Analyses of Self-Resonant Bent Antennas
by
Mohammod Ali B.Sc. (Electrical & Electronic Engineering). Bangladesh University of Engineering & Technology. Dhaka. 1987 M.A.Sc. (Electrical &: Computer Engineering). University of Victoria. 1994
-A. Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Electrical Computer Engineering
We accept this dissertation as conforming to the required standard
Dr. S.S. ^^chly. Supervisor (Dept, of Elec. &: Comp. Eng.)
______Dr. .1. Bomemann. Departmental Member (Dept, of Elec. & Comp. Eng.)
Dr. M. Okoniewski. Departmental Member (Dept, of Elec. &; Comp. Eng.
Dr. S. Dost. Outside Member (Dept, of Mechanical Engineering)
Dr. E.\'. .lull. External Member (Dept, of Elec. Eng.. University of British Columbia)
(c)Mohammod .A.H. 1997 University of \'ictoria All rights reserved. Thesis may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. 11
S upen’isor: Dr. S. S. Stuchly
A bstract
The primary' focus of this dissertation is on the analyses of self-resonant bent antennas. The need for the accurate characterization of such antennas due to their growing importance in present day wireless communications is the motivation for this work. To this end. several self-resonant bent antennas are analyzed which includes an inverted-L antenna (ILA). a meander-line dipole (MLD) antenna, a meander-line bow-tie (MLBT) antenna, a dual meander antenna, and a printed meander antenna.
.A. simple analytical model, based on the induced EMF method, is presented to compute the input impedance of the ILA. First, a sinusoidal distribution of current on the antenna, with zero current at the end is assumed, and then an expression for the input impedance is derived using the near-fields of the antenna. The accuracy of the formulation is verified by comparing the results computed using it with that from NEC [l] computation. Unlike the analytical solutions available in the literature, our proposed solution is not restricted to antennas that are electrically small. In addition the new formulation can be extended to treat other antennas, such as the T-antenna. the folded unipole antenna, and the loop-loaded monopole antenna.
The input impedance, radiation pattern, and gain of the MLD and MLBT anten nas are computed and correlated with their parameters. Input impedances of both antennas are computed using NEC. Simple analytical models are presented to com pute the radiation patterns of the MLD and the MLBT antennas. For each antenna, a sinusoidal distribution of current is assumed and closed-form expressions for the radiation fields are derived. The results computed using the analytical models are verified by comparing them with the results from the NEC computation. Since in each model the radiation pattern of an antenna is expressed in terms of ready to evaluate algebraic expressions, the computation of such pattern is fast and easy.
The input impedance and radiation characteristics of a dual meander antenna Ill
are computed using NEC. Similarly as before the input impedance, radiation pat tern, and gain of this antenna are also correlated with its parameters. The input impedance and radiation pattern of a planar printed meander antenna are investi gated using the Finite-Difference Time-Domain (FDTD) technique. The antenna is modeled on a dielectric substrate both in the presence and absence of a metallic ground plane. Characteristics of the antenna are examined as function of dielec tric constant, and substrate thickness. New results of input impedance, radiation pattern, and gain are presented which are vital for the design of such antennas.
Several novel applications of self-resonant bent antennas are described. First, a wide-band dual meander-sleeve antenna is designed, manufactured, and measured for application in dual frequency vehicular personal communication. The antenna can operate simultaneously in the 824-894 MHz and 1850-1990 .MHz bands of the PCS system. Second, an MLBT dipole is introduced as a feed for plane sheet reflectors. Numerical results computed using NEC show that the feed when used in front of a plane sheet reflector, results in superior radiation characteristics than a conventional dipole feed, namely, it reduces the reflector dimension by 46% for the same front to back ratio, beamwidth and gain. Finally, a compact plane sheet reflector antenna is described that uses an MLBT monopole feed. Since the antenna uses a monopole, a balun is not rec^uired. This antenna has a gain and half-power beamwidth of 8.4 dBi and 94^. respectively. IV
Examiners:
Dr. S.S. Stuchly. Supervisor (Dept, of Elec. & Comp. Eng.)
Dr. .1."Bomemann. Departmental Member (Dept, of Elec. &: Comp. Eng.)
Dr. M. Okoniewski. Departmental Member (Dept, of Elec. & Comp. Eng.)
Dr. S. Dost. Outside Member (Dept, of Mechanical Engineering)
Dr. E.\'. .lull. External Member (Dept, of Elec. Eng.
University of British Columbia) C on ten ts
Abstract ii
C ontents v
List of Figures xvii
List of Tables xviii
Acknowledgements xix
Dedication xx
1 Introduction 1 1.1 Motivation ...... 1 1.2 C o n trib u tio n s ...... 4 1.3 O u tlin e ...... 5
2 Definitions of Antenna Parameters and Literature Review 8 2.1 Definition of Param eters ...... 9 2.1.1 Radiation Pattern ...... 9 2.1.2 Radiation Intensity ...... 10
2.1.3 Directivity ...... 11
2.1.4 Input impedance ...... 11 CONTENTS vi
2.1.5 Efficiency ...... 11
2.1.6 Gain and Half-Power Beam w idth ...... 12 2.1.7 Bandwidth ...... 12 2.1.8 Field Regions ...... 13
2.1.9 Polarization ...... 14 2.2 Review of Literature ...... 15
3 The Inverted-L Antenna 22 3.1 C urrent D is trib u tio n ...... 24 3.2 The Induced EMF M ethod ...... 25 3.3 .A n a ly sis ...... 27
3.3.1 Results...... 32 3.4 Discussion ...... 38
4 The Meander-Line Dipole Antenna 42
4.1 Input Im p e d a n c e ...... 43
4.1.1 Computation Technique ...... 43
4.1.2 Results...... 44 4.2 Radiation Characteristics ...... 49 4.2.1 Fundamentals ...... 50 4.2.2 Current Distribution ...... 52 4.2.3 Radiation Pattern ...... 53 4.2.4 G a i n ...... 56 4.2.5 Results...... 56 4.3 Discussion ...... 62
5 The Meander-Line Bow-Tie Antenna 64 5.1 .Antenna C o n fig u ra tio n ...... 66
5.2 Input Impedance ...... 67 5.3 Radiation Characteristics ...... 71 CONTEXTS vii
5.3.1 Current Distribution ...... 72 5.3.2 Radiation Pattern ...... 74 5.3.3 Results...... 78 5.4 Discussion ...... S3
6 The Dual Meander Antenna 84 6.1 Input Im p e d a n c e ...... 85
6.2 Radiation Pattern ...... 88 6.3 Discussion ...... 89
7 The Printed Meander Antenna 92 7.1 .\ntenna Configuration ...... 94
7.2 FD TD M o d e lin g ...... 94
7.3 R e s u lts ...... 96 7.3.1 .A.ntenna on a Grounded Dielectric Substrate ...... 97 7.3.2 .A.ntenna on a Dielectric Half-Space ...... 107
7.4 Discussion ...... I l l
8 A pplications 114 8.1 Dual meancier-Sleeve A ntenna ...... 115 8.1.1 Introduction ...... 115 8.1.2 Design Considerations ...... 116
8.1.3 Numerical Results ...... 118 8.1.4 Experimental Procedure ...... 120
8.1.5 Experimental Results ...... 121 8.2 Plane Sheet Reflector .A.ntennas ...... 124 8.2.1 Introduction ...... 124 8.2.2 MLBT Dipole Feed ...... 127
8.2.3 MLBT Monopole Feed ...... 134 CONTEXTS viii
8.3 Discussion ...... 138
9 Conclusions and Future Work 140 9.1 C o n c lu sio n s ...... 140 9.2 Future Work ...... 143
Bibliography 145
Appendix A 154
Appendix B 157 IX
List of Figures
2.1 Spherical coordinate system ...... 9
2.2 (a) .A. zigzag, and (b) a meander antenna [7] ...... 17
2.3 Several meander antennas (a) N=2. (b) N=4. (c) N'=6 [S] ...... 18
3.1 .A.n inverted-L monopole ...... 23
3.2 An invert ed-L d ip o le ...... 23
3.3 ( a)A transmitting antenna fed by a voltage source ( magnetic current ).
and (b) an auxiliar." current distribution ...... 25
3.4 The input reactance of an inverted-L antenna as a function of k l = k-(h -I- L) with the radius of the wire («) as a parameter: //=40 mm. £=40 mm. and frequency range 100-1550 MHz ...... 33
3.5 The input resistance of an inverted-L antenna as a function of k l w ith the radius of the wire (a) as a parameter: //=40 mm. £=40 mm. and frequency range 100-1550 MHz ...... 34
3.6 Return-loss verstis frequency characteristics of an inverted-L monopole with the radius of the wire, a as a parameter. Other parameters of the antenna are /< = £ = 40 mm ...... 35 LIST OF FIGURES x
3.7 Return-loss versus frequency characteristics of an inverted-L monopole. Antenna 1: h = 20 and L = 60.3 mm. a = 0.1 mm. .A.ntenna 2: /? = £ = 40 mm. and a = 0.1 mm...... 36
3.8 Input impedance of an inverted-L antenna as a function of kh w ith kL as the parameter; wire radius a = 0.0005A...... 37
3.9 The input impedance of an inverted-L antenna as a function of kl: h = £ = 40 mm, and a = 0.1m m . frequency range 100-1875 .MHz. . . 39
3.10 Current distribution of an inverted-L dipole as a function of the wire length (21) ...... 40
4.1 The geometry of an MLD antenna: e\ = vertical segment length. e> =horizontal segment length, and £ = antenna length ...... 44
4.2 Shortening ratio. SR and resonant resistance. Rrcs of an MLD an tenna plotted as function of .V with ^ as the parameter. The length
and the radius of the wire are constant (length of wire. £^.,r, = 32
cm. and radius of wire, a = 0.325 mm)...... 46
4.3 Input impedance of a monopole meander antenna as a function of antenna length with the segment length, e as the parameter: .V = 2. a = 0.325 mm ...... 47 4.4 Input impedance of a monopole meander antenna as a function of antenna length with the number of meander sections. .V as the pa rameter: Lu_.,re = 16 cm . a = 0.325 mm. and £ = 8 cm 48 4.5 Input impedance of a monopole meander antenna as a function of antenna length with the radius of the wire, n as the parameter: Livire = 12.8 cm. -V = 4. and £ = 6.4 cm ...... 49
4.6 (a) .A. meander-line dipole antenna: (b) integration path ...... 50
4.7 Radiation pattern of an MLD antenna with £ as a parameter (.V = 4). 57 LIST OF FIGURES xi
4.8 0-poIarized power density pattern of a meander-line dipole antenna wdth L = 0.5A (.V = 4): wire radius a = 0.001 A...... 58
4.9 ^-polarized power density pattern of a meander-line dipole antenna w ith L — 0.75A (.V = 4) ...... 58
4.10 Relative power density and half-power beam width (HPBW) of the MLD antenna as a function of antenna length (N = 4 ) ...... 59
4.11 opolarized power density pattern of a meander-line dipole antenna with antenna length L as a parameter (.V = 4) ...... 61
4.12 o-polarized power density pattern of a meander-line dipole antenna with number of meanders (.V) as a parameter (L = 0.25A)...... 61
4.13 Gain of a meander-line dipole antenna as a function of antenna length L with N as a parameter ...... 62
5.1 A meander-line bow-tie (MLBT) antenna ...... 67
5.2 Input impedance of the monopole MLBT antenna with bow-tie angle Cl (degrees) as the parameter ...... 70
5.3 Input reactance of the monopole MLBT antenna with number of me ander sections (A') as the parameter ...... 70
5.4 Input resistance of the monopole MLBT antenna with number of meander sections (A*) as the parameter ...... 71
5.5 (a) .An MLBT antenna, and (b) integration path ...... 72
5.6 ^-polarized power density pattern of an MLBT dipole with wire length L^. as a parameter (A* = 4 and a = 20°). Solid lines- analytical results. crosses-NEC results ...... 79
5.7 opolarized power density pattern of an MLBT dipole with wire length as a parameter (A' = 4. and a = 20°). Solid lines- an alytical results. crosses-NEC results ...... 80 LIST OF FIGURES xii
•5.8 Radiation pattern of an MLBT dipole at resonance. Solid lines- an alytical results. crosses-NEC results ...... 81
•5.9 Gain of an MLBT dipole as a function of antenna length; .V = 4. . . 82
6.1 A dual meander antenna on a perfectly conducting infinite ground plane ...... 85
6.2 Computed input reactance of the dual meander antenna as a function of antenna length with ?•> as a parameter ...... 86
6.3 Computed input resistance of the dual meander antenna as a function of antenna length with co as a parameter ...... 87
6.4 Computed \'S\\’R frequency response of the dual meander antenna as a function of frequency with eo as a parameter ...... 87
6.5 El). E-plane pattern of a dual meander antenna: e\ = c-> = 0.8 cm. a = 1.25 mm. L = 6.4 cm. and .V = 4 ...... 89
6.6 Comparing the input impedances of a meander and a dual meander antenna. For both antennas L = 6.4 cm. a = 1.25 mm. .V = 4. c, = e-y = 0.8 cm. and for the dual meander alone w =0.5 cm ...... 90
6.7 Computed \'SV\'R frequency response of a dual meander and a meander. 91
7.1 A printed meander antenna ...... 95
7.2 A printed dipole anten n a ...... 96
7.3 The input impedances of a meander dipole in air and a meander dipole printed on a grounded dielectric substrate. The parameters of the antenna in air are: .V = 2. e = 12 mm. 2/ — 44 mm. and «’ = 4 mm. w hereas that of the printed are: e = 12 mm. 2/ = 44 mm. w = 4 mm. L< = 244 m m . tc, = 244 mm. G = 20 mm. = 2.1...... 99 LIST OF FIGURES xiii
7.4 The input impedance of a printed meander antenna as a function of frequency with fr as a parameter. Other parameters are: .V = 2. e = 12 mm. 21 = 44 mm. a- = 4 mm, = 244 mm. = 244 mm. ts = 20 m m ...... 99 7.5 The input impedance of a printed meander antenna as a function of frequency with as a parameter. Other parameters .V = 2. e = 12 m m . 21 = 44 m m . a- = 4 mm. = 244 mm. iv^ = 244 m m . fr = 2.1. . 100
7.6 Resonant resistance. Rres (T^) of a printed meander dipole versus sub strate thickness. Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. a - 4 mm. £.< = 244 mm. a\, = 244 mm. fr = 2.1...... 101 7.7 Resonant resistance. Rres (R) of a printed meander dipole versus sub strate thickness. Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. a- = 4 mm. £^ = 244 mm. = 244 mm. Cr = 4.0...... 102 7.8 .r(/-plane pattern of a printed meander antenna. Parameters of the antenna are .V = 2. e = 12 mm. 21 — 44 mm. ir = 4 mm . £^ = 244 mm. a\, = 244 mm. 0 = 20 mm. and Cp = 1.0 ...... 103 7.9 //z-plane pattern of a printed meander antenna. Parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm.ir = 4 mm . = 244 m m . u's = 244 mm. £, = 20 mm. and fp = 1.0 ...... 104
7.10 T//-plane pattern of a printed meander antenna at resonance (1.618 GHz). Parameters of the antenna are: .V = 2. e = 12 mm. 21 = 44 mm. «• = 4 mm. £^ = 372 mm. u\, = 372 mm. 0 = 30 mm. and fp = 2.1 ...... 105 LIST OF FIGURES xiv
7.11 //c-plane pattern of a printed meander antenna at resonance ( 1.618 GHz). Parameters of the antenna are: .V = 2. e = 12 mm. 2/ = 44 mm. IV = 4 mm. = 372 mm. = 372 mm. = 30 mm. and = 2.1 106 7.12 .r;/-plane pattern of a printed meander antenna at resonance (1.73 GHz). Parameters of the antenna are .V = 2. c = 12 mm. 21 = 44 mm. tc = 4 mm. I., = 372 mm. = 372 mm. ri = 40 mm. and Cr = 2.1...... 106 7.13 yc-piane pattern of a printed meander antenna antenna at resonance (1.73 GHz). Parameters of the antenna are .V = 2. c = 12 mm. 21 = 44 mm. u- = 4 mm. = 372 mm. = 372 mm. = 40 mm. and = 2.1 ...... 107 7.14 Resonant resistance. Rres (0.) of a printed meander dipole on a di electric half-space versus substrate thickness, Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. w = 4 mm. Ls = 244 mm. = 244 mm...... = 2.1...... 108
7.15 .ry-plane pattern of a printed meander dipole on a dielectric half space. Parameters of the antenna are .V = 2. c = 12 mm. 21 = 44 mm. y = 4 mm. = 372 mm. = 372 mm. 0 = 20 mm. and Cr = 2.1...... 109 7.16 yz-plane pattern of a printed meander dipole on a dielectric half space. Parameters of the antenna are .V = 2. c = 12 mm. 21 = 44 mm. u- = 4 mm. = 372 mm. y« = 372 mm. t, = 20 mm. and
Cr = 2.1...... 110 7.17 .ry-plane pattern of a printed meander dipole on a dielectric half space. Parameters of the antenna are .V = 2. c = 12 mm. 21 = 44 mm. y = 4 mm. = 372 mm. = 372 mm. 0 = 4: mm. and = 2.1.110 LIST OF FIGURES xv
7.18 yr-plane pattern of a printed meander dipole on a dielectric lialf- space. Parameters of the antenna are .V = 2. e = 12 mm, 21 = 44 mm, fi- = 4 mm. = 372 mm. ii\ = 372 mm. = 4 mm. and Cr = 2.1.111
