the field distributions of the qth modes excited by all the array elements are assumed to be the same. 3. 4 ExBerimental Setup The experimental setup for the measurement of the radiation patterns and the input impedance of an open-cavity radiator is sche- matically shown in Fig. 3. 3. The open-cavity radiator is placed inside an anechoic chamber, which is covered by microwave absorbers. The radius A of the cylindrical cavity is 10 cm and is equal to one free-Space wavelength under the operating frequency. The rim length 11+ [2 of the radiator is made adjustable for the experimental purposes. A movable receiving antenna is used to measure the radiation patterns of the radiator. The distance between the radiator and the receiving antenna is 50 cm (5 A0) when the rim length is ad- justed to be 10 cm. By rotating the position of the radiator, this receiving antenna can measure both the E-plane and H-plane radiation patterns. The primary radiator, namely, the dipole exciter or the dipole array excier, is excited by an R. F. oscillator at 3 CH7. and 42 f Door Ane choic Chambe r 6 'x6 'x6' ‘l_ Receiving Antenna Open- Cavity ‘ ’ ° Radiator Linear Pot. l r ,. _ . Slot Lin+ Portionl I X-Y Movable Recorder robe .- L—-.__.——I..—__J — —.~ [j Balun I l lOdB R.F. Sq. Wav pad Osc. Osc. 3 GHz 1 kHz Fig. 3. 3 Experimental setup for the open-cavity radiator. 43 7””!!7: ... - A)“ .,....4.41 ‘7 I’d"/// 4 h ..V .l p “‘ J ‘ — ’- I J- -‘: . _—\."-' 41 r ‘ ' I - . .1 1.. ‘ B..,S>_ . v Fig. 3.4 Open-cavity radiator with a dipole exciter inside the anechoic chamber. Fig. 3. 5 The experimental setup outside the anechoic chamber. 44 with a square wave amplitude modulation of l KHz. Fig. 3. 4 shows a receiving antenna and an open- cavity radiator with a dipole exciter all placed inside an anechoic chamber. A balun(13)lias been employed to convert a GR coaxial line to a balanced, shielded two-wire line which feeds the primary radiator. A slot has been cut over a portion of the shielded two- wire line and a movable probe has been inserted in the slot, for the purpose of measuring the input impedance of the primary radiator. A simple detecting system consisting of an amplitude detector and an SWR indicator has been used to measure both the radiation field and the input impedance. An x-y recorder has been used to obtain a direct plot of the radiation patterns. Fig. 3. 5 is a photograph showing the eXperiment setup outside the anechoic chamber. 3. 5 Comparison between Theory and ExPeriment Theoretical and experimental results on the radiation pat- terns and the input resistance of an Open-cavity radiator with a dipole or a dipole array exciter are obtained and compared in this section. In the theoretical calculation for the radiation patterns and the input resistance, all the propagating TE and TM waveguide modes are considered. Some of the cutoff modes have also been Considered in addition to the propagating modes to see their effect on radiation patterns. The effect was found to be insignificant 45 when the total rim length was over 0.6 A0. The theoretical results on the radiation patterns and the input resistance are calculated numerically by a CDC 6500 computer. Figures 3.6 to 3.17 show the E-plane (6:900) and H-plane (4): 00) radiation patterns of open- cavity radiators with various di- mensions, different locations of primary exciters, and various rim lengths of the cavity. The theoretical results (dotted line) and experimental results (solid line) are plotted together for easy com- parison. In all these figures, a satisfactory agreement between theory and eXperiment is observed. Figures 3. 6 to 3. 8 show the radiation patterns of an open- cavity radiator with a dipole exciter and a variable rim length. The dipole exciter with a half length of Ao/4 is located A0/4 away from the shorted end. The rim lengths of cavity for these three figures are 0. 8 A0, 1. 0A0 and 1. 2 A0 reSpectively. The effect of the cavity length on the H-plane pattern is found to be rather significant. Figures 3. 9 to 3.11 give the radiation patterns of open-cavity radiators with three different dipole exciters placed at the same position as the first three figures and with the rim length fixed at l. 0 A0. The dipole half lengths for these three figures are 0. 05 A0, 0.15 X0 to 0. 35 A0. The effect of the dipole length on the radiation pattern is not very significant. Figures 3. 12 to 3. 14 illustrate the radiation patterns of the radiators with'a dipole exciter placed at three different distances, 0.15 A0, 0.1 A0 and 0.05AO from the 46 shorted end of the cavity. The rim length is kept at l. 0 A0 for these three cases. The effect of the exciter location on the radiation pattern is found to be insignificant. Figures 3.15 to 3.17 show the radiation patterns of an open- cavity radiator with a two—element dipole array primary exciter for three different rim lengths. The driven element with a half-length of A0/4 is placed A0/4 away from the shorted end of the cavity. The parasitic element has a half length of 0. 22 A0 and is located 0. 25 A0 from the driven element. The three different rim lengths are 0. 8 A0, 1. 0 A0 and 1. 