8.1 .A. dual meander-sleeve antenna ...... 117 8.2 Computed input impedance of the dual meander-sleeve antenna vs frequency with the sleeve length / as a parameter; sleeve spacing 2S = 3.2 cm ...... 118 8.3 Computed input impedance of the dual meander-sleeve antenna vs frequency for 3rd resonance around 1930 .MHz ...... 119
8.4 Measured \'S \\’R frequency characteristics of the dual meander-sleeve antenna with sleeve length I (cm) as the parameter ...... 122
8.5 Measured H-plane pattern of the dual meander-sleeve antenna at 890 MHz. Scale: linear ...... 122 8.6 Measured H-plane pattern of the dual meander-sleeve antenna at 1890 MHz. Scale: linear ...... 123 8.7 Measured E-plane pattern of the dual meander-sleeve antenna at 890 MHz. Scale: linear...... 123 8.8 Measured E-plane pattern of the dual meander-sleeve antenna at 1890 MHz. Scale: linear ...... 124 8.9 .A. comer reflector antenna ...... 125 8.10 (a) MLBT dipole (Type A), and (h) MLBT dipole (Type B) ...... 128
8.11 Computed input impedance of the MLBT dipole ( Type B) as a func tion of frequency (computed using NEC): wire length. L^-ire = 2.95A. a = 120°. 21 = 0.28A. h = 0.48A. n = 0.0035A. and .V = 4 ...... 129
8.12 Horizontal-plane pattern of the MLBT dipole (Type B) at 835 MHz (computed using NEC) ...... 129 LIST OF FIGURES xvi
8.13 Elevation-plane pattern of the MLBT dipole (Type B) at 835 MHz (computed using NEC) ...... 130
8.14 Horizontal-plane pattern of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire 0.44A dipole and an MLBT dipole, respectively: S = 0.21A. H = 0.65A. L = 0.21A. and a = 0.05A...... 1.30 8.15 Return loss vs frequency characteristic for a plane sheet reflector. Reflector parameters are: L = 0.21 A. H = 0.65A. 5 = 0.21 A. and G = 0.05A. MLBT feed param eters are - a = 120°. 21 = 0.2SA. h = 0.48A. .V = 4. and a = 0.0036A. Dipole feed parameters are: h = 0.44A. and a = 0.0036A...... 131
8.16 Horizontal-plane patterns of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire 0.44A dipole and an MLBT dipole, respectively. Reflector height [H) for the dipole feed is 0.98A and that for the M LBT feed is 0.65A ...... 133 8.17 Horizontal-plane patterns of plane sheet reflectors with MLBT and straight-wire dipole feeds (computed using NEC). Reflector parame ters for the dipole feed are: L = 0.31A. S = 0.21A. G = 0.05A. and H = 0.81 A. Reflector parameters for the MLBT feed are: L = 0.21 A. S = 0.21 A. G = 0.05A. and H = 0.65A...... 133
8.18 Elevation-plane pattern of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire dipole and an MLBT. respectively. S — 0.20A. G = 0.05A ...... 134
8.19 .A. compact plane sheet reflector fed by an MLBT monopole. Param eters of the antenna are: h = 0.5A. a = 114°. I = 0.18A. h = 0.018A. and e = 0.074A. Radius of wire is 0.0005A ...... 135 8.20 Input impedance of the compact plane sheet reflector antenna (com puted using NEC) ...... 136 LIST OF FIGURES xvii
8.21 M easured R eturn loss versus frequency characteristics of the compact plane sheet reflector ...... 137
8.22 Horizontal-plane pattern of the compact plane sheet reflector: solid lines-computed. crosses-measured ...... 137 X V lll
List of Tables
2.1 Parameters and characteristics of the zigzag antenna in [<]: see Fig. 2.2 16 2.2 Characteristics of the meander antennas of [8]: see Fig. 2.3: I = 4.5 cm. a = 0.4 mm. and w = 0.3 mm. /u.,>f = 13.5 cm ...... 19
3.1 Expressions for the parameters in eqns. ( 3.24)-( 3.27) ...... 31
3.2 Expressions for the parameters in eqns. (3.28)-(3.31 ) 32
5.1 Parameters and characteristics of an .MLBT dipole: .V = 2. length of \vire=32 cm. and radius of \vire=0.325 mm ...... 68
5.2 Parameters and characteristics of an MLBT dipole: .V = 4. length of \vire=32 cm. and radius of \vire=0.325 mm ...... 69
7.1 Resonant resistance. Rres of a printed dipole antenna as a function of substrate thickness: comparison with the results in [34]. Parameters of the antenna are: = 2.1. 21 = 44 mm. u’ = 4 mm. ;r, = 244 mm. and Ls. = 244 m m ...... 97 XIX
Acknowledgements
I wish to express my deepest gratitude to my super\isor. Dr. S.S. Stuchly for his continuous encouragement and guidance shown throughout this research work and the process of writing this manuscript. The valuable discussions that we have had for long hours in your office has essentially been the guiding light for this work. I can not thank you enough for your patience and forbearence with me. I also express my sincere gratitude to Dr. M.A. Stuchly for her valuable suggestions in relation to the writing of this thesis. Special thanks are also due to Dr. Michal Okoniewski for his valuable suggestions as related to my work on the printed meander antennas. Michal, I certainly learnt a lot from many of our enthusiastic yet realistic discussions in the lab. thanks.
I also wish to thank Mr. Krzysztof Caputa for his day to day help in arranging the experimental setup and performing measurements. Special thanks go to Mark for many of our interesting and lively discussions as related to electromagnetics and philosophy, in general. Thanks to Elise. Mario. Dr. Chris. Bassey. and Mrs. Ewa Okoniewska. .A. hearty thank is also extended to all colleagues in the department from w hom I have received valuable suggestions and help. Special thanks also go to my buddies like Mahmood. Shahadat. Mahboob. Intekhab. and Miah Mahmood.
Finally. I wish to express my deepest gratitude to my wife. .Ayesha. who has practically relieved me from many of my family responsibilities and thus has made it possible for me to finish writing this thesis. She has been the inspiration and encouragement in this enlightening path of travel. I also thank my little ones. Orko and Nazia for enduring the lonely moments and for providing me with the energy and enthusiasm needed to accomplish this work. XX
To Nnzia. Orko. and Ayesha. the wonderful droplets of rain in an amaziny s ^ l t t i - mer evening of solitude. Chapter 1
Introduction
1.1 Motivation
With the recent advances in telecommunications, the need for small and low-profile antennas has greatly increased. The greatest demand for these antennzis is from mobile applications (vehicles) and from portable equipment. Small antennas are also in demand for applications, such as in base stations, in sea mobiles, and in aircrafts.
Depending on applications, there are differences in terms of antenna performance requirements. For example, in a base station antenna gain of 9 to 17 dBi may be required, whereas in a personal communication serv ices ( PCS) handset a gain of 5.1 dBi may be adequate in practice [2]. Similarly, for a base station antenna linear polarization is required, while in land-mobile to satellite communications circular polarization is used [2]. However, among all these differences in antenna performance requirements, antennas of small size are essential for such applications, for either mechanical reasons or cine to the miniaturization of electronic equipment in general.
-A. small antenna, as defined in this dissertation, is one in which its largest dimen sion is small compared to a similar type of a conventional antenna [3]. For instance, a monopole antenna smaller than 0.25A is considered as a small antenna [3].
It is well known that, as the size of the antenna is reduced, the efficiency tends to degrade and the bandwidth becomes narrower [4]. This happens because the input impedance of a small antenna consists of a small resistance and a relatively large capacitance or inductance. Thus, for a size-reduced antenna, matching of its input impedance with the characteristic impedance of the feeding transmission line is difficult. Nonetheless, this matching is important in order to make the antenna useful.
One of the effective ways to accomplish efficient matching is to attain self resonance of the antenna because such an antenna is purely resistive at the frequency of operation and hence no conjugate matching circuit is necessary [4]. Performance of a self-resonant antenna is not degraded by the losses in the matching circuit. The self-resonance of a small antenna can sometimes be achieved by the antenna itself, or by adding passive reactive loadings or active devices in the antenna structure.
For instance, a normal mode helical antenna (.NMH.A) is an antenna that can attain self-resonance without an additional matching circuit.
Other antennas that are self-resonant include the inverted-L antenna (IL.\) [5]. the meander antenna [6]-[9]. the zigzag antenna [7]. the meander zigzag monopole antenna [10]. and the sinusoidal antenna [Il]-[15]. .All of the above can be man ufactured by bending wires or metal ribbons according to the specific geometrical configurations.
One feature is common for all self-resonant bent antennas, i.e.. thev are smaller 3
than straight-vvire half-wave dipoles while operating in dipole configuration and are smaller than straight-wire quarter-wave monopoles while operating in monopole configuration. To measure the extent of size reduction a term called the shortening
ratio (SR) [7] is often used. The SR is defined in percent ;is
SR = — — X 100 11.1) O.oA where L is the length of a self-resonant bent dipole in wavelengths. The SR depends on the geometry and parameters of the antenna.
The self-resonant bent antennas have some inherent disadvantages, such as low resonant resistance (the real part of the input impedance at resonance), narrow bandwidth, and undesired cross-polarization. These limitations may become severe when the antenna parameters are adjusted to increase the sliortening ratio. Thus, a knowledge of the characteristics of the self-resonant bent antennas is vital for their design to be accurate and efficient. This knowledge can only result from the analyses of such antennas. This dissertation undertakes the task of analyzing several existing self-resonant bent antennas from a design point of view. The motivation behind this work is to fill the gap in the present state of knowledge pertaining to the analyses and design of such antennas. .Antennas that are investigated include
(1) the inverted-L antenna (ILA). (2) the meander-line dipole (MLD) antenna. (3) the meander-line bow-tie (MLBT) antenna. (4) the dual meander antenna, and (5) the printed meander antenna. 1.2 Contributions
The major contribution of this dissertation in relation to the antenna analysis, is the development of simple and accurate analytical models to study the characteristics of several existing self-resonant bent antennas. The second major contribution in relation to antenna design, is the design and development of a \ride-band dual meander-sleeve antenna for dual-frequency vehicular application in the personal communication services (PCS).
Specific contributions of the dissertation are as follows:
• Development of an analytical model to compute the input impedance of an
inverted-L antenna (ILA) using the induced EMF method.
• Development of simple analytical models for the computation of the radiation
fields of the MLD and MLBT antennas.
• Establishment of the correlation between the shortening ratio. SR. the res
onant resistance. Rr^s. and the cross-polarization of the self-resonant bent
antennas with their parameters using NEC and analytical techniques.
• Characterization of a planar printed meander antenna using the Finite-Difference
Time-Domain technique from a design point of view.
• Design and development of a wide-band dual meander-sleeve antenna for ap
plication as a vehicular antenna in both bands (824-894 MHz and 1850-1990
MHz) of the personal communication ser\ices (PCS). To design the antenna
numerical modeling is performed using NEC and to confirm its proper opera
tion measurements are conducted using an HP 8720C vector netwrok analyzer. • Design of a novel MLBT dipole as a feed to plane sheet reflectors for application
in base stations using NEC.
• Design and development of a compact plane sheet reflector antenna that uses
an MLBT monopole as a feed using NEC and experimental techniques.
1.3 Outline
C hapter 2 presents the definitions of various antenna parameters, such as radia
tion pattern, radiation intensity, input impedance, efficiency, gain and half-power
beamwidth. bandwidth, field regions, and polarization. It also presents a brief re
view on self-resonant bent antennas.
Chapter 3 describes the analysis of the inverted-L antenna (ILA). .After a brief
introduction and an overview of the literature the current distribution is given, followed by a description of the induced EMF method. .An expression for the input impedance of the ILA is derived. The accuracy of the newly derived expression is verified by comparing the results computed using it with that from NEC [1] computation. Finally, the advantages and limitations of the new formulation are discussed.
Chapter 4 analyzes the meander-line dipole (MLD) antenna. .A review of the present state of knowledge is presented first, followed by an analysis of its input impedance. Input impedance is computed using NEC. The dependence of the shortening ratio. SR. the resonant resistance. Rres- and the cross-polarization on the antenna parameters is demonstrated. Input impedance graphs as a function of antenna length with three different parameters are also given. Next, a simple analytical model is presented for the computation of the radiation
characteristics of the MLD antenna. The current distribution along the antenna is
described, followed by a detailed derivation of the radiation fields which are given
in closed-form. The analytical model is verified by comparing the results computed
using it with that from NEC computation. Radiation characteristics of the MLD
antenna is discussed as function of various antenna parameters. Finally, the chapter
is closed by a discussion of the analysis and results.
Chapter 5 describes an analysis of the meander-line bow-tie (MLBT) antenna.
Similarly to the MLD antenna the input impedance of this antenna is also computed
using NEC. and antenna characteristics such as. the SR. Rres- and cross-polarization
are correlated with the parameters of the antenna. To study the radiation charac
teristics a simple analytical formulation like the one for the MLD is presented. The radiation pattern and gain of the antenna are computed and discussed as function of its parameters.
C hapter 6 presents an investigation of the dual meander antenna. The antenna is analyzed using NEC. Input impedance, radiation pattern, and gain results are presented. The advantages and limitations of the antenna are also discussed.
Chapter 7 describes the characterization of planar printed meander antennas using the Finite-Difference Time-Domain ( F DTD) technique. Two cases are consid ered: ( I) antenna printed on a grounded dielctric substrate, and (2) antenna printed on a dielectric half-space (no ground metallization present). Input impedances com puted using a gap-excitation model are presented in graphical forms with both the dielectric constant, and the substrate thickness. as parameters. The dependence of the resonant resistance on the substrate thickness and the dielectric constant, is demonstrated. The radiation pattern and gain of the printed meander antenna are also presented followed by a brief discussion.
C hapter 8 examines the potential applications of self-resonant bent antennas.
First a dual meander-sleeve antenna designed using NEC is described. Experimental results for this antenna are also presented. Application of meander-line bow-tie
(MLBT) antennas as feeds to plane sheet reflectors is discussed. Novel designs are proposed for application in base stations, followed by a discussion.
Chapter 9 closes the dissertation with a few concluding remarks. C hapter 2
Definitions of Antenna
Parameters and Literature
R eview
In this chapter first some basic antenna parameters, such as the radiation pat tern. radiation intensity, directivity, input impedance, efficiency, gain, bandwidth, field regions, and polarization are introduced. .A.11 definitions and mathematical ex pressions are taken from [16] unless otherwise mentioned. The definitions inside quotation marks come from the IEEE Standard Definitions of Terms for .Antennas
(IEEE Std 145-1973) [17].