2 A0 reSpectively. It is observed that the H- plane pattern is greatly improved with a dipole array exciter com- pared with the case of a dipole exciter. The experimental result of input resistance of an open- cavity radiator is compared with the theoretical input resistance, while the eXperimental reactance is not checked due to lack of theoretical reactance. Table 3.1 shows the comparison between eXperimental and theoretical resistances of an open-cavity radiator with a dipole exciter which has a half length of 0. 25 A0 and placed at a distance of 0. 25 AC from shorted end of the cavity. The rim length is varied from 0.6 A0 to 1. 2 X0. Table 3. 2 shows the same comparison as Table 3.1 for a same radiator with a dipole exciter of a 0. 32 A0 half length. Table 3. 3 shows the theoretical and eXperi- mental input resistances of a same radiator with a dipole array exciter with dimensions described in Fig. 3.15 to Fig. 3.17. 47 Table 3. l EXperimental Input Impedance and Theoretical Input Resistance of an Open- Cavity Radiator with a Dipole Exciter h=0.25>\ , I = 0.25X . o l o Rim Length Experimental Theoretical Input L: lli+£2 Input Impedance Resistance 0.6 A0 86.5 +jl9.5 60.97 0.8 A0 71.1 +j82.4 67.69 1.0AO 78.7 +j53.4 70.73 1.2 A0 89.6 +j86.3 70.99 Table 3. 2 Experimental Input Impedance and Theoretical Input Resistance of an Open Cavity Radiator with a Dipole Exciter h: 0. 32 A0, I 1z: 0. 25 X0. Rim Length EXperimental Theoretical Input L: 11+22 , Input Impedance Resistance 0.6 A0 153.8 +jl63.8 188.12 0.8 A0 148.5 +jl65.2 208.10 1.0)\o 175.2+j199.5 217.76 1.2AO 146.2 +jl72.8 218.5 Table 3. 3 Experimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Dipole Array Exciter ho: 0. 25 A0, hlz O. 22 X0, £1: 0. 25 X0 and d1: 0. 25 A0. Rim Length Experimental Theoretical Input L: 11-1-12 Input Impedance Resistance 0.8A0 52.l+jll8.2 56.6 1.0A0 68.6 +j71.2 60.6 1.2).0 57.6 +j121.9 49.3 48 In these three tables, a qualitative agreement is obtained between theory and eXperiment. 3. 6 Conclusion A theoretical analysis on the radiation and circuit properties of an open-cavity radiator with a dipole or a dipole array exciter has been carried out in this chapter. Theoretical results have been con- firmed by eXperimental results. Concerning the radiation patterns, a few points of interest are as follows: (a) The radiation patterns of a radiator are quite independent of the length and the location of the dipole exciter. This implies that a proper exciter may be chosen to improve the matching with the driving line while keeping the desired radiation patterns unchanged. (b) The rim length of the cavity has a rather significant effect on the H-plane pattern. (c) A radiator with a, two-element dipole array exciter gives very desirable radiation patterns both in the E-plane and the H-plane. No side lobes appear in the patterns. A radiator with this exciter may prove to function better than usual backfire antenna with a dipole exciter and a small reflecting plate. Among these figures on the radiation patterns, rather large disagreements between theory and eXperiment are recorded in some cases. The sources of discrepancy are believed to be due to: (a) negligence of the diffraction at the radiator aperture, (b) inaccurate calculation of the reflection of the propagating modes at the aperture, 49 and (c) the effect of the cut-off mode fields Specially for the cases of short cavity rims. For the input impedance of the open-cavity radiator, the present analysis yields only the theoretical input resistance which is in qualitative agreement with the eXperimental results. Generally speaking the input impedance is not strongly dependent on the cavity dimensions. From the results presented in this chapter, it is concluded that the radiation property of the open-cavity radiator is essentially controlled by the cavity dimensions while the circuit property of the radiator is primarily determined by the geometry of the exciter. These characteristics may lead to the advantages of separate controls of the radiation and circuit properties and, therefor e, an easier design of an open-cavity radiator. ——_l T I A: 0 A 1— I 1211 I Zh=05 __ — —'—.z .i. I 11:0.25A '1" l L 0.8 ...—L xprime talR —— — h reticlReul "I o // 4&4 - H” 131‘" 1‘1” o ..... I - .4 I I \ I 616111 l-—:>—-1 —-' T A _i_.-z%.— —-1—.- ll 1'——( A: J‘HZJ : 2h: 1‘1—‘l I L 1____1 ...—....L 1 I“ _ II ‘66 (We. : “53-“ 9, t->—-1 _.( ...)— ”MW! 3» II N“ ‘ /’ ’4 $633) 111 4:. ._ _ / ...». Z4 55 I]; I [I 1/ \ 1 3 ‘51.; 4:1!) ~£§fi O p 1 “ “Its. 56 Fig. 3. 12 Radiation patterns of an open-cavity radiator with a dipole exciter (h = 0. 25 X0, 11: 0. lSAo, L = 1.0).0). 57 R wt .m Re flR t1 n .e m i r e D. x E L7; Tr: “\_ gm... =o.1ox, L=1.0A). O O open-cavity radiator with a 1 l ofan (h: 0.25). , o dipole exc ter Fig. 3.13 Radiation 59 I A=1.0).o I _ I 2h=0.5x,2h=0.22). hl 211 l o o p o J- _._ l—r—v-z I = 0.25 x0, d1: 0.25 A0 1 _ ' I *r‘trl : L=°'“o if j? ‘ ‘11 I/////’.¢ ‘ . 3 ; fl .....‘T‘f Fig. 3. 15 Radiation patterns of an open-c radia w‘th avity tor 1 a dipole array exciter (ho: 0. 25 A , 111: 0. 22). , 11: 0.25 x0, d1: 0.25 x0, L: 0.8%). ° 60 Fig. 3. 16 Radiation patterns of an 0 -ca radiat with a dipole array exciter (ho: . Z h1= 0. 22 A , 11: 0.25 x0. d1: 0. 25 x0, ). ° pen 0 5 L=l 61 A=1.0)\° 2h=0.5)., 2h=o.22). o o l o I =0.25A,d=0.25). l o o L=1.2). o l Experimental Result ... ... _. _ Theoretical Result Fig. 3. 17 Radiation patterns of an open-cavity radiator with a dipole array exciter (h = 0.25 A , h = 0. 22 A , o — a. o .- 11- 0.25 x0, d1.— 0. 25 x0, 1. -1.2°>.