Next a literature review on self-resonant bent antennas is presented. This in cludes a brief description of previous work on the normal mode helical antenna
(NMH.A). the invert ed-L antenna (IL.A). the meander antenna, the zigzag antenna. 9 = 0
0=90 0=0
Figure 2.1: Spherical coordinate system, the sinusoidal antenna, and the dual meander antenna.
2.1 Definition of Parameters
2.1.1 Radiation Pattern
The radiation pattern of an antenna is defined as the "graphical representation of the radiation properties of the antenna as a function of angular coordinates. In most cases, the radiation pattern is determined in the far-fie Id region and is represented as a function of the angular coordinates. Radiation properties include radiation intensity, field strength, phase or polarization."
Radiation pattern of an antenna can be either a power pattern or a field pattern.
Referring to Fig. 2.1, a two dimensional pattern is obtained by fixing one of the angle
{6 or o) while varying the other. Keeping o constant, and varying d (Q < 0 < 180°) 10
gives elevation patterns. Similarly, keeping 6 constant, and varying o (0 < o < 2t)
gives azimuthal patterns.
The performance of an antenna is often described by using two principal plane
patterns which are called the E-plane and the H-plane patterns. The E-plane p at
tern for a linearly polarized antenna is defined as "the plane containing the electric
field vector and the direction of maximum radiation." Similarly, the H-plane p at
tern is defined as "the plane containing the magnetic field vector and the direction
of maximum radiation."
Radiation pattern can be broadly clssified as. isotropic, directional, and omnidi
rectional. .An isotropic radiator is defined as "a hypothetical antenna having equal
radiation in all directions." .An example of such a radiator is a point source although
physically unrealizable, it can be a useful reference for expressing the directive prop
erties of practical antennas.
.A directional antenna is one "having the property of radiating or receiving elec
tromagnetic waves more effectively in some directions than in others". .An omnidi
rectional pattern is defined as one "having an essentially nondirectional pattern in a given plane of the antenna and a directional pattern in any orthogonal plane."
2.1.2 Radiation Intensity
Radiation intensity in a given direction is defined as "the power radiated from an antenna per unit solid angle. The unit solid angle dO. is given by sint9 dO do.
Radiation intensity can be expressed as
CiO.o) ~ ^[|Eo(g.o)|- -f- \EJ6.o)\-] (2.1) where Eo. E^ = far-zone electric field components of the antenna, and y = intrinsic 11
impedance of the medium. The total power radiated by an antenna is given by
Prad = [ [ U smO (16 do (2 .2 ) 70 Vo
2.1.3 Directivity
The directivity of an antenna is defined as "the maximum value of the directive gain
in the direction of its maximum value." Thus
D„ = i^ (2.3) ^ rad
where Do is the directivity. Umax is the maximum of the radiation intensity obtained
from ( 2 .1). and Prad is given by ( 2.2 ).
2.1.4 Input impedance
-■\ntenna input impedance is defined as "the impedance presented by an antenna at
its terminals or the ratio of the appropriate components of the electric to magnetic
fields at a point." The input impedance consists of a resistance and a reactance of
which the resistance comprises of a radiation resistance. Rr and a loss resistance.
Rf..
2.1.5 EflSciency
.\ntenna efficiency accounts for the losses at the input terminals of the antenna and within the structure of the antenna. Losses in an antenna may occur due to the mismatch between the feeding transmission line and the antenna, and also due to conductor and dielectric losses. 12
In general, the overall efficiency may be written as e, = where e, is the total overall efficiency, is the reflection efficiency and is expressed as (1 — |r |- ) where F is the reflection coefficient. e<- is the conduction efficiency, and is the dielectric efficiency.
Since and ej cannot he separated [16]. it is convenient to write c, = 1 —|r|~ ) where e^d is the antenna radiation efficiency. If an antenna has a loss resistance of
Rc and a radiation resistance of Rr. e^d =
2.1.6 Gain and Half-Power Beamwidth
.A. usefid measure to describe the performance of an antenna is to specify its gain.
Power gain of an antenna in a specific direction is defined as "dr times the ratio of the radiation intensity in that direction to the net power accepted by the antenna from a connected transmitter." The maximum gain is defined iis
6'o = CfDo- (2.4)
"In a plane containing the direction of the maximum of the beam, the angle between the directions in which the radiation intensity is one-half the maximum value of the beam" is defined cis the half-power beamwidth of an antenna.
2.1.7 Bandwidth
The bandwidth of an antenna can be defined as "the range of frequencies within which the performance of the antenna, with respect to some characteristics, conforms to a specified standard."
The characteristics of the antenna can be pattern, input impedance, beamwidth. 13
gain, efficiency, polarization, beam direction etc. The term range of frequencies'
may mean the ratio between the upper and the lower frequencies for a broadband antenna. For narrowband antennas, the bandwidth is usually expressed as a per cent of the center frequency. Thus for a broadband antenna D\V = and for a narrowband antenna D\V = x 100 where / f . fr.- and fr are the upper, lower, and the center frequencies.
2.1.8 Field Regions
The space surrounding an antenna is usually subdivided into three regions. These regions and their outer limit are discussed below.
Reactive Near Field Region This region is defined as "that region of the field
immediately surrounding the antenna where the reactive field predominates."
The outer boundary of this region for all but small antennas is specified by
R = 0 . 6 2 where A is the wavelength and D is the largest dimension of the
antenna, and R is the distance from the surface of the antenna to the outer
boundary.
Radiating Near Field This region is defined as "that region of the field of an
antenna between the reactive near-field region and the far-field region wherein
the angular field distribution predominates and wherein the angular field dis
tribution is dependent upon the distance from the antenna. For an antenna
focused at infinity, the radiating near field region is sometimes referred to as
the Fresnel region on the basis of analog}' to optical technolog}'. If the antenna
has a maximum dimension which is ver}' small compared to the wavelength,
this field region may not exist." The outer boundar}' of this region for most 14
antennas is specified by Ro =
Far-field region This region is defined as "that region of the field of an antenna
where the angular field distribution is essentially independent of the distance
from the antenna. If the antenna has a maximum overall dimension D. the
far-field region is commonly taken to exist at distances greater than from
the antenna. A being the wavelength. For an antenna focused at infinity, the
far-field region is sometimes referred to as the Fraunhofer region on the basis
of analog}' to optical terminolog}."
2.1.9 Polarization
The polarization of a radiated wave is defined as “that property of a radiated elec tromagnetic wave describing the time-var\'ing direction and relative magnitude of the electric field vector; specifically, the figure traced as a function of time by the extremity of the vector at a fixed location in space, and the sense in which it is traced, as observed along the direction of propagation."
Polarization may be classified as linear, circular, and elliptical. Let us consider the instantaneous electric field of a plane wave, travelling in the positive z-direction.
[18]
S = dj-EJocos(.i,'t — .ic -I- Ox) -f cos(w’t — .iz 4- Oy). (2.5)
The magnetic field is related to the electric field of (2.5) by the intrinsic impedance of the medium.
Linear Polarization A time-harmonic field is linearly polarized at a given point in
space if the electric (or magnetic) field vector at that point is always oriented 15
along the same straight line at ever}' instant of time. This can happen if the
field vector (electric or magnetic) possesses (a) only one component or (h) two
orthogonal linearly polarized components that are in time phase or 180° out
of phase.
Circular Polarization Circular polarization can be achieved only when the mag
nitudes of the two components in Eqn. (2.5) are the same and the time-phase
difference between them is odd multiples of ~j'2.
Elliptical Polarization Elliptical polarization can be obtained (a) when the mag
nitudes of the field components in (2.5) are not equal and the time phase-
difference between the two components in (2.5) is an odd multiple of ~j'l or
(b) when the time-phase difference between the two components is not equal
to multiples of ~ /2 irrespective of their magnitudes.
2.2 Review of Literature
Although a self-resonant bent antenna can be of virtually any imaginable geometrical shape, the ones that are mentioned and studied in the literature include the normal mode helical antenna ( XMHA). the invert ed-L antenna (ILA), the meander antenna, the zigzag antenna, the sinusoidal antenna, and the dual meander antenna. In the following we provide a brief account of the previous work as related to the analysis and design of these antennas.
The normal mode helical antenna (NMHA) is an antenna which radiates in the direction normal to the helical axis. The radiation pattern of a small NMHA is the same as that of a small straight-wire dipole [19]. The study of this antenna dates 16
Table 2.1: Parameters and characteristics of the zigzag antenna in [7]: see Fig. 2.2.
2Ia-,rc (A) 2Z «r(A ) r (°) Rres (H SR (%) Gain (dBi) H PBW (M 0.50 0.45 129 65 10 2.1 ±80 0.58 0.38 81 46 24 2.0 ± 82 0.67 0.33 59 37 34 1.95 ±84
back to the 1940's when Kraus [19] first introduced the helical antenna. However,
due to its attractive self-resonance property, and small size, the NMHA has been
found useful for application in mobile communications and hence has been studied quite extensively [19]. [4].
The inverted-L antenna (ILA) is a low-profile antenna that has found application
in missile telemetn.', and rockets [5]. .-Vs the name suggests the antenna consists of a small vertical conductor on top of which there is a horizontal conductor. In the literature the IL.A. has been studied using both analytical and numerical techniques.
Closed-form expressions for the radiation pattern of this antenna are available in
[4]. King and Harrison Jr. [5] have presented an analysis for computing the input impedance of the ILA. The analysis is based on transmission line theor>' and is restricted to the situation kh Hu [20] have derived a closed-form expression for the input impedance of the ILA. This expression is restricted to antennas that are electrically small. Nakano et al. [7] described a zigzag and a meander antenna. The zigzag antenna described in [7] is shown in Fig. 2.2a. Its geometrical parameters are e and r. The antenna was analyzed in [7] using the Method of Moments (MoM) which was verified by experimental results. Results from [7] are summarized in Table 2.1. IL . . . © - ■ i n ..... (b) Figure 2.2: (a) .A. zigzag, and (b) a meander antenna [/] From Table 2.1 it can be seen that as r decreases the SR increases and Rres decreases. For an SR of as much as 34%. the Rres is about 37 V.. The gain of the antenna is close to the gain of a straight-wire half-wave dipole of 2.16 dBi. The half-power beamwidth. HPBW" is also close to that of a straight-wire half-wave dipole of 78°. However, the same cannot be said for the resonant resistance. For instance, the value of the resonant resistance for a zigzag antenna is about 37 fl for a shortening ratio of 34%. which is significantly smaller than the resonant resistance of a straight-wire half-wave dipole of about 70 fl. The meander dipole of Fig. 2.2b resonates at a length of 2Lax = 0.35A (where 2I^.,re = 0.70A. c = 0.0133A) with Rres = 43 n and SR of 30%. The half-power beamwidth (HPBW) and the gain are ±84°. and 1.95 dBi. Hashed and Tai [ 8] introduced another class of meander antennas which are shown in Fig. 2.3. The shortening factor defined in [9] states that if a bent antenna of length I and a conventional antenna of length Iq have the same resonant frequency. IS A. I (a) (b) (c) Figure 2.3: Several meander antennas (a) N=2. (h) X=4. (c) X=G [ 8]. the size reduction (SR) is. — I X 1009f ( 2 . 6 ) In when both the antennas are manufactured from the same diameter wire. The size reduction for these antennas [as defined in ( 2.6 )] depends primarily on the number of sections per wavelength (X) and the width of the rectangidar loops (fc). Some results from [8] are summarized in Table 2.2. .\ccording to the numerical results of Table 2.2 a size reduction of 36% or more means that the resonant resistance is 20.7 Q or smaller whereas the experimental results show that for a size reduction of 29% or more. Rres is 22 Q or smaller. Despite the differences in the numerical and experimental results, it may be predicted from Table 2.2 that if someone intends to manufacture the dipole counterpart of Fig. 2.3. 19 Table 2.2: Characteristics of the meander antennas of [ 8|: see Fig. 2.3: I = 4.5 cm. a = 0.4 mm. and w = 0.3 mm. = 13.5 cm. Param eters SR(%)SR(%) R r e s (H) R r e s (H) (Numerical) (Experimental) ( Numerical) ( Experimental) .\' = 2 41 41 11.5 13 .V = 6 37 33 19.3 21 .V = 10 36 29 20.7 22 and is interested to achieve Rres = 50 Q. the size reduction may be less than 30%. Wong and King [10] proposed a height-reduced meander zigzag monopole con figuration that consists of a driven element fed from a coaxial line and one or more closely spaced open sleeves. Both the driven elements and the open sleeves were m ade from 2.0 mm diameter wire. The dual zigzag configuration was chosen to reduce cross-polarization. The antenna was designed to operate in the frequency range of 250-750 .MHz. The height and width of the antenna were 13.97 and 11.43 cm. Throughout the frequency range of 250-750 MHz. the \'SW'R was less than 5.5:1. The radiation pattern of the height-reduced meander zigzag monopole on a ground plane of diameter 121.9 cm was similar to the pattern of a straight-wire quarter-wave monopole upto 500 MHz. The pattern showed small side lobes for frequencies above 500 MHz. The input (\'SW’R bandwidth) and the radiation (pattern and gain) character istics of two classes of bent wire antennas e.g. the sinusoidal and the meander were measured experimentally as function of their shortening ratios (SR's) in [13]. The design variables for these configurations were defined and then correlated with the antenna characteristics. The results of the input characteristics show that for both types of antennas the bandwidth becomes narrower as the shortening ratio increases. For a shortening ratio larger than 30%. V'5U'/? > 2.0. However, it was found that 20 for the same shortening ratio the sinusoidal antenna had better \ ’S\\'R frequency characteristics than the meander. Measured radiation patterns ( both H-plane and E-plane ) of both classes of amten- nas show that these are similar to the pattern of a straight quarter-wave monopole. The half-power beam width's (HPBW "s) of these bent antennas are somewhat wider (42° — 44°) than that of a straight quarter-wave monopole (39°). The gain of the bent antenna was found to be decreasing with the increase in SR (4.96-3.84 dBi for a sinusoidal when the SR increased from 18% to 38.82%). The gain of a bent antenna is found to be smaller than a straight cjuarter-wave monopole (Ô.1 dBi). \'u and .Tu [21] proposed two meander dipole antennas for possible application in personal communication network (PCN) handsets. Both the antennas were printed on Duroid 6010 (e^ = 10.2: thickness=0.064 cm) substrate. The ground conductor of the substrate was etched off. For further size reduction, the second meander antenna was covered by another layer of dielectric. The length of the first antenna was 11.2 cm at 880 MHz. The size reduction and bandwidth of this antenna were 34.1% and 7.3% respectively. The length of the second meander was 9.2 cm with size reduction and bandwidth of 45.9% and 5.0% respectively. To calculate size reduction ( 1.1) was used. Nakano et al. [22] studied a printed zigzag dipole using the Method of Moments (MoM) [22]. The antenna geometry' without the substrate is shown in Fig. 2.2a. It was assumed that the antenna consisted of a thin wire printed on a grounded dielectric substrate of infinite extent. The thickness of the substrate was O.IOI 6 A0. The resonant resistance and the shortening ratio for this antenna were 20 Q and 30%. respectively. The cross-polarization was found to be below -45 dB. A study of a printed meander dipole based on the MoM was presented in [23]. 