°). CHAPTER 4 OPEN-CAVITY RADIATORS WITH TRANSMISSION LINE EXCITERS 4. 1 Introduction In this chapter, the radiation and circuit properties of an open-cavity radiator with a transmission line type exciter are studied. A thin conducting wire is placed closely in the front of the shorted end of the cavity. The wire and its image form a section of a transmission line. With a proper termination, a traveling wave of current can be excited on the transmission line. The waveguide excitation theory has been employed to deter- mine the fields excited inside the cavity. The radiation fields are calculated based on the aperture field. The Poynting vector method is used to determine the radiated power and the radiation resistance. Theoretical and eXperimental results on the radiation patterns and the radiation resistance are obtained and compared. 62 63 4. 2 ExPansion Coefficients and Radiation Resistance of an Open- Cavitj Radiator with a Transmission Line Exciter 4. Z. 1 Geometry The geometry of an open-cavity radiator with a transmission line exciter is shown in Fig. 4.1. A section of thin conducting wire with a length of 2h is located on y-z plane. The total current flowing in the wire is IO and the frequency is 6). If the Spacing between the conducting wire and the shorted end, I 1’ is small, the conducting wire and its image form a section of a transmission line with a characteristic impedance of ZC. If this section of transmission line is terminated with a resistor of Zt: Zc/ 2, a traveling wave of current can be excited in the wire . Mathematically, this current _) can be represented by a current density Ja such as, _, A 'jpo(y+h) Ja(x, y, z) = y 106 (z)6 (x) e for -h E y _<_ h (4.1) where 10 is the input current at y = -h, and [30 is the wave number in the free-Space. The two short ends of the transmission line will be ignored in the theoretical analysis. The circular cylindrical cavity is the same as that defined in the previous chapter. 4. 2. 2 EXpansion Coefficients The expressions for the eXpansion coefficients for the qth mode excited in the open-cavity radiator have been given in Chap- ter 2 as, 64 Resistive TerminatO“ I zt | I D1 V I Coaxial o l 10 Line I I I I zz-ll z=0 z_!2 z (a) + I V 2 1 2t " I V+ ZC 1 t .. Z 1 I 9 I (b) Fig. 4.1 Geometry of an open-cavity radiator with a transmission line exciter and the equivalent circuit of the transmission line exciter. 65 r NII—N B .. Zq <1 q q rlquq-1 r N +64 c __ lq q q ‘ -1 q r‘quZq dv C-II 1,15... q a M = ' -) 4 25 (e xh )~ st C.S.q q where LE+T dv q a q 25 (:xK)-gds c.s.q q The numerators of Mq and Nq for the case of a transmission line exciter can be found to be \ - ‘3 ‘22 _. _. Jflq A -JBO(y+h) st “Tm/=3 5 (e-e )e .y16(z)6(x)e dzds q a q 0 c.s. £1 ‘ ... II -jBO(y+h) = I 3 e .y 5(x)e ds (4.2) 0 Cl c. s. and ' —> + —> ' -> A -jfio(y+h) '—> _ a ‘SVE ~Jadv = I 5 e y6(x)e ds:‘SE oJadv (4.3) q 0 c.s. q v q where 3 ds is the surface integration over the cross-section of C.S. the waveguide. Since (:qxhq) . 2 : expressed as ‘ —> A e . y5(x)e 5C.S. q _1_ (Z Z q .3), M andN canbe q q C1 q -15 (y+h) ds I Z M = N = O q _ q q 2 5 c.s. We define INq and IDq (4.4) :.2 ds ( q) as follows: 66 -J'I30(y+h) 1 _, A INq = 5 eq. y 6 (x)e ds (4. 5a) c.s. : x (g o: )dS (4051)) 1m 3.. q q Converting to cylindrical coordinates, eq. (4. 5a) becomes .h 37+ -j(3 (r sin¢ +h) I =3 5 e (r,¢)o(iisin¢+cos¢)6(rcos¢)e O rd¢dr Nq 0 —1r q ‘h _o_* A A -jfi (r sin¢+h) =3 5 e o(rsin¢+¢cos¢)5(rcos¢)e O rdcbdr 0 --77 q _h ,Tr _* A A —jf30(r sin¢+h) +3 5 e . (r sinq n(cq) (I) (b) n is even I _ 2 COS (EI)e-Jfioh ,h Jn(kcqr)51n(30r dr Nq‘ J k 2 3 r cq o J (k r) e = - n cq cos n4) rq k r e =J'k rsinn ¢q n(cq) IDq is given in eq. (4.11) (ii) TM Modes: (4. 12a) (4.126) (4. 12c) (4.13a) (4.136) (4.13c) The transverse components of the electric field of the qth TM mode are cos n4) rq: J;(kc r) - q 8111 no e _ J (kcqr) -51nn¢ (bq— k r 69 and -jfi h ‘h -j£3 r c0591 jfi r cosfl I = e o 3 J'(k r)[e 0 2 -e o ZIdr Nq n cq . n'rr . nn' 0 Sin— -31rr— 2 Z -j(30h .h -jsin(3 16666323 2 2 e 5 J'(k r) ° dr (4.14) n cq Si DTT 0 cos Bor n—Z 2 .A n2 Jn(kcqr) 2 : t d 4' IDq IT‘S [ 2 r +rJn (kcqr)] r ( 15) o cq Choosing the prOper set of INq and IDq’ we have for (a) n is odd nTr -jfioh "h = 2 ' — ' INq Sln 2 e 50 Jn(kcqr)cos (30r dr (4.16a) = I k ' erq Jn( qu)51nn¢ (4.16b) n Jn(kcqr) e = cos no (4.16c) ctq kcq r (b) n is even nn -jBoh h : _Z° -— I ' INq J cos 2 e S Jn(kcqr)sm(30r dr (4.17a) _ I erq— Jn(kcqr)cos n4) (4. 17b) 4 n Jn(kcqr) e : .. sinncI) (4.17C) ¢q kCq r Whil‘e IDq is given in eq. (4. 11). Equations (4.12) to (4. 17) give the proper expressions for INq and the fields in an open-cavity radiator with a transmission 70 line exciter. The integrations for INq can be carried out numerically by a computer. 4. 2. 