21 The geomet r}' of the antenna without the substrate is shown in Fig. 2.2b. Similarly to the analysis given in [ 22] it was assumed that the antenna consisted of a thin wire printed on a grounded dielectric substrate of infinite extent. The thickness of the substrate was O.IOIGAq. The radius of the wire was 10“ 'Ao and the element length of the meander antenna e was taken to be 0.0133Ao [Fig. 2.2b|. The input impedance of the printed meander antenna was computed with the dielectric constant of the substrate. Cr as a parameter. For f.r = 4.0. and 6.05 the resonant resistances were 13 and 32 Q. respectively. For f.r = 2.0. and 6.05 the shortening ratios were 47 and 68 %. respectively. oo C hapter 3 The Invert ed-L Antenna Electrically small monopoles have drawn the interests of antenna designers over the decades due to their size and conformity with geometries like aircrafts, rockets, missiles etc. Small monopoles are also used in low frequencies when the size of a resonant quarter-wave monopole becomes prohibitively large. In the lossless case, the input impedance of a small monopole consists of a small radiation resistance and a large capacitive reactance. Such an antenna is inefficient and is difficult to match to standard transmission lines, with characteristic impedances of 50 and 75 Q. The small radiation resistance may be increased and the capacitive reactance can be reduced or cancelled entirely by adding an additional segment of conductor of length L as shown in Fig. 3.1. The antenna shown in Fig. 3.1 is called an inverted-L antenna (ILA) [5]. It consists of a vertical element of height h and a horizontal element of length L. King and Harrison .Ir. [5] have presented an analysis for the ILA which is based 23 V 7777777 Figure 3.1: .A.n inverted-L monopole. 2h / h- Figure 3.2: .A.n inverted-L dipole on transmission line theop.'. The analysis is restricted to the situation where kh In this thesis we derive an expression for the input impedance of an ILA using the induced EMF method [24]. To obtain this expression, a sinusoidal distribution of current that drops to zero at the antenna end is assumed and closed-form expressions 24 for the near-fields of the antenna are derived. The closed-form expressions are then used to derive an expression for the input impedance of the antenna. Z. The expression for Z contains integrals that can be easily evaluated using a suitable numerical integration routine. While the transmission line model of King and Harrisson .Ir. [5] is restricted to kh To verify the accuracy of the proposed formulation, the input impedance com puted using it is also compared with that computed using .\'EC [ij. 3.1 Current Distribution We consider the dipole mode of the inverted-L antenna shown in Fig. 3.2. .Assuming the IL.A [Fig. 3.1] to be radiating on a perfectly conducting infinite ground plane, its input impedance should correspond to half the impedance of the antenna shown in Fig. 3.2. W hat follows now is a derivation for the input impedance of an IL.A based on the dipole mode of the inverted-L antenna, which consists of four wire elements [Fig. 3.2]. We call the lower and upper horizontal elements antenna elements 1 and 4. and the lower and upper vertical elements antenna elements 2 and 3 respectively. The antenna is fed by a delta-gap voltage and is manufactured from a wire that is thin and perfectly conducting. .Assuming a sinusoidal distribution of current along 25 7b N A/a fa) (b) Figure 3.3: (a)A transmitting antenna fed by a voltage source (magnetic current), and (b) an auxiliary current distribution. the antenna that drops to zero at both ends, the current distribution for the vertical elements (r-directed) of the ILA can be expressed as /osin(A-/ + k:') for < 0 (3.1 /osin(A7 — k:') for c' > 0 where Iq is the maximum amplitude of current, k = -c’^^/oCo = / = L + li. fio = 4n X 10“ ' H/m, and e,) = 8.83 x 10“'- F/m. Similarly, the current distribution for the horizontal elements (elements parallel to the //-axis) can be expressed as /( //) = Fosini kl — kij' — kb). !3.21 3.2 The Induced EMF Method The induced EMF method was introduced by L. Brillouin [25] and elaborated by A. -A.. Pistolkors [26] and P. S. Carter [27]. The method involves a self-impedance formula which can be derived with the aid of the reciprocity theorem. This classical method is described in detail in references, such as [28], [24], and [16]. Thus, a brief overview of this method should be sufficient here. However, the application of this 26 method for computing the input impedance of the invert ed-L antenna is dscribed in detail in subsequent sections. The derivation of an expression for the input impedance of an antenna in the in duced EMF method can be carried out by the application of the reciprocity theorem and what follows here is similar to the derivation presented in [28]. For example, in Fig. 3.3a a perfectly conducting antenna is shown which is located in free space and is fed by a voltage source (magnetic current .V7"). On the other hand, the case in Fig. 3.3b shows an auxiliary' current density ./* flowing in free space and occupying the same volume (inside surface S) as the case in Fig. 3.3a. -Application of the reciprocity theorem to the two situations in Fig. 3.3 yields J = J - y /7*..\7“df. (3.3) Within S. E" = 0 and the tangential component of E“ is zero on S. Thus y^/7*.-\7“dr = J E^../"dc. (3.4) The left-hand side of (3.4) can be expressed in terms of I *, the applied voltage and /*. the total auxiliary current passing through the terminal region. Thus - = y E ^ .E -d f. (.3.5) The applied voltage may be expressed as = /“Z" where is the terminal current in the transmitting case (Fig. 3.3a) and Z“ is the antenna impedance. Therefore, Z" = - - - L f E \.Ë 'd r. 3.6) / “ / * J Now. if ./^ = = .7* and consequently = I. and also if E is the field of the induced current ./ flowing in free space. (3.6) becomes Z “ = ~~p j E.Jdv. (3.7) The advantage of (3.7) is that the impedance given by it is stationan- with respect to small variations in the current density .7. That is if the assumption in J involves a first-order error, then the impedance given by (3.7) contains a second order error. .A. proof for this can be found in [24]. To apply the induced EMF method for calculating the input impedance of a straight wire antenna we may consider the skin eflfect and assume that the current is concentrated on the wire surface. Thus we have a cylindrical sheet of current the field of which we need to compute. However, we actually assume a straight current filament instead of cylindrical current sheet because for such a thin filament the expression for the field in (3.7) is easier to derive as compared to that for a cylindrical sheet of current [28]. For the current term J in (3.7) we assume another filamentary current source located at a distance a (radius of the wire) from the previous current filament. Now we take the dot product of the field and the current as given by (3.7) and integrate it over the domain of the antenna. 3.3 Analysis Now that the induced EMF method is described, an expression for the input impedance of an inverted-L antenna can be derived. Following (3.7). the expression of input impedance for a thin IL.A can be written as [24] Z = - , .^ ■>77 [ E'"(.s)[{.s)(l.s (3.8) Iq sin" A7 7u-,rf.s where E"^(s) is the scattered field, and I(s) is the current distribution along the antenna. We use Pocklington's integral equation to calculate E'^s). The equation for an 28 arbitrarily bent wire antenna is [29] 1 r 0 0 . ■ •s£'*(.s) = -— :------/—n— (3.9) A-J^'€q Jwirei Os Ds' R where .s. s' are unit vectors lying along the wire at the observation and the source points respectively, and R is the distance from the source to the observation point. In general, all primed variables indicate source coordinates and all unprimed variables indicate observation coordinates. Rewriting (3.8) Z = - -.^-.;-ry[ / E'"{ + [ E*(c)/(r)dr + /q Sin- A / Jy=f. Jz=-h f E^(z)Ii = ) f h + [ E^!i)I(i/)ih/] (3.10) 7j=o -'y =0 The first integral in (3.10) indicates that the observation point is any point on antenna element I where : = —h and /(//) = /osin(A/ — kij — k-fi). The term E'^i/) in this integral contains the summation of the y components of fields for antenna elements 1. 2. 3. and 4. Closed form expressions of these fields are derivable using (3.9). Similarly, the second integral in (3.10) indicates that the observation point is any point on antenna element 2 where // = 0 and I(z) = /osin(Ad + k-z). The term E^{z) in this integral contains the summation of the : components of fields for antenna elements 1. 2. 3. and 4. The other two integrals in (3.10) have similar meanings. The guidelines that may be followed in order to compute the input impedances of antennas that have elements parallel and perpendicular to the y-axis. are: 1. Number the antenna elements. The total number of antenna elements gives the total number of observation elements. Consider a single observation element at a time 29 2. Calculate the fields radiated from each element with reference to the specific observation element using (3.9). Take a summation of all such fields. Multiply the result from summation with the current distribution at that observation point and integrate it over the domain of the observation element 3. Repeat step 2 for all observation elements. .\dd all the results from integration. This final result represent the value of the integral in (3.8) Rewriting (3.10). the input impedance of the inverted-L dipole of Fig. 3.2 can be expressed as Z = — [Zi 4- Z-> -I- Z;j + Z|] (3.11) Iq sin" kl where Z\. Zo. Z3. and Z, are the 1st. 2nd. 3rd. and 4th integrals in (3.10). It may be shown that Z, = Z; and Z-> = Z*. Thus Z = - .V,..[2(Z, +Z,)|. (3.12) /,) sin- kl where Zi — I [El -l- E j 4- E;} 4- E|]/()sin( A / — kij — kh)(ljj (3.13) J1)=I. and Z'2 = f [Ej 4- E(5 4- E 7 4- Es]/osin(A4 + kz )dz (3.14) J z = where El = — f ^ ■ ■■ sin(A-/ - kl/ - kh)(li/ (3.15) A-j^'eo { OyOi/ J R 2 = 3- ^ r + A:-h/.ÿ| ^-^sin( W + kz’)dz' (3.16) Jz'=-h Oiiuz' RrjuJ^Q J:'=-hJz'=-h [ dy Oiiuz' dz RrjuJ^Q I a E 3 = ——I 4- k-y.z'\ ^ — ■ sin(A/ - kz')dz' (3.17) 4-/u,'Co 2.-'=o I dyOz'ay R 30 I ’J [ - T y i / + 4 “ - ^'''''"' ' 3 'SI £■-. = — / I 1 ^— — sin( k-l - kl) - kh )(li/ (3.19) Jy'=r. [ 0 : d i / J R Ec = [' ( - r f sin{kl + kz')fh’ (3.20) À-j^'(Q Jz’=-h [ 0 : 0 : ' J R In r:'=h f 0 0 •> - 1 £7 = — ^ / + k^E:' \ — — sm{kl - k:'),!:' (3.21) A-ju:enJ:'=o [ 0 : 0 : ' J R En = -— r — / l ~ ' ^ T 7 ^ 1 p sin(A.-/ — A ;/ — A )f/(/' (3.22) 4-j..t.'eo I 0 :0 i/ J /t and /?= -k (v -- 4/)- 4- (: -- (3.23) The input impedance of the ILA in Fig. 3.1 is half of what is given by (3.12). The integrals in (3.15)-(3.22) can be solved analytically. The analytical treat ment of these integrals is presented in .Appendix .A. Using the results from .Appendix .A and simplifying them. El = -j30/o L> |/7i + ^ ( j + j - Pi (.7.3 + (3.24) Ri AT?, Pfi , , Pr> '3.231 ç-jkl çjkl — — ( P 9 ~ P ii)H— — (Pi2 — Pio)i (3.26) 31 Table 3.1: Expressions for the parameters in eqns. (3.24)-( 3.2i = \la- + {ij — L)~ 73 = cos(A7 — AT — A7i) ^ — Ri'~h t R-2 = \/a- + if- 7i = sin{ A7 — AT — A7; ) P7 = /îji-i-A i + //- + /;•- 75 = sin kl PS = ^ Al 7g = sin( kl — kh ) (• - J A ( + A ) Ra = \jfi- + (;/ — £)■ + 4/;'- f . = ^ ~ +•->/. I = cos( A7 — kh ) Pli - fuiru-h) _ ,-lkR^ f/o = sin( kl — kh ) P i = «s P'- ~ ruiiu^h) q-2 = % E , = - 7.30/0 (.3.2 ; The parameters in eqns. (3.24)-(3.27) are defined in Table 3.1. Similarly, the other field components are ipven by Rr, = — j30(c + /')/o[^-^(i + — riio A/tio /?n A-E., (3.28) E,3 = - 730/0 I"' - + kR £7 = —730/0 «T + T E 11 .72 » 7 , . = —730(7 — h)/o[^-^(7 + . p ~ + “ 16 A E|G Ru ^ kR II 32 Table 3.2: Expressions for the parameters in eqns. (3.28 )-( 3.31 R.J = ^a-’ + P + (r + /0- »1 " 7 = % R n R\o = ^a- + {: + h )- ll2 flio" s = fll« Ç. — /2g — L 1 /?I2 = \/ a- + « 9 = “ .3 Rg ( Rg — r ) Rie( Ris — 7. ) ^^—jkiRg-hL) « 1 Rgi Rg-^f. ) " t o = Ris ( Rm + {■ /?ir, = y»- + Z- + (z — A )- « 6 fjT = COS k l R i 2 + ------«lü)] (3.31 - ^11 ^11 Expressions for parameters in eqns. (3.28)-(3.31) are given in Tables 3.1 and 3.2. 3.3.1 Results The formulation proposed in this chapter is tested for a straight-wire dipole antenna of length 0.5A. T he input impedance of this dipole is computed using 1 = 0. h = 0.25A. and a = 0.00025A. The result obtained is 73.13 + _/'42.45 Q, which is the well established input impedance of a thin wire half-wave dipole [16]. The input impedance of the inverted-L antenna is computed using the proposed induced EMF formulation. For comparison the input impedance of the same an tenna is also computed using the Numerical Electromagnetic Code (NEC) [1]. The dependence of the input reactance on A:/ is depicted in Fig. 3.4 with the radius of the wire. «. as a parameter. It is apparent that the results computed using the induced EMF formulation are in good agreement with that computed using NEC. especially, for A7 < 2.1. The antenna resonates near A7 % 1.58 for both a = 0.1 and 33 600 200 -200 a = 0 .1 mm -600 Induced EMF a3 -1000 + +++ + + NEC -1400 -1800 77777777 -2200 0.1 0.9 2.3 2.5 kl Figure 3.4: The input reactance of an inverteci-L antenna as a function of kl = k{h + L) with the radius of the wire {a) as a parameter: //=40 mm. 1=40 mm, and frequency range 100-1550 MHz. 34 200 180 160 .77777777 140 S 120 U 20 0.1 0.3 0.5 0.7 0.9 2.1 2.3 2.5 M Figure 3.5: The input resistance of an inverted-L antenna as a function of Â7 with the radius of the wire (a) as a parameter: h=40 mm. £=40 mm. and frequency range 100-1550 MHz. 1.25 mm. The input resistance of the IL.A. as computed using the induced EMF method and NEC is shown in Fig. 3.5 as a function of k/ with the radius of the wire, n as the parameter. The resistance computed using the induced EMF method does not depend on the radius of the wire, hence we have the same curve for i)oth rz = 0.1 and 1.25 mm. .Also it is clear that the the input resistance computed using the induced EMF method and NEC are the same for A7 < 2.1 (for both a = 0.1 and 1.25 mm). However, the results from the NEC computation show that the wire radius affects the input resistance for A7 > 2.1. 