3 Radiation Resistance In Chapter 2, the radiation resistance of the primary radiator has been obtained by calculating the total radiated power and then dividing it by a half of the square of the input current of the primary radiator. For the case of a transmission line exciter, the radiation resistance is different from the input resistance because of the presence of the terminal impedance Zt. The formula we derived in Chapter 2 gives only the radiation resistance. From the equivalent circuit of this transmission line in Fig. 4.1(b), the input impedance of the exciter may be expressed as P + 2j63(Wm- We) 2 = I (4.18) in l - I I 2 o o where P! is the real power radiated by the radiator plus the loss in the terminal resistor, and (Wm- We) is the stored energy in the transmission line, the cavity and the transmission line terminator. If we define the radiation and terminal resistance as Rr and Rt, reSpectively, then the real power, PI’ is equal to l =3 r l ’3‘ t = — - h 4. PI 2 1010 R + 2 I( )1 (h)R ( 19) The first part of P is the radiated power and is the same as I that defined in Chapter 2, therefore the theoretical radiation resistance 71 can be calculated from eq. (2. 29). The second part of PI can be calculated based on the wire current eXpressed in eq. (4.1) and a measured value of the terminal resistance Rt. In this case, it is found that the loss due to the radiation is small compared with the loss at the terminal resistor. In other words, Rr is small compared with Rt. 4. 3 Comparison between Theoretical and Experimental Results The experimental setup for measuring the radiation field and the input impedance of an open- cavity radiator with a tr ans- mission line exciter is almost identical to the setup used for the case of a radiator with a dipole exciter. A GR precision slotted line is us ed to substitute the balun and the shield pair line for measuring the input impedance. In the course of measuring the input impedance, we can only measure the total input impedance which includes the impedances due to radiation and due to termination of the transmission line. To measure the radiation resistance we conduct one more experi- ment as follows: The Open end of the open-cavity radiator is covered by a perfect conducting plate and the length of cavity is properly adjusted to avoid the resonance. The input resistance under this condition should be due to the loss at the terminal resistor of the transmission line only. If the total length of the transmission line is half wave length, the difference between two measured resistances mentioned above is the radiation resistance of the primary radiator. 72 Table 4. 1 shows the experimental and theoretical radiation resistances as functions of the cavity length Table 4. 1 Experimental and Theoretical Radiation Resistance of an Open- Cavity Radiator with a Transmission Line Exciter I = 0.045 A0, 2h : 0. BAD. l Lit-11+! 2 Theoretical Radiation Experimental Radiation (in A0) Resistance Resistance 0.6 5.13 7. 6 0. 8 6.17 9. 1 1.0 7. 58 9.6 l. 2 5.14 6. 6 1.4 4. 92 4. 7 1.6 5. 93 8.1 1. 8 6. 72 9.1 2. 0 6. 50 8. 5 2. 2 5. 50 7. 6 The theoretical results of the radiation patterns of an open- cavity radiator with a transmission line exciter are obtained from the formulas in Chapter 2, while the experimental results are measured by a setup discussed in Chapter 3. Since the Spacing between the conductor and the shorted end of the cavity is kept small, only the case of I = 0. 045 A0 is considered. In Figs. (4. 2) l to (4.4), the theoretical (dotted line) and eXperimental (solid line) results are presented and compared. In these figures, the trans- mission line has a length of 0. 5 A0 and the dimensions of the cavity are the same as the previous case in Chapter 3. A satisfactory agreement was obtained between theory and eXperiment. 73 4. 4 Conclusion In this study, the radiation fields and the radiation resistance of an open-cavity radiator with a transmission line exciter have been obtained theoretically and experimentally. A satisfactory agreement between theory and experiment confirm the accuracy of the present theoretical analysis. For the radiation patterns a better agreement between theory and experiment is obtained for the case of a longer cavity length. The small value of the radiation resistance of this radiator suggests a low radiation efficiency. The strong point of this radiator is its broadband nature. Because of the resitance termination of the trans- mission line exciter, the input impedance of this radiator is quite frequency independent. 6“ SI“?! , ~N\)I/%;é‘ 175’“ " , If- ~‘ , V“ CHAPTER 5 OPEN-CAVITY RADIATORS WITH CIRCULAR LOOP EXCITERS 5. l Introd uction This chapter is devoted to investigate the radiation fields and the input resistance of an open-cavity radiator excited by a cir- cular loop. A circular 100p is assumed to be either in a transverse plane or in a longitudinal plane. The waveguide excitation theory and Stokes' theorem are used to find the eXpansion coefficients of the waveguide modes which are excited in the cavity. The aperture field is then determined. The expansion coefficients of the propa- gating modes are also used to determine the input resistance of the primary exciter. Experimental and theoretical results for radiation fields and input resistance are obtained and compared. A satisfactory agreement is obtained between theory and experiment. The effects of the cavity length and the loop size on the radiation fields and the input resistance are the main concerns of this analysis. 77 78 5. 2 Expansion Coefficients and Input Resistance of an Opgn-Cavity Radiator with a Circular Loop Exciter Placed in a Transverse Plane 5. Z. 1 Geometry Figure 5. 