35 -5 -9.5 dB CQ-10 -2-15 a= O .I mm —25 • /77777T5 800 900 1000 1100 1200 Frequency (MHz) Figure 3.6: Retum-loss versus frequency characteristics of an inverted-L monopole with the radius of the wire, n as a parameter. Other parameters of the antenna are h = L = 40 mm. The bandwidth of the ILA can be determined from the input impedance of the antenna and the characteristic impedance of the feeding transmission line. .Assuming a perfect match with the feeding transmission line at the resonant frequency and using the input impedance data shown in Figs. 3.4 and 3.5 the bandwidth of the ILA is calculated. The frequency range over which the antenna operates is determined from the ret urn-loss ( RL = 20 log,o p. where p is the magnitude of the complex reflection coefficient) versus frequency characteristics shown in Fig. 3.6. .A. return- loss of -9.5 (IB (\'S\VR=2) is considered to be the upper limit in order to determine the frequency range of operation. As it can be seen from Fig. 3.6. the antenna with the thicker wire has a bandwidth of 84 MHz (8.9%) and the one with the thinner wire has a bandwidth of 50 MHz (5.3% ). The comparatively wider bandwidth of the thick ILA is expected because it is well known that a thick antenna is less frequency 36 - 5 -9.5 dB ÇQ-10 - 2 - 1 5 h=20 and L=60.3 mm -20 - 2 5 8 0 0 9 0 0 1000 1100 1200 Frequency (MHz) Figure 3.7; Retum-loss versus frequency characteristics of an inverted-L monopole. Antenna 1: h = 20 and L = 60.3 mm. a = 0.1 mm. .Antenna 2: h = L = À0 mm. and a = 0.1 m m . sensitive than a thin antenna when the input impedance is concerned [16]. The handwidths of two IL.A's resonant at the same frequency (942 MHz) are compared in Fig. 3.7. The height, h of one antenna is 20 mm (0.06A) and the other is 40 mm { 0.12A). Let us call the one with h = 0.06A. antenna 1 and the one w ith h = 0.12A. antenna 2. The handwidths of antennas 1 and 2 are 3% and ô.3'/f. respectively [Fig. 3.7]. From Fig. 3.7. it is clear that by increasing I the resonant height of an inverted-L antenna can he reduced. The consequence, however, is narrow bandwidth (antenna 1 as compared to antenna 2). Next, the input impedance of an inverted-L antenna is computed using the in duced EMF method as a function of k-fi. Impedances are computed for hL = 1.27 and 0.8 with the radius of the wire, a = O.OOOoA.Tlie results are shown in Fig. 3.8. The first set of curves represent antenna .A for which k:L = 1.27 and the second set of curves represent antenna B for which k L = 0.8. It is apparent that antenna .A 200 150 100 kL=0.8 50 kL=l.27 = - 5 0 - -100 Resistance -1 5 0 Reactance -200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 kh Figure 3.8: Input impedance of an inverted-L antenna as a function of kh w ith kL as the parameter: wire radius a = 0.0005A. 38 resonates when kh % 0.32 and antenna B resonates when kh % 0.79. This means that the resonant heights {h) of antenna .A. and B are 0.05A and 0.13A, respectively. Thus a larger kL leads to a shorter antenna. Fig. 3.8 clearly demonstrates that the resistance at resonance for antenna .A is much smaller than antenna B. 4.9 and 20.6 V.. respectively. Small resistance at resonance is indeed a disadvantage when impedance matching and efficiency are concerned. .Also, according to Fig. 3.8. the shorter the antenna the narrower is the bandwidth. Thus there is always a compromise necessary between the advantage of size reduction and performance characteristics when designing short antennas. 3.4 Discussion The input impedance computed using the induced EMF method approaches infinity as kl approaches ux. where n = 1 .2 ,3 ...... This can be explained with the help of Eqn. (3.12). according to which, if kl approaches n~. the denominator of (3.12) approaches zero and hence, the input impedance approaches infinity. The depen dence of the input impedance of an inverted-L antenna on kl is shown in Fig. 3.9. It is evident that the impedance computed using the induced EMF formulation is in good agreement with that from NEC computation as long as kl is not near x. Note that the region of main interest is the antenna resonance region, since the antenna is expected to operate over a band of frequencies centering the resonant fre quency. From Fig. 3.9 it can be seen that the induced EMF results are in excellent agreement with the NEC results for a bandwidth of about 76% near the antenna resonance region [shown using rectangular boxes]. Thus, while the impedance band width of the inverted-L antenna is only a few percent [Figs. 3.6 and 3.7]. the percent 39 2000 Induced EMF ^ 1000 ~ + NEC 'V u-1000 76% bandwidth 0.3 0.6 0.91.2 1.5 1.8 2.1 2.4 2.7 3 k l 2500 3 2 0 0 0 e 1500 induced EMF 76% bandwidth •3; 1000 - + NEC 500 0.3 0.60.9 2.1 2.4 2.7 k l Figure 3.9: The input impedance of an inverted-L antenna as a function of kl: h = L = ÀQ mm. and a = 0.1mm. frequency range 100-1875 MHz. bandwidth over which the input impedance can be accurately computed is probably more than what is needed in most practical cases. The problem of infinite impedance is caused by an error in the assumption for the current distribution. Since the assumed distribution of current is purely sinusoidal, as kl approaches n~. the current at the feed-point approaches zero, and hence the impedance approaches infinity. In reality, for kl = n~. the current at the feed- point is small, but not zero and thus the input impedance is large, but not infinite. This can be confirmed from Figs. 3.10a and 3.10b. In these figures we compare the sinusoidal distribution of current with the current distribution computed using NEC for an inverted-L antenna. We consider I = 0.35A and / = 0.5A and plot the current distribution for an inverted-L dipole in Figs. 3.10a and 3.10b. respectively. It is 40 0.8 o 0.6 0.4 O) Assumed (sinusoidal) NEC -0.3 - 0.2 - 0.1 0.1 0.2 0.3 Length of wired) S 0.8 Assumed (sinusoidal 0 0.1 0.2 0,3 0.4 0.5 Length of wired) Figure 3.10: Current distribution of an inverted-L dipole as a function of the wire length {21). 41 apparent that the current distributions are ver}' similar in the vicinity of the feed- point when I = 0.35A or kl = 2.1 [Fig. 3.10a). However, for / = 0.5A or kl = - [Fig. 3.10b). although the assumed current is zero at the feed-point. the one computed using NEC is not. Thus the deviation of the input impedance computed using the induced EMF formulation from that computed using NEC is due to the deviation of the assumed input current from the one computed using NEC. The expression for the input impedance for an IL.A. i.e. Eqn. (3.12). may be further simplified in terms of Si and Ci integrals [16). However, since an efficient nu merical integration routine is the only thing that is needed to obtain the final result, further simplification is not intended. The advantage of the proposed formulation lies in the fact that it gives simple expressions which can be readily programmed into a computer. L nlike the Method of Moments technique it does not require comput ing the elements of the generalized impedance matrix, matrix storage, and matrix inversion [30). However, the method is limited to input impedance computation of IL.A. S or IL.A. like configurations for which n < 0.025A and kl < 2.1. For thicker (n > 0.025A) antennas the purely sinusoidal distribution of current is still representative but not accurate [16). .A.lso. it was shown that for longer antennas (kl > 2.1) the current distribution is somewhat different than the assumed purely sinusoidal distribution. Thus for thicker and longer antennas more accurate techniques like the Method of Moments should be used. The formulation presented here may be used to compute the input impedance of a straight-wire dipole or a straight-wire monopole by simply making L = 0. .A.Iso. it may very well be used to derive expressions for the input impedance of other antennas like the T. the folded unipole, and the loop-loaded monopole. 42 C hapter 4 The Meander-Line Dipole A n ten n a The antenna shown in Fig. 4.1 is called a meander-line-dipole (MLD) antenna. Such an antenna is essentially a repetitive structure that contains a number (.V) of identical sections with each section consisting of four segments. The antenna, originally described by Nakano et al. [7] can be used as a substitute for a straight- wire half-wave dipole as because its radiation characteristics are similar to that of a dipole. In addition since an MLD is always smaller than a half-wave dipole it may also be suitable to use MLD's instead of straight-wire dipoles where size reduction is important. For instance, consider a collinear array of half-wave dipoles for application in a base station. Replacing the dipoles by MLD's can reduce the length of the array without deteriorating its peronnance. The study presented in [7] describes only a single MLD design. The parameters and characteristics of the antenna are: antenna length= 0..35A. wire length= 0.70A. 43 segment length, e, = e» = 0.0133A. and Rres = 43Q. Thus, it is apparent that, at present, there is not much information available in the literature about this antenna. An attempt, therefore, is taken to analyze the MLD antenna with the objectives of examining its input impedance, radiation pattern, and gain as function of its parameters. 4.1 Input Impedance 4.1.1 Computation Technique The analytical model described in chapter 3 may be extended to derive an expres sion for the input impedance of the MLD antenna as well. However, since the MLD antenna usually consists of many more segments than an ILA. an analytical for mulation of its input impedance would involve large number of parameters. For instance, consider an MLD antenna with A segments. For this antenna there are .V observation elements, and for each observation element, there are .V field com ponents. This indicates that .V x .V integrals are to be solved, which would make the entire solution more complicated. Because of the complicated nature of the analytical treatment, we decide not to pursue it any further. To compute the input impedance of the MLD antenna we decide to use the Numerical Electromagnetic Code (NEC) [I]. 44 I J - e - Figure 4.1; The geometry of an MLD antenna: Pi =vertical segment length, e-2 =horizontal segment length, and L = antenna length. 4.1.2 Results -A.n MLD antenna shown is in Fig. 4.1. It consists of conducting wire segments that are either on the y or z axis, or are parallel to the y or : axis. The segments that are on the y axis or are parallel to the y axis may he called horizontal segments. The length of each horizontal segment is co- Similarly, the segments that are on the z axis or are parallel to the z-axis may he called vertical segments. The length of each vertical segment is c ,. We know from the study of the input impedance of the IL.A. that its resonant length is smaller than a straight-wire quarter-wave monopole. It happens because there is a horizontal conductor of length L on top of the vertical element of the IL.A [Fig. 3.1]. It may he recalled th at for a fixed h (length of the vertical conductor), increasing L decreases the resonant frequency of the antenna. In other words, it 45 decreases the resonant length of the antenna. In light of the above, looking at the geoinetr}' of the MLD antenna [Fig. 4.1]. one can guess that its resonant length would decrease if c> increases. Similarly, it may also he possible to guess that the input impedance of an MLD antenna with a fixed wire length would be a function of the ratio of the horizontal segment length to the vertical segment length f^). the number of meander sections (.V). and the radius of the wire (a). Note that for a straight-wire dipole. = 0. To examine whether the above assumptions are true and in case if they are. to investigate the dependence of the input impedance of the MLD antenna on the above parameters we conduct a study. First the resonant length and the resonant resistance of the MLD antenna are investigated. Two cases are considered: ^ = 1. " ' 1 and ^ = 2. In both cases the number of meander sections. .V is the variable and the wire length. Lu.,re and the wire radius, a are constant. 32 cm long wire with 0.325 mm radius is considered. For .V =2. 4. 8. and 16 the input impedance of the MLD is computed as a function of frequency. The resonant frequency of each antenna is determined from the input impedance data. The wavelength corresponding to the resonant frequency is used to normalize the length of the antenna. L. Finally. (1.1) is used to calculate the shortening ratio. The resistance corresponding to the resonant frequency (the resonant resistance. Rres) and the shortening ratio are plotted as functions of .V in Fig. 4.2. These results demonstrate that an increase in ^ increases the shortening ratio. However, it is also clear that any such increase decreases the resonant resistance of the antenna. For example, for .V = 2. and ^ = 1 the shortening ratio and resonant resistance are 46% and 32 Q. respectively, whereas for .V = 2. and ^ = 2 the shortening ratio and resonant resistance are 62% and 17 H. respectively. 46 7 0 60 30 20 10 N Figure 4.2: Shortening ratio. SR and resonant resistance. of an MLD antenna plotted as function of .V with ^ as the parameter. The length and the radius of the wire are constant (length of wire. Lu.-,re = 32 cm. and radius of wire, a = 0.325 mm). For comparison, consider a straight-wire dipole that is 62‘X shorter than a half wave dipole. This makes the length of the dipole to he Ü.19A. Consider that it has the same wire length and wire radius as the MLD antenna described above. Calculating its input impedance using the formulation presented in chapter 3 gives 7.4 — 7909.7 n. Comparing this with the MLD antenna that has a 62% shortening ratio and 17 D resonant resistance, it is clear that the straight-wire dipole is not only non-resonant (has a large capacitive reactance) but also its resistance is relatively small (7.4(1). Next, the input impedance of a monopole meander antenna is computed as a function of its parameters. Such an antenna radiating on a perfectly conducting infinite ground plane is considered. Two cases are considered: (i) .V constant. C[ = e-2 = e as the parameter, (ii) I and Lu.-,re constant and .V as the parameter. Computed input impedance of a monopole meander antenna is plotted as a func- 1000 500 u S 0 -500 e= 1.3 cm 10) CE e=2.0 cm -100C .1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna length (A.) 15001- § g 1000- 6=1.3 cm I I 6=2.0 cm .2 500- 0.15 0.2 0.5 0.55 Antenna length (X) Figure 4.3: Input impedance of a monopole meander antenna as a function of an tenna length with the segment length, e as the parameter: .V = 2. « = 0.325 mm. tion of the antenna length in Fig. 4.3 with the segment length e as the parameter. .-Antenna param eters that are constant are: number of m eander sections. .V = 2. and wire radius, n = 0.325 mm. Given these circumstances, the resonant length of the antenna is 0.I5A irrespective of the segment length, e. .An increase in e shifts the second resonance toward the left on the abcissa. It also increases the positive and negative peaks of the curves. The resonant resistance. for c = 1.3 cm is IS.5 Q and that for e = 2.0 cm is 17.0 Q. Thus there is a slight decrease in Rres with an increase in e. .Also, com puted input im pedance as a function of the antenna length is plotted in Fig. 4.4 with the number of meander sections. .V as the parameter. .Antenna parameters that are constant are: wire length. Lw.rc = 16 cm. and wire radius. n = 0.325 mm. It is apparent that an increase in .V. increases the resonant length 48 1000 N=4 N=8 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna length (X) N=2 1000 N=4 N=8 M 500 0.4 0.45 0.5 0.55 Antenna length (X) Figure 4.4: Input impedance of a monopole meander antenna as a function of an tenna length with the number of meander sections. .V as the parameter: — IG cm. a = 0.325 mm. and L = S cm. of the antenna. In other words, it decreases the shortening ratio as defined in ( 1.1 ). However, an increase in .V. increases the Rre.^. For example, for .V var\ ing as 2. 4. and 8 the Rres varies as 17. 