1 shows the geometry of an open-cavity radiator with a circular loop exciter placed in the transverse plane. The loop is made of a thin conducting wire with a radius of d. The loop is located in a transverse plane at z = 0 and with its center on the z- axis. A cylindrical coordinate (r, 4), z) is used in the analysis. The current distribution for this circular loop can be mathematically expressed as I cosfl dIW-M) J = g o o a cos 80d " 6 (r-d)6 (z) -1r 5 (I) f II (5.1) where I0 is the input current at (d, 0, O) and $0 is the wave number in the free-space. Fig. 5.1 Geometry of an open-cavity radiator with a circular loop exciter placed in a transverse plane. 79 The circular cavity is the same as that defined in Chapter 2. This cavity is shorted at z 2 -I1 and has an open end at z : I g. 5. 2. 2 EXpansion Coefficients The eXpressions for the expansion coefficients for the qth waveguide mode in the open-cavity radiator are FZM+N B: qq -1 quFZq F1N+M C _ 99 q rquZq- Z E-.Jdv 29.8 E+.Jdv q q a q a M = ,__ _> and N = _V_’ _, q 23 (e . e )ds q 23 (e . e )ds 6.st q c.s. q l where The numerators of Mq and N for the case of a circular loop placed in a transverse plane can be found as .—F —5 Io DIZ I _‘> ngz A 5‘,qu Jadv : “COSflOdTT ‘8! 3C (eq- Czq)e o ¢COSBOd(TT- hp!) 9 so 1 5(r-d)5 (z)dsdz IO ,A .TI’ : W 50 -fleq¢COSBOd(fl-|¢‘)0(r—d)rd¢dr 10 IT = cosfiodn S... eq¢(r:d)c°sfiod‘"‘1¢1’d'd‘l’ (5' Z) '->+ —> '—>.. -> 0 ' 5vK- Jadv = ‘8 Eq - Jadv = m5 eq¢(r:d)cosf30d(n- (Cb‘ki (III) 0 -TI' (5. 3) 80 This leads to qu Nq and then Cq can be obtained as I“ +1 I Z d I‘ +1 I lq o q lq Nq CzN I‘ 1“ 1:2cosfidrrr I“ 11 (5'4) q q lq Zq o lq Zq Dq where TT : :d d .. INq Sweqqph )cosfiO (Tr ‘¢‘)d¢ (5.5) and . a * .A.w 2 2. I =3 (e-e)ds:‘S §[(e )+(e )]rd¢dr (5.6) Dq c.s.q C1 0 -Tl' 1“} cbq The value of IDq is the same as that obtained in Chapter 3. INq should be evaluated separately for the TE and TM modes. (i) TE Modes: The transverse electric fields for the qth TE mode are n . erq — k r Jn(kcqr)smn¢> (5. 7a) CC} e = J' k r cosn 5. 7b ¢q n< Cq > ¢ ( ) Substituting eqs. (5. 7) into eqs. (5. 5) and (5. 6), INq and IDq can be obtained as .1r INq : 5.1T Jl'n(kch)cos n4) cosfiodhr- ‘¢|)d¢ 17 TI' : J'(k d)[cosf3 dug Cost3 d¢C05n¢d¢+SinB (”S sinfi dl¢|° n cq o -w 0 0 1T 0 . cos ncp d¢] 25 d = 02 2 sinfiodn Jykch) (5.8) (F3 d) -n o 81 and 1 - n Jzk J'Zk d 5 -nS‘[Z n(qu)+rn(cqr)]r (.9) (ii) TM Modes: The transverse electric fields for the qth TM mode are: : ' k ' erq Jn( qu)51nn¢ (5.10a) L a - n J (k )c 5 10b) e¢q — kcqr n qu osncb ( . Therefore, INq and IDq for the TM mode can be evaluated to be 11 n : d - INq S k d Jn(kCq )cosnocosfiodhr |¢l)d¢ -Tl' cq ZnfiO : k [(3 (1)2 n2] smfiodTr Jn(kch) (5-11) cq o and 'A n2 2 Z : ' 2, IDq 1T5 [ 2 Jn (kcqr) + r In (kcqr)] dr (5.1 ) o kcqr Up to this point, INq and IDq for the TB and TM modes are evaluated. The calculations ofINq and IDq are carried out numerically by a CDC 6 500 computer. 5. Z. 3 Input Resistance After the eXpansion coefficients are completely determined, the input resistance of the loop can be obtained by using eq. (Z. 29) developed in Chapter 2. 82 5. 3 ExEnsion Coefficients and Inflt Resistance of an Open-Cavity Radiator with a Circular Loop Exciter Placed in a Longitudinal 212.12 5. 3. 1 Geometry and EXpansion Coefficients Figure 5. 2(a) shows the geometry of an open-cavity radiator with a circular loop located in the y-z plane or a longitudinal plane. The circular loop has a radius of d and its center is located at the origin of the cylindrical coordimtes (r, q), z). The cavity is the same as the previous case and it is shorted at z : -21 and open at z 2 £2. A new cylindrical coordinate system (r', 9, x) is used to des- cribe the circular loop exciter as shown in Fig. 5. 2(b). For sim- plicity, the radius of the circular loop is assumed to be small compared with the wavelength. The current Ta for such a small loop can be assumed to be A 'J' = I 5(r'-d)5(x)9 (5.13) a 0 where I0 is the input current at (d, —% , 0). The case of a more general current distribution on a larger loop will not be considered here to avoid mathematical complexity. In order to find the eXpression for Cq, MC1 and Nq are evalu- ated first. Substituting eq. (5.13) into the eXpressions for Mq and N , we have C1 + IIZ .211- .A + A E . J dv = I .8 3 ‘8 E - 96(r'-d)é(x)r'dr'd9dx ' q a o q 410 o °->+’ .—p+—> :15)Eo9dd9=1 §E - 12 (5.14) ocq ocq ..J ‘9‘."14',‘ ll? ‘ 83 >—‘| i i 1.361%. i l I z=0 z=-ll (a) Geometry of the radiator (b) Geometry of the circular loop exciter Fig. 5. Z Geometries of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane. ' l.r A.) ".1 v . a—ih Jl- - ua=,‘ 84 where _, A f : d . d6 9 Using Stokes' Theorem and a Maxwell equation, eq. (5. 14) becomes ' —>+ —-> 8' —>+ " ' —> iEonv=I§EodI:I\S(VxE)ds q a 0 c Cl 0 5 Cl A = -jwp.10§ li+~nds S q (5.15) A A where s is the total area enclosed by the 100p c and n = x or the unit vector normal to 5. Similarly, we get "‘ 5. 16) q ( iii = (+E’ +3 )e (2.2b) q - q zq Therefore Cq can be rewritten as F1N+M -jwpl Z . -j(3 7. Cq: I‘qI‘q lq O ,_. _. [I‘lq5 hqe q-xds 1 2— 1“ I“ -l-Z e-eds s q q (lq Zq ) S q q c.s. . jfiz . -> q H -5 h e oxds] (5.17) s q Since hq is a function of r and <1) only, the integration with reSpect to 2 can be simplified and C becomes 85 jwpIOqZ (l-I‘ ) lq '-s 1‘ Cq= , h cosfiquxds 25 (e.eq)ds(I" -1) 2 c S q lqr q JO) poIquu - I‘lq) INq 2(1" 1" _ l) I (5.18) lq Zq Dq where INq and IDq are defined as n '—> A I : Sh cosfl z-xds (5.19a) Nq s q q and I = . 