18.5. and 21 Q. respectively. To observe the effect of the wire radius on the input impedance, a meander monopole with e=O.S cm. .V = 4. and = 12.8 cm is considered. Wire radii of 0.325. and 1.25 mm are considered. The results are shown in Fig. 4.5. Increasing the wire radius increases the resonant length [Fig. 4.5] of the antenna. The resonant length of the antenna changes from 0.16A to 0.18A while the wire radius changes from 0.325 mm to 1.25 mm. This is clearly opposite to what happens with a straight- wire dipole or monopole. For a straight-wire dipole or monopole, an increase in wire radius decreases the resonant length of the antenna [30]. .\lso. for the meander antenna here, the change in wire radius changes the resonant resistance. Rres- For 49 a=0.325 mm a=1.25 mm 0.25 0.3 0.35 0.4 0.55 Antenna length (A.) — a=0.325 mm - - a=1.25 mm 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna length(A.) Figure 4.5: Input impedance of a monopole meander antenna as a function of an tenna length with the radius of the wire, n as the parameter: = 12.8 cm. -V = 4. and L = 6.4 cm. example, for a = 0.325 mm. /?re.s = 19 Q. and for a = 1.25 mm. Rres = 23 H. 4.2 Radiation Characteristics Since the MLD antenna consists of short vertical and horizontal conductor segments that give it a repetitive geometr\'. it may be possible to derive closed-form expres sions for its radiation fields. In this thesis, we present an analytical formulation that gives the radiation pattern of an MLD antenna in easy to calculate closed-form algebraic ecpiations [31]. The formulation is based on an assumption that the cur rent distribution on the antenna is purely sinusoidal. This is a valid assumption for radiation fields for wires that are infinitely thin and perfectly conducting [16]. To derive analytical expressions for the radiation fields of the MLD antenna 50 i! \i &- Ir 3'\ e _L 1 4' (a) (b) Figure 4.6; la) .A. meander-line dipole antenna: (b) integration path. we assume that each segment of the antenna is equal. This assumption would significantly simplify the analysis. We also assume that the segment numbers start from the center of the antenna and increase as shown in Fig. 4.6. Segments in the upper half-plane (c > 0) have unprimed tag numbers, whereas those in the lower half-plane {: < 0) have primed tag numbers. The wire has a length of Lu.-,re = 4.Vc. The length of the antenna is £, = 2Xe = L^.,re/'2. 4.2.1 Fundamentals Consider an actual-source antenna with sources contained in a volume I '. .4.11 sources are harmonic with an angular frequency u.'. There are no sources outside C. The origin of the coordinate system is inside I'. The source coordinates are denoted by x'. f/. z', and the observation point coordinates are .r. y. r. .A.t a remote point 51 where the distance R from the source to the observation point satisfies the inecpiaiity R » > max^/x'" + if - + z'-. in the far-field region the expressions for the fields are as follows [24] — / w /t n e Eo(O.o) = ------;------Wo(B.o) (4.1) 4 " r Eo{()-0) = "------n'j(fl.o) (4.2) 4~ r where WoiO.o) = [cosO cosoJj.{x . if. z') + cosO sinoJ,fx'. if. z') — sinff.Lix'. if. z')]e^'"''^' dx'di/dz' (4.3) \V,^(0.o) = [— sin oJj.(x . if. z') + cosoJ^ix'. if. z')]e^^''^'dx'di/dz' (4.4) M = x' sin 0 cos o + if sin B sin o 4- cos B (4.5) where r is the distance to the observation point from the origin, k = is the free-space wave number, //o = 4 - x 10~' H/m is the permeability of free-space. f-o = 8.85 X 10"'- F/m is the permittivity of free-space, and ./j.. are the x. ij. and : components of current density of the sources. The (^-polarized or vertically polarized and the o-polarized or horizontally po larized power patterns are given by [24] Pr.oiB.o) = ^[ ■_-^]|H o(^-o)|- (4.6) Pr.oi(^.o) = ^[(l^]l^^o(^.o)[- (4.7) where W'oiB.o). and ll'^(^,o) are given by (4.3) and (4.4), and r] = 377 Q. is the intrinsic impedance of free-space. In most cases one is interested in either the relative field intensity or the relative power density radiated in diflferent directions. Thus for relative Eoi^.o) p attern . \V q{9.o ). and for relative EJ kO-o) pattern. U'oi^.o) can be used. Similarly, for the power density patterns Pr.oif^-o) and Pr_o(^-o). the term in (4.6) and (4.7) can be suppressed. 4.2.2 Current Distribution To apply the procedure outlined in the previous section to the MLD. the current distribution on the antenna has to be known. Since it is well known that the current distribution for a thin wire antenna is sinusoidal, it is reasonable to assume that the current for the MLD also has a sinusoidal distribution with zeros at both ends. The current for segments oriented along the z-axis can be expressed as /osin(AT — kz' — nke) for segments 1. 3. 5. ... etc. (4.8) /osin(A L -t- k:' — nke) for segments I'. 3'. o'. ... etc. and the current for any segment oriented along the y-axis can be expressed as /osin(A-£ -f- ky' — nke) for segments 4-4'. 8-8'. 12-12' etc. ly = (4.9) /osin(££ — kl/ — nke) for segments 2-2'. 6-6'. 10-10' etc. where / q is the maximum amplitude of current, and n = 0.1.2.3.4.... for segments 1-1'. 3-3'. 5-5'. 7-7'. 9-9' etc. (4.10) n = 1.5.9.13.17.... for segments 2-2'. 6-6'. 10-10'. 14-14'. 18-18'etc. (4.11) n = 4. 8.12.16. 20.... for segments 4-4'. 8-8'. 12-12'. 16-16'. 20-20' etc. (4.12) 53 4.2.3 Radiation Pattern As the current distribution in the antenna is known, it can now he used to calculate the radiated fields. Since the x component of the current is zero, and since A/ = y sin <9 sin o + cos (4.13) following the path of integration shown in Fig. 4.6b. defined in (4.3) can be expressed as \ \ gi 6. o) =■ \\ Oy[d. o) -\- \\ 0 .{d. o) (4.14) where (2.V4-1) \Vg.{B.O) = S / D-2 sm{kL — k z '— nke](lz'\y'^y^ + rn = 1.3.5.... y / D-iS\n{kL + k z '— nke\dz'\y-=y^ (4.15) m' — I'.3'.5' .. J-' = PI and ( I .V - 2 ) \Voy[B.o) = ^ .4| / .4-.sin{ I - A-y - + m=2.6.10.... ry'=Q /. -4i / .4-) sinjA 'I + A'//'— uA'c + = 1.8 . 12.... ry'=0 4 1 f A-2sin{kL — ki/ — nke\tli/\.'--:^ + '=2'.6'.10'.... “'V'='’ I t.V )' .4, f A-2sm{kL + k i / — nke}(li/\->=-.^ (4.16) m '=l'.8'.12'.... where in and in' indicate the segments over which integrations are performed. is the constant z-coordinate of any segment that is oriented along the (/-axis, y, is 54 the constant «/-coordinate of any segment that is oriented along the r-axis. pi is the lower limit of integration. is the upper limit of integration, and .4, = locosOsinoe’^' (4.17) .4.) (4.18) Z?, = -/nsin^e^^'^'^'"""'"'^ (4.19) 3, ^t.-'coso ( 4 .2 0 ) Each of the integrals in (4.15) and (4.16) can be calculated using / sin( Jx -h '■ )dx = — {a sin( Jx + *■ ) — J cos( Jx + * )}. ( 4.21 ) J o - -h J- Integrating and simplifying (4.15) and (4.16). (2.\>1) (2,V*1)' \\i).(0.o)= ^ (iiiUoU-.i — U\U\Ury) + ^ {««1 — ;/i (/(«(7) (4.22) T7i= 1.3.3.... rn'= I'.3'.3\... where «1 = -/o sin (4.23) pjkp,, cosO - CO.,‘ D] H;j = jk cosO sm(—kpu + kL — nke) -t- k cos{—kp„ -f- kL — nke) ( 4.25) f,jkpi cosO «7, = jk COS 0 sin{—k Pi 4- kL — nke) -I- kcos{—kpi -h kL — nke) (4.27) «6 = jkcosOsinikpu + kL — nke) — kcos{kpu + kL — nke) (4.28) Uj = jk cos6 sin{kpi -f kL — nke) — kcos{kpi 4- kL — nke) (4.29) oo and (IV-2) W'oyid.O) = YL + t’3 - t’l - f’-.) + m=2.6.10.... ( Î.V) Y 1 f’i(fi - r-, - f,j + rr)-f rn= (.S.12.... ( L V -2r y i i’l( —1’-2 — r;i + f'l + I'-,) + I I.V)' ^ ri(-t'i + 1>, + I-,; - t’7) (4.30) m=l'.SM 2'.... where -'-H rS S ki-'---- f-2 = jÂ-sin^sino sin( —Ae 4- AI — nk:e) (4.32) i-:i = A-cost -k-e + k-L - nke) (4.33) ('i = jksin0sino sin{kL — nke) (4.34) = kcosikL — nke) (4.35) r,i = jksmOshio sin(Ae + kL — nke) amOsmc (4.36) ('T = kcosike + kL - nke) (4.37) .A.ccording to (4.4). the opolarizeci component of the field is radiated by the segments that are oriented along the y-axis. Following similar procedure it can be shown that \V,^(f),o) is given by (4.30) with an exception of c, in (4.31) being substituted by 56 4.2.4 G ain Gain of the MLD antenna is calculated using the method described in [16] G' = 101og,o— (4.39) where Pr = / I [Pro(()-o) + Prc,((^-o)]sin0d9(lo (4.40) Jn Jo and f max = max[Pr.o(i9. o) + Pr.o(9. o)j (4.41 ) where Pr,o{9.o) and PrA^-o) are defined in (4.6) and in (4.7) [the term : 7^ [tj'j in (4.6) and (4.7) and the term Iq in (4.23). (4.31). and (4.38) are suppressed]. Equation (4.40) can he simplified as •>- - At .V Pr = + Pr.oio,.9j )] sindj (4.42) . = 1 7=1 where M represents the number of discrete o points and .V represents the number of discrete 0 points on the surface of a sphere. 4.2.5 Results .A. meander-line dipole antenna with four sections was considered. Equations (4.14). and (4.22)-(4.38) were then used to calculate the relative (9-polarized and the o polarized power densities. For comparison, the same power densities were also com puted using the Numerical Electromagnetic Code (NEC) [1]. The ^-polarized pattern of a meander-line dipole in the o = 0°-plane is shown in Fig. 4.7 with antenna length E as a parameter*. The analytical and the numerical ^Radiation pattern and gain plots with \-arions line styles such as solid, dotted, dash-dot rep- 0/ -20 -'0 \0(dB) -30 90 180 Figure 4.7: Radiation pattern of an MLD antenna with L as a parameter (.V = 4). results are in good agreement. The beam is narrower for a longer antenna as ex pected and has side lobes appearing when the antenna length exceeds about Ü.65A. The ^-polarized radiation pattern when observed in a plane other than o = 0° or o — 180° is slightly wider (observable below -20 clB). The effect of wire radius on the radiation pattern of the MLD antenna can be explained with the help of Figs. 4.8 and 4.9. In Fig. 4.8 the radiation pattern of an MLD antenna with L = 0.5A is shown. .As mentioned, the analytical results (solid lines) are based on a thin-wire assumption. The NEC results (crosses), represent an antenna that has a finite wire radius {n = O.OOIA). It is apparent that the patterns look ver>' sim ilar [Fig. 4.8]. Increasing the antenna length to 0.75A results in a multilobe pattern as shown in Fig. 4.9. The appearance of the side lobes and the emergence of a multilobe resent analytical results and discrete points such as pluses, circles, and stars represent numerical results 58 0 (dB) 90 90 180 Figure 4.8; ^-polarized power density pattern of a meander-line dipole antenna with L = 0.5A (.V = 4): wire radius a = O.OOIA. ++ + + + uO (dB) _30 -20 -10 90 90 + + + + ++ + + + + NEC ((PO.OOlX) ^ ^ N E C (very thin) Analytical (very thin) 180 Figure 4.9: ^-polarized power density pattern of a meander-line dipole antenna with L = 0.75A (.V = 4). 59 m -9 L=0.25X:HPBW=86° , iL=0.50X;HPBW=70° -15 L=0.65X;HPBW=48® - 18 , 40 60 80 100 120 140 160 180 Thêta - degrees Figure 4.10: Relative power density and half-power beam width (HPBW) of the MLD antenna as a function of antenna length (N=4) pattern is the result of the phase reversal of the current in the antenna. For the antenna with L = 0.75A, the radiation pattern is affected by the change of the wire radius. .Although the analytical and NEC results are identical for a very thin antenna, the NEC results for an antenna with a finite wire radius {a = O.OOIA) show some differences. The direction of the main beam and the side-lobes remain the same but the intensities of the side-lobes decrease and the nulls are replaced by low-level radiation as the antenna becomes thick. The half-power beam width of the MLD antenna decreases with increasing an tenna length [Fig. 4.10]. From Fig. 4.10. the half-power lieam width is 86° for an antenna of length 0.25A. 70° for an antenna of length 0.50A. and 48° for an antenna of length 0.65A. For comparison, the half-power beam width of a straight-wire 0.5A long dipole is 78° [16]. The o-polarized pattern of the meander-line dipole is shown in Fig. 4.11. .Ac cording to (4.38). the o-polarized pattern has a coso variation. Therefore, it is 60 maximum when o = 0° anti is minimum when o = 90'^. The worst case of cross- polarization is shown in Figs. 4.11 and 4.12 [i.e. cross-polarization computed in the o = 0° plane]. Cross-polarization is well below -25 tlB for antenna lengths ranging from O.lOA to 0.65A (for both N=4 and N=6) [Figs. 4.11 and 4.12]. The pattern in Fig. 4.11 shows that for a constant number of meanders (.V = 4). increasing the antenna length increases the cross-polarization. The pattern in Fig. 4.12 shows that for a constant antenna length {L = 0.25A in this case), increasing the number of meanders decreases the cross-polarization. Cross-polarization for the MLD antenna at resonance is also computed. Consider the graph shown in Fig. 4.2. The maximum shortening ratio. SR is 62%. This occurs when eo/ei = 2. while the other parameters are = 32 cm. .V = 2. and a = 0.325 mm. The cross-polarization for this case is -IS.7 dB. In contrast the smallest SR in Fig. 4.2 is approximately 30%. The other parameters that correspond to this case are = 32 cm. .V = 16. and a = 0.325 mm. The cross-polarization computed is -51.7 (IB. Thus a larger SR essentially results in larger cross-polarization. Calculated gain as a function of antenna length is shown in Fig. 4.13. Depending on the number of meander sections there is some difference in gain. Gain increases as the antenna length increases and it becomes maximum when the antenna length is approximately 0.71 A. For antenna lengths larger than 0.71 A gain decreases. The gain of the MLD antenna is greater than the gain of a straight-wire dipole of the same length (Fig. 4.12 and Ref. [16]). However, the maximum gain for an MLD antenna of length 0.71 A is about 4.8 dBi which is smaller than the maximum gain of 5.1 dBi for a straight-w ire dipole of length 1.25A [16]. 61 -10 (D -5 0 -6 0 L=0.25X o - o - e - o L=0.50X -7 0 L=0.65X -80 60 80 100 120 140 160 180 Theta - degrees Figure 4.11: o-polarized power density pattern of a ineander-line dipole antenna with antenna length I as a parameter (.V = 4). -10 ^-20 m -31 -401- (S -60 —o N=4 - • N=6 -7 0 -80 20 40 60 80 100 120 140 160 180 Theta - degrees Figure 4.12: o-polarized power density pattern of a meander-line dipole antenna with num ber of m eanders (.V) as a param eter [L = 0.25A). 62 pT' 4.5 m 3.5 2.5 N= 6 0.2 0.3 0.4 0.50.6 0.7 0.8 Antenna length, A. Figure 4.13: Gain of a meander-line dipole antenna as a function of antenna length L with N as a parameter. 4.3 Discussion The analytical formulation presented for the MLD antenna can he used to calculate the radiation characteristics of meander-line dipole antennas with any number of sections. Since the principles described here are relevant to thin MLD antennas, it may be expected that some limitations apply to a thick antenna. However, the difference in the radiation pattern of MLD antennas as a function of wire radius is also demonstrated in Fig. 4.9. The effect of the wire radius on the radiation pattern of wire antennas is discussed in [16] where it is concluded that the pattern is essentially unaffected by the thickness of the wire in the regions of intense radiation. Also, with the increase in wire radius the minor lobes diminish in intensity and the nulls are substituted by low-level radiation. The study of the input impedance of the MLD antenna shows that the resonant 63 resistance and shortening ratio strongly depends on both ^ and .V when the length and the radius of the wire are fixed. Typical range of the shortening ratio is 30-6291 while that of the resonant resistance is 42-17 Q. Thus, for a resonant resistance of 50 f2. the shortening ratio is clearly less than 3091:. Increasing the radius of the wire decreases the shortening ratio and increases the resonant resistance. Radiation pattern results show that at resonance the £<;-pattem for the MLD antenna resembles the pattern of a straight-wire half-wave dipole. The resonant MLD. however, has cross-polarization which is below -30 dB for 5/? < 3091. The cross-polarization increases when L increases ( .V constant ) and it decreases when .V increases {L constant). The HPBW of the MLD with .V = 4 varies as 86. 70. and 48° for L = 0.25. 0.5. and 0.65A. respectively. The gain of the antenna steadily increases as L is increased. The peak gain occurs at a longer antenna length when .V is increased. 64 C hapter 5 The Meander-Line Bow-Tie A n ten n a The study of the meander-line dipole (MLD) antenna presented in the previous chapter shows that the resonant length of such an antenna is smaller than that of a straight-wire half-wave dipole. The study also shows that the radiation pattern of the MLD antenna is similar to the pattern of a straight-wire dipole with some cross-polarization present. In addition, the MLD antenna has larger gain than a straight-wire dipole of same length. However, since the MLD antenna is expected to operate at its first resonance, its gain lies between 1.7-2.1 dBi which are the gain of an infinitesimal dipole and a half-wave dipole, respectively. The results from the previous chapter show that the MLD antenna has the limitations of low resonant resistance and the presence of cross-polarization. For instance, the resonant resistances for an MLD antenna are 17. 32. and 41 F>. while the shortening ratios are 62. 46. and 30%. respectively. Increasing the shortening ratio 65 decreases the resonant resistance creating a problem for impedance matching. For example, consider a 50 Q coaxial line for the above three antennas. The s at resonance are 2.9. 1.5. and 1.2. respectively. Thus the shorter the antenna the larger is the \ ’S\\'R. .A. large \'S \\'R not only decreases the amount of power transmitted to the antenna but also effectively reduces the bandwidth which in this case is defined as the the frequency range of operation of the antenna within a specific \'S\\'R . For an MLD antenna with constant wire length, increasing the number of mean der sections (.V) increases the resonant resistance and decreases the cross-polarization. This however, decreases the shortening ratio. Thus, when the design constraints, such as the resonant resistance, and cross-polarization are to be strictly maintained, the shortening ratio obtainable becomes limited to a certain value. The above problem may be alleviated either by printing the antenna configu ration on a dielectric substrate, or by searching for new antenna geometries with superior characteristics. The first of the above two choices is explored in chapter 7. where a printed meander antenna is analyzed using the finite-difference time-domain (FDTD) technique. The second choice, i.e.. the task of searching for new antenna geometries with superior characteristics is indeed a tedious one. since there may be numerous other bent antenna configurations which may some way or the other be better than an MLD antenna. .As because the primar." objective of this work is to analyze several existing self-resonant bent antennas employing a combination of an alytical and numerical tools, a study that analyzes a large number of bent antenna geometries with the objective of optimizing their characteristics is outside the scope of this dissertation. Nevertheless, such a study can be a future research topic. Therefore, in the following we present an analysis for the 3rd self-resonant bent antenna under consideration which we call the meander-line bow-tie (MLBT) an 66 tenna. The MLBT antenna is similar to the trapezoicial-tooth antenna ciescril)ed in [24]. However, while the trapezoidal-tooth antenna is a log-periodic structure, the MLBT is a self-resonant linear one. Since the MLBT antenna looks similar to a meander and a bow-tie antenna [Fig. 5.1] we call it a meander-line bow-tie ( MLBT) antenna. Preliminary' results of our study of this antenna can be found in [32]. The analysis of the MLBT antenna is carried out in two phases. First, the input impedance is computed as a function of various antenna parameters using NEC [1]. To compute the radiation pattern and gain of the antenna a simple analytical model is presented. Similarly to the previous model described in chapter 4. this one also presents the radiation fields in closed-form algebraic equations. The validity of the analysis is verified by comparing the results computed using it with that from NEC computation. 5.1 Antenna Configuration .A.n MLBT antenna is shown in Fig. 5.1. The parameters of this antenna are; e, the length of each conductor segment that is inclined from the :-axis by an angle a/2: .V. the number of meander sections: I. the antenna length: and a. the bow-tie angle. The number .V for the MLBT antenna gives the total number of conductor segments in the antenna. For example, in each meander section there are 4 conductor segments. Thus for the antenna shown in Fig. 5.1 the number of meander sections is 2. Similarlv to the IL.A. and MLD the MLBT antenna can also be manufactured V z A a / ■y ■ / / Figure 5.1: A meancier-line bow-tie (MLBT) antenna. from wire or metal ribon. The antenna length and wire length are given by a / — 2A> cos —o (5.1 and a Lu.- = '2e .V + ( 2 sin - ) P 5.2) p = \ 5.2 Input Impedance Since the resonant resistance. Rres and shortening ratio. SR are two important characteristics of a self-resonant bent antenna, their dependence on the parameters of the MLBT antenna are investigated first. .A. 32 cm long wire with 0.325 mm radius is considered for the above purpose. For a constant number of meander sections ( X). the bow-tie angle (a) is varied and the input impedance of the MLBT antenna is computed using NEC as a function of frequency. From computed input impedance data the resonant frequency and Rres are determined. The resonant frequency is then used to express the length of the antenna in terms of the wavelength. Finally. 68 Table 5.1; Parameters and characteristics of an MLBT dipole: .V = 2, length of wire=32 cm. and radius of \vire=0.325 mm. a (degrees) c (A) MA) .FA % R re s (H) 15 0.0931 0.3694 26.1 58 20 0.0864 0.3404 32 52 30 0.0761 0.2940 41.2 42.6 35 0.0720 0.2748 45 38.6 45 0.0643 0.24 52 32.2 (l.I) is used to calculate the SR. The results of the computations are presented in Tables 5.1 and 5.2. Table 5.1 presents the results for an MLBT antenna with .V = 2. It can be seen that inceasing a from 15 to 45° increases the SR from 26.1 to 529(. However, it decreases the Rr,s from 58 to 32.2 Q. Similarly, Table 5.2 shows that for .V = 4. the SR increases from 33 to 60% while a increases from 15 to 45°. In the same circumstances, the Rres decreases from 48.4 to 22.2 Q. It is apparent that if one is interested to design an MLBT antenna with Rres ~ 50 n. then the SR he or she obtains is approximately 32%. It may l>e recalled that for an MLD antenna, for Rres ~ 4:2 D. the SR % 30%. Thus, the MLBT has larger Rres than the MLD while they have the same SR. The above statement holds for any value o( SR presented in Fig. 4.2 and Tables 5.1 and 5.2. For instance, according to Fig. 4.2. the resonant resistances for an MLD antenna are 40. 33. 27.and 17 D while the shortening ratios are 32. 42. 45 and 60%. For the same shortening ratios the resonant resistances for the MLBT antenna are 52. 42. 38. and 22.2 f>. respectively. Next the input impedance of the MLBT antenna is computed as a function of a and .V. .A. monopole MLBT antenna radiating on a perfectly conducting infinite ground plane is considered. To obtain the impedance for the dipole configuration 69 Table 5.2: Parameters and characteristics of an MLBT dipole: .V = 4. length of wire=:32 cm. and radius of wire=0.325 mm. n (degrees) e (A) 1 (A) 5-A % R re s (^-) 15 0.0422 0.3350 33 48.4 20 0.0386 0.3038 39.2 42.2 30 0.0330 0.2548 49 32.4 35 0.0309 0.2358 52.8 28.6 45 0.0275 0.2000 60 22.2 shown in Fig. 5.1. the impedance computed is to be multiplied by 2. First, we consider the following parameters as constant: .V=2. a = 0.325 mm. and c = 1.0 cm. Then we vary a and compute the impedance as a function of frequency. We express the length of the antenna in wavelengths corresponding to the frequencies of computation. Computed input impedance as a function of antenna length is plotted in Fig. 5.2 with a as a parameter. The reactance characteristic shows that as a is increased, the resonant length of the antenna decreases. For instance, the resonant lengths of the antenna are 0.15A. and 0.12A for a = 24°. and o = 34°. respectively. Increasing a also increases the positive and negative peaks of the curves. Applying the resonant lengths of the antenna above to the resistance characteristic shown in Fig. 5.2. we find that for a = 24°. Rres = 19.7 Q whereas, for a = 34°. = 15.0 Q. Finally, the dependence of the input impedance of the MLBT monopole on .V is examined. For this purpose, a. e. and a are kept constant and the input impedance is computed as a function of frequency. The results of this computation are shown in Figs. 5.3 and 5.4. Note that an increase in .V decreases the resonant length of the antenna. In other words, according to Eqn. (1.1) it increases the SR. However, it is also clear that the peaks of the curves increase (both for Fig. 5.3 and Fig. 800- 300 g -200 -700 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna length (X) 2000 m 1500 1000 a = S'r 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna length (X) Figure 5.2: Input impedance of the monopole MLBT antenna with bow-tie angle a (degrees) as the parameter. 1500 1000 500 -500 — N=1 ■ - - N=2 - - N=3 -1000 -151 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Antenna lentgh(X) Figure 5.3: Input reactance of the monopole MLBT antenna with number of mean der sections (.V) as the parameter. 2500 N=1 - - N=2 - - N=3 2000 - 1500 % 1000 500 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Antenna lentgh (X) Figure -5.4: Input resistance of the monopole MLBT antenna with number of mean der sections (.V) as the parameter. •5.4) while .V is increased. In addition it can also be observed that an increase in .V makes the variation of the impedance with the antenna length more abrupt. Both of the later two situations indicate narrowband antenna characteristic. 5.3 Radiation Characteristics To compute the radiation characteristics of the MLBT antenna we present an an alytical formulation. Similarly to the model described in chapter 4 a sinusoidal distribution of current on the antenna is assumed. Closed-form expressions for the radiation fields are then derived using the principles described in the previous chap ter. The results computed using the proposed formulation are verified by comparing them with the results obtained from NEC computation. The geometry to describe ■y (a) (b) Figure 5.5: (a) .A.n MLBT antenna, and (b; integration path. the current distribution on the antenna is shown in Fig. 5.5a. The path of integra tion to derive the closed-form expressions for the radiation fields is shown in Fig. 5.5b. 5.3.1 Current Distribution The current distribution on the MLBT antenna shown in Fig. 5.5a can be described as follows. The current for any segment oriented along the y-axis can be expressed as sin{f>-L -h k(i/ — n ’esin f ) — nke} for segments 2-2'. 6-6'. ... etc. / = dy-^sin{kL — A(// -b n'esin %) — nke} for segments 4-4'. 8-8'. ... etc. ( 5 .3 ) where Iq is the maximum amplitude of current. L = and n = 1 .3 .5 ---- for segments 2. 6. 10. . . . etc. and 2'. 6'. 10'. . . . etc. (5.4) n = 2 .4 .6 ---- for segments 4. S. 12. ... etc. and 4'. S'. 12'. ... etc. (5.5) The current for segments 1. 5. 9. ... etc. I = dy—=sin{AT — nA-(2.V, — l)esin — [+ \/2 sm ? 2 4-n j —-p= sin {AT ------— — uA ( 2 .\, — 1 )e sin —} 15.6) T2 cos ^ 2 and that for segments 1'. 5'. 9'. ... etc I = d y - ^ s in ( A T — uA-(2.V, - 1 )c sin ^ + \/2 sm § 2 "To j —^ sin (AT 4 ------— — nA ( 2.\ < — 1 )c sin —} 15.7) v/2 cos t; 2 where n = 0.2.4.... for segments 1. 5. 9. ... etc. and T. 5'. 9'. ... etc. (5.8) and .V, is the meander section in which the specific segment is located, e.g. for segments 1 to 4. .V, = 1. for segments 5 to 8. .V, = 2. for segments T to 4'. = 1. and for segments 5' to 8'. .V, = 2. Similarly, the current for segments 3. 7. 11. ... etc. .la.r. ki/ . a I = o»—= sin (A l 4- —— — — 2.\Jrnesm — \ + V2 sm ? ' 2 4-nj—ÿ= sin{AT ------— — 2.\;A ne sin —} (5.9) V 2 cos 7^ 2 and that for segments 3'. 7', IT. ... etc. r ' -A) * r / r I ^ .(X o \ 1 I I = «y—7= sm{A,'L + ^ — - — 2.\^kne sin —} + V2 sm ^ 2 -\-n.—■= sin{A:Z. 4------— 2.\«Anesin —} (5.10) sj'l cos 2 where n = 1,3. 5.... for segments 3. 7. II. ... etc. and .3'. 7', 11'. ... etc. (5.11 ) 5.3.2 Radiation Pattern Eo Component of Field .A.S the current distribution on the antenna is known, it can now he used to calculate the radiation fields. Since the x component of the current is zero, and since M = î/sin(9sino 4-r'cos/9. (5.12) following the path of integration shown in Fig. 5.5b. for the .MLBT dipole. \Vo(0. o) defined in (4.3) can be expressed as \\'o(0.o) = H oy.(^.o) 4- n oy(/9.o) (5.13) where Uoy-(^.o) = [ ^sini9 / sin{AL ------^ - Cl}(h'+ m = 1.3.9.... V- cos-j y cos ^ sin o f sin {AT -----r ^ — Cl}(h/] + m = l..3.9.... V 2 ■Jy'=yi S i n ^ ( I \ — t ) T r:'=:, L-~' [ ^ ---- & sin^ I sin (AT 71^ - C2}dc'4- I 0 cos(9 sin o f sin{kL + ^ — C2fr/v'] + m=3.7.ii.... v ‘2 A'=yi sm ^ ( l.\ —3|' T I [ V — &sin<9 I sin {A I H---- - ^ — Cl^f/c' + cos<9 sin o f sin{k-L------7- ^ — C l }f/z/] + sin k-z' cos (v y. cos^sino f sin{A-£ + — C 2 }(/;/](5.14 A'=v, Sin ^ where m and rn' indicate the segments over which integration is performed, iji is the y-coordinate of the lower limit of integration. (/„ is the y-coordinate of the upper limit of integration. :/ is the z-coordinate of the lower limit of integration. is the z-coordinate of the upper limit of integration, and -\/i = _/'A'{sin^sinotan — + cos^^} (5.15) -\/o = jA-{sin^sino + } (5.16) .1/ 3 = j A ( — sin <9 sin o tan ^ 4 - cos ^} (5.17) = _/A-(sin^sino — ^ } (5.18) Cl = kn( 2.V, — 1 )e sin — (5.19) C-2 = 2.V,Av?e sin — (5.20) anti (t.V-2) \Voy{9.o) = ^E M q [ MryS'm{fcL + k(y'- n'-esin^) — nke\(!i/+ m=m=2.6.10....2.6.io.... Jy'=yi 2 ( I.V-2).V-2) Ey a/6 f A/-, sin {A-1 — A ( (/ + n-’esin ^ ) — nk:e\di/ + r«=i.s.i2.... -'y '-y i (t.V-2)' ■ ■ ry =y« . q A/e / A/3 sin{A /, — A-( y — ir e sin —) — iiA e}i/(/ + ' <' Jy'=yi 2 y i A/o / A4sin{A^-£ + A-(i/' + n ’esin — nk-e}(hl'ô.21) m=2'.6'.I0'....■VIV in' Jy'=yi — where A/, = IÔ.22) and A/e = c o s ^ s i n (-5.23) V 2 Integrating (5.14). (5.21) and rewriting (5.13) ( I.V-.l) \ \ o ( 9 . o ) = Pl(P2 ~ P:}) + Pl(Pô — Po) + m = 1 ..1.0.... ( I.V-1) y i Pt(Ps ~ Po) + PioiPi1 — P 12) + m=.J.7.11.... ( I.V-.;,' y i PdPi.i — Pm ) + Pi(Pi.i — Pie)+ '7l = l'..l'.0'.... (I.V-!)' y i Pt(PI7 — Pis) + Plo(Pl9 — P2 0 ) + 'n=.3'.7'.ir.... (I.V -2) i.v y i 7i<7> - <7i<7.3 4- 9i9i — 9193 + m=2.6.10.... m= I.S.12.... (I.V-2)' (.I.V)' '7i<7_> — Î 1Ç.3 + 51 — 9i 9.3 (5.24) m=2'.6'.10'.... rn= l'.S'.I2'.... The variables in (5.24) are given in Appendix B. They are expressed in ready to calculate simple algebraic equations in terms of known antenna parameters. Eo Component of Field For any antenna of known source distribution, the E^, component of field is given by (4.4). For the MLBT antenna W'oiO.o) = ^ —& coso [ sin{A:£ r ^ — Cl}(li/] + m = i..3 .9 .... V - s i n 7 E -% c o s o J sin{k-L + - C2}dt/] + m =.3.7.11.... V - 2 ( t.V-.3)' E -&COSO J sin{A-I - - Cl }d.i/] + '..3'.9'.... V - yy = E cosoj ' ' sin {AT + ^ - C2}di/] + =3'.7'.II'.... V - ■^y = y 2 ‘ ry'=y,. .. 52 -M; / .\/5 sin(AT 4- A (// — u'-Vsin —) — nke}di/ 4- =2.6.10.... ■^y'=y - '(I.V -2) ry'=yu 52z -My / .1 /3 sin {AT — A ( ;/ 4- n'esin — ) — uAe}di/ 4- m =-I I.S. s 1212.... Jy'-yi 2 ( I.