3.? ds 5,191) i Dq 3C 5( q q) ( ) g The value of IDq is the same as that in the previous chapter, but the calculation for INq is quite complicated. In the cylindrical coordinates (r, 4;, z), x = r cos 4) = 0 implies that 4): (Zn-l )TT/ 2 where n is the integer. Also it and liq can be expressed as A A A x = r c084) - (p sin¢ (5. 20a) -+ A A A h = rh + h 5. 20b q rq (b 4K1 ( ) _,_ A Therefore, the scalar product of hq- x on the S surface is equal to -+ A box , _, A . q : hq(x:0).x = (hrqcoscp -h¢qsin¢1 (Zn—LL)” ¢:_____ onS = - h sin¢ (5.21) M 2 n-l ¢_(___ )1. ‘Since r = xcos q; + y sin¢, we have 11- ¢ = 3 r = Y (5.22) ¢=~31 1' = -y 2 86 Refer to Fig. 5. 2(b), r' along the path C is CXpressod by Z 22 2 r':y+z:d OI‘ Zzi'd-y (5.23) Substituting the above relations in the eXpression for INq' it leads to I zghocosfiZoxds s ‘1 C1 ‘2 Z 2 2 3d! -Y ,‘O d—y -‘S S h (43:3, r:y)COSB Zdzdy +\ S‘ h ¢q Z q .. cm 0 _ ’ Z_y2 -d J Z'YZ -cos(3 zdzdy q (4):"; 9 rz’Y) d 2 ‘ 3 Tr . B; So [hcquz-ZI) ' h¢q(¢:'2')] 5m Bq' d -3, dy (5'24) The final expressions for the eXpansion coefficients Cq for the TE and TM modes are obtained as follows: (i) TE Modes: The fields for the qth TE mode are _ _1_ . hrq — - Z Jn(kcqr)coan> q n Jn(kcqr) h : sinn¢ Z k r M q Cq INq for the TE mode can be obtained as d k 2 n , 3n1r . nTr Jn( ch) . Z Z : —— - (sm - sm—) ___________ 5mg " d “V dy fiq Y q Zk Z Z qu o I Nq . nTr 4n51n—2- ,d Jn(kcqy) . Z Z = _---——-—-———[3 Z k 5 ———-——-—-—y Sinfiq d -y dy (5.25) q q Cq 0 87 With Zq: mp/fiq, Cq for the qth TE mode becomes Z sin(nn/2) F .dJ (k Y) " 2 2 . “1 ——n-——C—q——sinfi »d -y dy I I‘ F -l y q cq Dq lq 2q 0 (5 26) C =j2nI ‘1 (ii) TM Modes: The fields for the qth TM mode are n Jn(kcqr) hrq : _ 72- 37—7- COS mp q Cq h - —1- J' (k r)sin n4) (pq _ Z n cq q Following the same process as in the TE mode case, INq and C for the qth TM mode can be obtained as nn’ 4sin— d ...____- 2 Z Z : - —-———-—— ' k ' c1 - d 5.27 INq (3 Z S Jn( qu)smfiq\/ Y Y ( l q q 0 and 7 Z l" l d .. . Z - . 2 2 Cq : jZ-IO oZ 512(nrr/ ) r qu 1 ‘8 Jukc y)sinfi .d -y dy zq Dq lq 2q 0 q q (5.28) Up to this point, the eXpansion coefficients Cq for the TE and TM modes are completely determined. 5. 3.2 Input Resistance After the eXpansion coefficients are completely determined, the input resistance can be evaluated using eq. (2. 29) developed in Chapter 2. ‘5”).2. 0.". .- , v ‘YV "i‘flm-I 88 5.4 Comparison between Theory and EXperiment In this section, theoretical and eXperimental results of the radiation patterns and the input resistance for an open-cavity radiator with a circular loop exciter placed in a transverse plane or in a longi- tudinal plane are obtained and compared. The eXperimental input resistance for the case of a small loop placed in a longitudinal plane is not presented here because it is so small that it is very hard to conduct the measurement. Figures 5. 3 to 5.8 show the radiation patterns of open-cavity radiators with various cavity length and two different circular loops placed in transverse planes of the cavity. The theoretical results (dotted line) and experimental results (solid line) are plotted together for easy comparison. The E-plane (¢:900) and H-plane (c1): 00) radiation patterns are presented in these figures. In all these figures, a satisfactory agreement between theory and experiment is obtained. Figures 5. 3 to 5. 5 show the radiation patterns of the radiators with a circular loop of 0. 09 X0 radius and placed at O. 25 X0 from the shorted end of the cavity, and with the cavity length of 0. 8 X0 1.. 0 X0 and 1. 2 A0, respectively. Figures 5.6 to 5. 8 show the radiation patterns of the three radiators treated in Figs. 5. 3 to 5. 5 but the size of the circular loop is increased to have a 0.19 X0 radius. Com- paring Figs. 5. 3 to 5. 5 with Figs. 5.6 to 5. 8, it is observed that the effect of the size of the loop exciter on the radiation patterns seems 89 rather significant. It is also observed that in the case of a larger loop exciter, the effect of the cavity length on the H-plane pattern is found to be quite outstanding. Since the theoretical analysis on the radiation and circuit properties of an Open-cavity radiator with a circular loop exciter placed in a longitudinal plane is based on the assumption that the f... 5‘ 100p is small and has a uniform current distribution, only the case i of a small loop with a radius of O. 06 X0 is investigated. Figures 5. 9 to 5. 11 give theoretical and eXperimental radiation patterns of the radiators with a small circular 100p as mentioned above. The center of the loop is placed at O. 25 x0 away from the shorted end of the cavity and the cavity lengths are set to be 0. 8 KO, 1. 0 x0, and l. 2 X0, reSpectively. It is observed that the radiation patterns in these figures are broader than those produced with a loop placed in a transverse plane. The eXperimental results on the input resistance of an open- cavity radiator with a loop placed in a transverse plane are compared with the theoretical results. Table 5.1 shows the comparison be- tween theoretical input resistance and eXperimental input impedance of a radiator with dimensions specified in Fig. 5. 3 and with the cavity length varied from 0.6 X0 to 1. 2 1&0. Table 5. 2 gives the same comparison for a radiator described in Fig. 5.6. In these two tables, a qualitative agreement is obtained between theory and experiment. The agreement is better for the case of a larger loop exciter. 90 The theoretical input resistance of an open-cavity radiator with a small circular loop exciter placed in a longitudinal plane is very small. It ranges from one to three ohms when the radius of the loop is 0. 06 RC and the cavity length is varied from 0.6 X0 to 1. 2 X0. It is very hard, if not impossible, to measure this small input resistance using a conventional driving line. For this reason, no experimental input resistance is available for comparison with theoretical results . Table 5.1 EXperimental Input Irnpedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Circular Loop Exciter Placed in a Transverse Plane, d : O. 09 X0, f = 0. 25 X . 1 o Cavity Length EXperimental , Theoretical L: [1+ 12 Input Impedance Input Resistance 0.6 X0 330.4 + j481. 7 484.1 0.8 RC 344.3 +j4ll.6 557.9 1.0)x0 406.7+j381 671.3 1.2k0 315.8 +j356.2 481.6 Table 5. 2 Experimental Input Impedance and Theoretical Input Resistance of an Open-Cavity Radiator with a Circular Loop Exciter Placed in a Transverse Plane, d : 0. 19 X , o f = 0.25 x . l o Cavity Length Experimental Theoretical Input Impedance Input Resistance 0.6 X0 259. 0 + j253.6 292. 9 0.8k0 342.7+j190.5 362.3 1. 0 X0 373. 6 + j95. 2 446. 3 1.2X 254.0+j152.9 309.9 0 91 5. 5 Conclusion A theoretical analysis on the radiation and circuit properties of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane or in a transverse plane has been presented in this chapter. Most of the theoretical results have been confirmed by the eXperimental results. Concerning the radiation patterns, some facts of significance are pointed out as follows: (a)'Ihe E-plane radiation pattern of the radiator is quite independent of the cavity length when the exciter is placed in a transverse plane. (b) The cavity length has a rather significant effect on the H-plane pattern. (c) The size of the circular loop exciter when placed in a transverse plane tends to have a rather significant effect on the radiation characteristics of the radiator. It appears that a good radiation pattern can be realized by a proper choice of a loop exciter. (d) For the radiation with a circular loop exciter placed in a longitudinal plane, the radiation resistance is usually small and the radiation patterns are less directive. This radiator may have a less value in practical applications. Among these figures on radiation patterns, rather large dis- agreements between theory and eXperiment are recorded in some cases. The sources of these disagreements are believed to be due to the same reasons mentioned in Sec. 3. 6. Concerning the input resistance of the radiator with a loop exciter placed in a transverse plane, theoretical input resistance 11 f.“ .- 5' S - . , ‘— ‘ h - ‘J—‘il. 1- 92 is in a qualitative agreement with the experimental input impedance. The agreement is better for the case of a larger loop exciter. Generally speaking, the input impedance is not strongly dependent on the cavity dimensions. 93 f 1 A = l. 0 X A ' ° I 1 = 0.25 x i | l o _ __ ——:—>z d ._. 0. 09 x0 .. l _ ~——-— L 0 Experimental Result 0, _. _ , ——-- — Theoretical Result (a) E-plane , . 9'" ' ""J q ,1) 7:12“: , . V . z . . - L r ....... it 4 1 yrs, . 1'27 777777777 94 A=l.0k o = 2 1 0. 5X0 d=0.09X o L=l.0)\ o Experimental Result -— — -— — - Theoretical Result Fig. 5.4 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d = 0.09 k0,.L = 1.0 X0). 95 96 A=1.0l 0 1 =0.25). -l o F' ‘ z d=0.l9X ' O ‘— —.. = . x 11 | L 08 O r?" .1. I J l fir; .1 , R a circular loop exciter placed in a transverse plane (d = 0.19 x0. L = 0.8 X0). 0 Experimental Result W {.0 i _. g ' —— — — — Theoretical Result . hr 9 > 7 . adi tion patterns of an open-cavity radiator with a 97 ' A = l. 0 X o | 1 = 0.25 x l o "' " —‘—’z d=o.19>. I o 1 '4 L = 1.0 x 0 Experimental Result 0° ___-_Theoretical Result _. w" s, .- ‘i {‘3‘ I; ‘ , x a 'v y A I. n W- - ‘ ’f” '3“ *7/31" 56>..." . b »-‘~‘\i' ’ »' . ,.-_' l-—:>—-{ —' Fig.1 5. 7 Radiation patterns of an open-cavity radiator with a circult loop exciter placed in a transverse plane (d = 1.9 X0, L = 1.0 ha). 98 I A = l. 0 X l O I I = 0. 25 X I l o .__.+ fi-r-z d = o. 19 x .. o 11" i L = 1. z x ‘L_____1 ° 00 Experimental Result . __ -. .,______"___‘\ __ _. .... Theoretical Result ‘ (a) E-plane ' J V “ 7Q" \ ‘\ ll}. \ i 1 ‘ v s b h “ \ % fl 4’ 2 \W/ <55» ‘3 ‘ 3‘“; ;- “1%M ‘ i : 3 ~\ 5% -—=: l-—:>-f1 \ In.) Fig. 5. 8 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a transverse plane (d = 0.19 lo, L =1.‘Z x0). 99 I——— A 10x I‘ll ' _—. I 1 025x L_ l s. 's‘ V \x. «“ , l/v . age? «WW 0 fivfifir‘fi‘f‘ UMQO %%§#% “W o.& W3 i .2 Liz) Fig: 5.9 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane (d = 0.06 X0, L = 0.8 x0). 11$; as 03“ \‘_‘_ “A74.— 100 1 . A=1.ox I 1 I ° ___.g . 11:0.25xo L-fl“ ——}—’z d=0.06xo ' : L=1.ox I O L______. [...—1.---.4 o ExPerimental Result 07 - - - — —Theoretical Result ‘ - ane & w fix?" ~., A . .I' Ii... , ' -\;-. _> ddA v‘fi‘wfiflw ' ., ._ a”, 3“ ill W o’ 9 “I r I. WW @’ 0:" 9 iii?» fit “a é” a??? ‘3'“- %§ ‘ 43394” All EMW/fiafig ’6/ — " ~ ' ‘ ’4“ 1 I; -- Ii: 3j:,_;‘--I /’: j} V E“ ‘7 '1'? : Fig. 5(10 Radiation patterns of a n circular loop exciter placed in a longitudinal plane (d = 0.06 lo, L = 1.0 x0). open-cavity radiator with a Paw: 101 ‘———_—__1 A=l.0X 11 | 0 I 1 =o.zsx l l 0 .%2d —J—-.I z «1:0.06x0 i ; L=1.zx I O Fig. 5. 11 Radiation patterns of an open-cavity radiator with a circular loop exciter placed in a longitudinal plane (d = 0.06 to, L = 1.2 X0). APPENDD( NORMAL MODES IN CIRCULAR WAVEGUIDING STRUCTURE Part I - Normal TE Modes r] Figure A.1 shows the geometry of a cylindrical waveguide ‘ with a circular cross [section of radius a. In view of the cylindrical geometry involved, cylindrical coordinates are used in the analysis. ‘F 1":m We assume that the waveguide is made of a perfect conductor and filled with a dielectric. The general equations for TE, or H, modes are 2 V h +kzh = 0 (A.1) t z c z L_-____-___4 V ’ Fig. A.1 The circular cylindrical waveguide 102 103 where Z Z 2 RC = k +7 (A.Z) and h in the longitudinal component of the H field in the waveguide. z The transverse components are 3'21th (A.3) t 2 1:2 F3 k c .- _, .2h/\ 3 A4 et—i zxt (.) The boundary condition in this problem is =1 ah L” Z 8n (r=a) = 0 (A5) where k : CON/HE is the propagation constant in the medium and kC : ZirfCVuE is the cutoff wave number with fC being the cutoff fre- h . quency for a certain waveguide mode, 2 : Jim is the field impedance for TE modes and 7 is the propagation constant for the waveguide mode. Using the separation of variables method, a solution for eq. (A. 1) is cos n¢ hz(r.¢>) = C Jn(kcr) (A6) sin n4) Where n is a positive integer. Subject to eq. (A. 5), we have 3h? (rza) cos n4) 74 = C'J'(k r) = 0 n n c . smn¢ or J'(k r) = o (A.7) n C 104 Table A. 1 below shows the ith root of .11;(p;1 I) = 0. The eigen- k ' b values c,n are given y pl n! c, nfl — a (A°8) Table A.1 Values of p; 1 for TE Modes , I I I I I I I I pol p11 p21 p31 p42 p51 p61 1 3.832 1.841 3.054 4.201 5.317 6.416 7. 501 2 7.016 5. 331 6.706 8.015 9.282 10.520 11.735 3 10.173 8.531 9.969 111.346 12.682 13.987 15.265 4 13.324 11.706 13.170 14.580 16.202 17.375 18.640 If q has been used as a mixed index of n, I covering all of the TE modes and normalize the fields by j f1 C = 1, eqs. (A. 3) and q 0C1 (A. 4) will lead to n sinnq) erq = k r J (kC r) (A. 9a) cq n q —cosn¢ cosn¢ e = J’(k r) (A. 9b) qu n cq sinnd) k cos n4) . cq h =-J——.I(k r) (A.9c) zq W n cq sinncp 1 cos no h = - -— J'(k r) (A. 9d) h rq Z n cq sinncp q sin n4) = J (k 1.) (Ao9e) h (pq k n cq -cosn¢ q cq 105 The propagation constant, Bq’ for the qth TE mode is 2 2 i 2 2 i 'y : - k 2 : k .. k 2 : q ( Cq ) J( Cq) JBq 1 (L) 1 CL) 1 ~ 2 — cq 2 — r = 3 1 - Cq ‘3 = L3 1 - 2 A. 0 o flq w(H€)[ (03)] [ ((0)1 (1) The cutoff frequency and cutoff wavelength are __1_ P21 E _ 2 _ _ wcq — kcqfllE) — v a (A. 11) 2 xcqz ?v— = 3:2 (A.12) cq pq where v is the velocity of light in the medium. The field impedance ! for this particular mode is h j__ooE_ QB (1)ch l CC1 'i 13 z—7 ='[1-(w)13=r.[1-< Mama;- q q (A. 13) where I; is the field impedance of the medium. If the dielectric in the waveguide is air, then 2; = {,0 : 1201r ohms. Figure A. 2 illustrates the field distribution of some TE modes in the guide. From these field distributions it is possible to deter- mine a proper location to place the primary exciter. In general, the primary exciter is placed in a location in such a way that the field of the exciter matches best with the field of the desired wave- guide mode. 106 Fig. A. 2. Field configurations in a circular waveguide for TE modes. 107 Part II - Normal TM Modes The general equations for TM modes are V26 +kze = 0 (A.14) t z c z where 2 2 2 kC = k +7 (A.15) The boundary condition is r: F ez(r:a) : 0 (A. 16) and the transverse fields are ...7_ et: + 2 Vtez (A.17) k c -> 1 A -—> h:+——zxe (A.18) t — e t Z where 28: J; 1816 Using the same technique as for the TE modes we get cosncb e (r,¢>) = D J (k r) (A.19) z n c . Slnncb where D is a constant and n is a positive integer. The boundary condition, ez(r:a) : 0, implies that J (kca) : 0. n Table A. 2 below shows the 1th root of Jn(pn£) : 0. The eigenvalues kc, n! are given as P k = .115 (A. 20) C, n! a Table A. 2 Value of pn 108 for TM Modes 1 I pol p11 p21 p32 p41 p51 p61 1 2.405 3.832 5.136 6.830 7.588 8.771 9.936 2 5.520 7.016 8.417 9.761 11.065 12.339 13.589 3 8.654 10.173 11.620 13.015 14.372 15.708 17.030 4 11.792 13.323 14.796 16.221 17.667 18.962 20.308 If q has been used as a mixed index of n, 2 covering all of the qth TM modes and the fields are normalized by letting 'YD qq k Cq = 1, eqs. (A. 17) and (A. 18) become cos n¢ e = J '(k r) rq n Cq . S111 114) n -31n n4) e = r J (kC r) (bq cq n q cos no r k cos ncp e = - Cq J (k r) Zq 7q n cq sin n4) n -51n no h = e J (kC r) rq k r Z n q cos no Cq q 1 cos no h 2 -""'—e' J' (kc 1‘) (bq Zq n q sin no The propagation constant, fiq, for the qth TM mode is, 2 = k- fiq ( k 2 Cq )5_ (1) 1311-4 cq)2]% (L) (A. 21a) (A. 21b) (A.ZIC) (A.21d) (A. 216) (A. 22) if“ 109 The cutoff frequency wcq and cutoff wavelength XCq are -4 Pg = k 2 = k = -— A.23 wcq “1046) V cq V a ( ) 2 xngL23—“iz—Tr—a (A.24) C (J) cq cq pq The field impedance of qth TM mode is defined as 7 (1) 1 (1) 1 e_.__q_._£ _ .2322- _ .232: Zq — jwe "'cmz [1 ( <0 )1 —€[1 ( (D )1 (A.25a) {3 or ZS: 139' (A.25b) Some typical field distributions of TM modes are shown in Fig. A.3. 110 1d configurations in a circular waveguide for . 3. Fie TM modes. Fig (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) REFERENCES H. W. Ehrenspeck, ”The Backfire Antenna, a New Type of Directional Line Source, " Proc. IRE, Vol. 48, pp. 109-110, January 1960. H. 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