\ —2)' ry'=y„ 52 d/; / .1/3 sin {AT — A-(.i/ — u 'c sin ^ ) — uA 4- m=i'.s'.i2'.... ■’y'=y< - ( IV —2)' ry'=y„ 52 -^^7 / A/3 sin{AT 4- A-(.;/ 4- n'csin ^ ) — nAc}T/.3.25) m=2'.6'.10'.... yy'=yi 78 where A/; = (5.26) v ‘- Integrating and simplifying (5.25) ( I I.V-t I T T ' / /I T ' COS O % ■» COS o W o i f f - O ) = 2_ ;r—TTylPô -pc) + 2 l , irjlPtt - Pi--’ ) + ( l.v-;})' _ (I.V-I)' COS O ^ COS o Z ^(Plô - Pl6) + ^ -2 , - P-21)) + m = l'..5'.9'.... *2 -‘•'2 m=.r.7'.l I'.... ^2 + I (I.V-2) ^ i.v % ^ 91 » % '^7'** I Z, p-— y(92 - %) + 2^ p - ' 2(91 - + m=2.6.10.... ^2 "T" - 6 m=I.S.I2.... ''2 “r (I.V-2)' , (I.V)' ^ pirpW 2 - %) + ;.2 , '.2**?' ~ 93) (5.27) rn=2'.6'.10'.... 2 + -^G m=l'.8'.l2'.... ^2 + '’G The variables in (5.27) are given in .Appendix B. 5.3.3 Results To study the radiation pattern of the MLBT antenna shown in Fig. 5.5. its # and o-polarized power densities are computed considering .V = 4 and a = 20°. Equations (5.24) and (5.27) are used for this purpose. For comparison the same power densities are also computed using NEC. The pattern for the MLBT antenna is shown in Fig. 5.6 with the length of the wire. I„. as a parameter. The results computed using the proposed analytical formulation are plotted using solid lines whereas those computed using NEC are plotted using crosses. It can he seen from Fig. 5.6 that the results are in good agreement. There is some disagreement visible between the analytical and NEC results when L^- = 1.2A. This is probably due to the deviation of the assumed purely sinusoidal current distribution from the actual - a o ,3 0 2 70 I •90 180 Figure 5.6: ^-polarized power density pattern of an MLBT dipole with wire length Lu.- as a parameter (.V = 4 and a = 20°). Solid lines- analytical results, crosses-XEC results. one. Such deviation increases as the length of the wire is increased [Fig. .3.10]. The o polarized pattern of the MLBT antenna is shown in Fig. 5.7 with as a parameter. Note that the results obtained from the analytical formulation is in good agreement with the results from XEC computation when L„. = l.OA. The agreement, however, is not so good for = 1.2A. This may be due to the same reason described above. Fig. 5.7 shows that for constant .V and n. increasing the wire length increases the cross-polarization. Next the cross-polarization of the MLBT antenna is computed at resonance. The geometrical parameters of the antenna are those given in Tables 5.1 and 5.2. For .V = 2. the cross-polarization is computed for a = 15° and a = 45°. The parameters and characteristics for these two cases are given in the 1st and the 5th rows of Table 5.1. respectively. The cross-polarization observed for a = 15° is below -29 dB while 80 -10 -3 0 a j-4 0 -5 0 œ -6 0 -7 0 -8 0 20 60 Thêta (degrees) Figure 5.7: opolarized power density pattern of an MLBT dipole with wire length Lu.- as a parameter ( .V = 4. and o = 20°). Solid lines- analytical results, crosses-NEC results. that for a = 45° Is observed below -48 dB. Thus an increase in a decreases the cross-polarization while .V is fixed. Cross-polarization is also com puted for .V = 4. .Again a = 15° and a = 45° are considered. The parameters and characteristics for these two cases are given in the 1st and the 5th rows of Table 5.2. respectively. The cross-polarization observed for a = 15° is below -35 dB while that for a = 45° is observed below -49 dB. Note that the cross-polarization for ,V = 2. and o = 15° is below -29 dB while that for .V = 4 and a = 15° is below -35 dB. Thus, for a constant a. Increasing .V decreases the cross-polarization. Two representative results are also shown in Fig. 5.8 for .V = 4 and = 32 cm. Solid lines indicate analytical results while the crosses Indicate results obtained from the NEC computation. Note that the Eo power density patterns in both cases SI 3 3 0 0 3 033 300. v60 300, 60 0(dBl -10 P(dB) -30-20 -10 271 90 240 120 240 120 210 150210 150 180 180 06= 15 degrees a =45 degrees e=0.0422k, L=0.335X ; SR=33% e=0.0275k; L=0.20X; SR=60% &es =^48.4ÇI R ^^^=22.20. Figure 5.8: Radiation pattern of an MLBT dipole at resonance. Solid lines- analyt ical results. crosses-NEC results. are similar to that of a straight-wire half-wave dipole. The cross-polarizations in both cases are suppressed below -30 dB. It can also be observed from Fig. 5.8 that for a = 15^ the SR and cross-polarization are 33% and -35 dB. respectively. However, for a = 45° the SR and cross-polarization are 60% and -49 dB. respec tively. Thus, unlike the MLD antenna an increase in the SR does not increase the cross-polarization. The half-power beam widths as observed from the radiation pattern computation are 80 and 86 degrees, for the 1st and 5th row entries in Table 5.1. respectively. Similarly, those for the 1st and 5th row entries in Table 5.2 are 82 and 86 degrees respectively. Thus, the half-power beamwidths for a resonant MLBT antenna lie between 78 and 90 degrees which are the half-power beamwidths of a straight-wire half-wave dipole and an infinitesimal dipole, respectively [16]. 82 4.51 0L=I5 degrees Oi=45 degrees 3.5 CÛ ■o .S ^ ^ 2.5 1.5- 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Antenna length, (A.) Figure 5.9: Gain of an MLBT dipole as a function of antenna length: .V = 4. From the radiated power densities we compute the gain of the MLBT dipole following the principles described in chapter 4. We consider two antennas with the following parameters: (1) a = 15°. e = 2.428 cm. / = 19.26 cm. = 32 cm. and -V = 4. (2) a = 45°. e = 1.373 cm. I = 10.1479 cm. L„. = 32 cm. and .V = 4. The results are shown in Fig. 5.9. wherefrom it can he seen that for both a = 15° and o = 45° the gain steadily increases with the antenna length. However, for a = 15°. the peak gain occurs at an antenna length of 0.8GA whereas for a = 45°. tiie peak gain occurs at an antenna length of 0.68A. 83 5.4 Discussion The study of the input impedance of the MLBT antenna shows that an increase in the SR decreases the Rres- The situation is similar to that of the MLD antenna. However, the Rres obtainable for an MLBT antenna is larger than that for an MLD antenna while they have the same SR. For instance, for an MLBT antenna with SR % 30%. Rres ~ 50 n. whereas for an MLD antenna with SR % 30%. Rres ~ 42 P.. Pattern results show that the resonant MLBT antenna has an Eo pattern similar to the pattern of a straight-wire half-wave dipole. Depending on the SR the half power beam width fluctuates between 80 and 86 degrees. The cross-polarization observed at resonance is below -30 dB. The gain of the antenna steadily increases as the antenna length is increased. The peak gain occurs at a shorter length when a is increased (other parameters are kept constant). The MLBT also has smaller cross-polarization as compared to the MLD antenna. For instance, the cross-polarization for the MLD antenna is below -18.7 dB for a shortening ratio of 62% while that for the MLBT antenna is below -49 dB for a shortening ratio of 60%. 84 C hapter 6 The Dual Meander Antenna The dual meander antenna shown in Fig. 6.1 is similar to the meander zigzag monopole antenna described in [10]. The antenna proposed in [10] was designed to operate within the frequency range of 250-750 MHz. The wide-hand operation of this antenna was made possible by adding parasitic sleeves with it. .A. dual meander antenna was also described in [11] where its \ ’S\\'R frequency characteristic was measured. In light of the above it is apparent that further analysis is needed to provide a complete account of the characteristics of the dual meander antenna. Similarly to the analysis of the meander and MLBT antennas, this also must include the input impedance, radiation fields, and gain of the antenna as function of its parameters. Once again, NEC is used to examine these characteristics [33], The parameters of the dual meander antenna shown in Fig. 6.1 are: e,, e^, Hie number of dual meander sections (.V), ii\ and the antenna length (L). 85 w 1 L T ground plane /777777777T? T Figure 6.1; .A. dual meander antenna on a perfectly conducting infinite ground plane. 6.1 Input Impedance To study the input impedance of the dual meander antenna, first an antenna with L = 6.4 cm. tc = 0.5 cm. .V = 4. a = 1.25 mm. and ci = 0.8 cm is considered. The input impedance of this antenna is computed as a function of frequency with e-i as the parameter. The frequency is varied from 505 to 2578 MHz. The length of the antenna. L is normalized to the wavelength that corresponds to the frequency of computation. Computed input reactance as a function of antenna length is shown in Fig. 6.2 with Co as a parameter. This figure shows that for e-) = 0.8 cm (antenna A), the first, second, and third resonances occur when the lengths of the antenna are 0.175A. 0.31A. and 0.52A. respectively. From Fig. 6.3. the resistances corresponding to these resonances are 25.2 H, 274 Q. 17.3 H. respectively. For co = 1-6 cm (antenna B). the first, second, and third resonances occur when the lengths of the antenna are 0.14A. 0.24A. and 0.39A. respectively. The resistances corresponding to these resonances 86 3 0 0 200 1st 2 nd 1st : 2 nd 3rd -200 — e 2 =0 .8 cm — e 3 = 1 .6 cm -30g^ 0.2 0.3 0.4 0.5 Antenna length, À. Figure 6.2: Computed input reactance of the dual meander antenna as a function of antenna length with eo as a parameter. are 18 Q. 300 Cl. 6.5 Q. respectively. Since the resonant length of the antenna A is 0.175A. its shortening ratio is 30% [Eqn (1.1)]. Similarly, the shortening ratio for the antenna B is 44% and its resonant length is 0.14A. Antenna B has relatively small resistances compared with the antenna A at both 1st and 3rd resonances. This is undesirable because when fed by a 50Q coaxial line antenna B will have larger \'S\\'R than antenna A. This situation is shown in Fig. 6.4 which clearly demonstrates that antenna A has significantly smaller \'S\\'R than antenna B over the entire frequency range. Thus although antenna B has the advantage of larger shortening ratio its \'S\VR frequency response is worse than antenna A. The effect of the number A on the input impedance of the antenna is examined next. The number A' is decreased from 4 to 2 with the antenna length {L) and wire radius (a) kept constant. It is found that the resonant length of the antenna decreses as A' decreases. The resonant lengths of the antenna are 0.175A and 0.15A 4 0 0 — e 2 =0 .8 cm 350 — e 0 =1 .6 cm 300 a 250 m c 200 S I '50 50 0.2 0.3 0.4 0.5 Antenna length, X Figure 6.3: Computed input resistance of the dual meander antenna as a function of antenna length with e-, as a parameter. 14 12 — e 2 —0 . 8 cm — e 0 = 1 . 6 cm 10 oc g > 500 800 1100 1400 1700 2000 2300 Frequency, MHz Figure 6.4: Computed \'SW R frequency response of the dual meander antenna as a function of frequency with C) as a parameter. 88 for .V = 4 and .V = 2. respectively. In other words, a reduction in .V. increases the shortening ratio. However, it is also observed that as .V is reduced from 4 to 2 the resistance at resonance also decreases from 25.2 Q to 18.6 Q. 6.2 Radiation Pattern To compute the radiation fields of the dual meander antenna the following param eters are considered; .V = 4. n = 1.25 mm. ic = 0.5 cm. and L = 6.4 cm. For ci = e-2 = 0.8 cm. the first resonance of the antenna occurs at 820 .MHz. The horizontal-plane Eo pattern at this frequency has been found to be uniform. The elevation-plane Eq pattern is shown in Fig. 6.5. The pattern is similar to that of a straight-wire quarter-wave monopole. The cross-polarization ( Fo-component ) for this antenna is suppressed below -65 dB. For comparison the radiation fields of a meander monopole with p\ = e-t = 0.8 cm. n = 1.25 mm. .V = 4. and L = 6.4 cm is also computed. The cross-polarization for this antenna is approximately -36 dB while it is resonant at 845 MHz. Thus the dual meander is found to be generating significantly less cross-polarization than the meander. .\n increase of e> from 0.8 cm to 1.6 cm shifts the resonant frequency of the antenna from 820 to 650 MHz. The radiation patterns {Eo horizontal and elevation plane) does not change. The cross-polarization is still below -65 dB. The effect of the number of meander sections ( .V ) on the radiation fields is also studied. With L = 6.4 cm. and a = 1.25 mm. reducing the number of meander sections (.V) from 4 to 2 changes the resonant frequency from 820 MHz to 700 MHz. The cross-polarization is below -60 dB. Note that the cross-polarization for .V = 4 is below -65 dB. 89 0 30 30 60 60 90 90 -30 -20 -10 O(dB) Figure 6.5: Eo. E-pIane pattern of a dual meander antenna: e\ = = 0.8 cm. a = 1.25 mm. L = 6.4 cm. and .V = 4. 6.3 Discussion The NEC-based analysis presented above shows that the dual meander antenna is similar to other resonant bent antennas, e.g. the IL.\. MLD. MLBT etc. in terms of its shortening ratio, resonant resistance, and radiation pattern. One distinct advantage of this configuration, however, is its low cross-polarization. The reason for this may be the dual meander geometry' which makes the overall Eo component of field smaller. .-knother distinct advantage is its wide-band impedance characteristic. This can be explained with the help of Fig. 6.6. Comparing the impedance frequency re sponse of the meander monopole and the dual meander monopole it is clear that the variations of the impedance are much smaller for the dual meander antenna than the meander monopole. This characteristic clearly resembles the comparison of the input impedance characteristics of a thin and a thick dipole, as the overall magnitude variation of the input resistance or the input reactance of a thick dipole with frequency is lower than a thin dipole. Based on this observation we can predict that the dual meander would have wider frequency bandwidth than the meander 90 500- Resistance Meander — Reactance 350 E 200 T3 CL -100 Dual meander -250 -400 500 800 1100 1400 1700 2000 2300 Frequency (MHz) Figure 6.6: Comparing the input impedances of a meander and a dual meander antenna. For both antennas L = 6.4 cm. a = 1.25 mm. .V = 4. c, = co = 0.8 cm, and for the dual meander alone «• = 0.5 cm.. 91 Meander CE CO > Dual meander 0.5 0.8 1.1 1.4 1.7 2 2.3 Frequency (GHz) Figure 6.7: Computed \'S\\'R frequency response of a dual meander and a meander, monopole. In Fig. 6.7. the \'S\VR of the meander and the dual meander are plotted verszis frecpiency. .A 50 Q coaxial line is selected as a feed and the input impedance data plotted in Fig. 6.6 were used to calculate the \'S \\’R. It is evident from Fig. 6.7 that the dual meander operates from 0.5 to 2.5 GHz for a \'S\VR< 6 whereas the meander operates over the same frequency range for a \'S\\’R< 12.5. 92 C hapter 7 The Printed Meander Antenna In chapters 3 to 6 a number of self-resonant bent antennas are analyzed. This includes the inverted-L antenna, the meander-line dipole antenna, the meander-line bow-tie antenna, and the dual meander antenna, each of which can be manufactured by bending a wire or metal ribbon. These antennas can also be printed on dielectric substrates, to create more practical, planar configurations. The printed antennas have the advantages of low-profile, small size, and increased directivity (when a ground metallization is present ). The presence of a dielectric material and a metallic ground plane, however, affects the characteristics of a printed antenna [34]-[46]. To examine this effect, accurate characterization of such antennas is necessan.'. .Analysis of printed dipole antennas using the Method of Moments (MoM) is well documented in the literature [34]-[43]. The analysis is based on the MoM solution of Pocklington’s integral equation. Such a solution gives the current distribution on the antenna, and once the current distribution is known the radiation pattern and the input impedance can be computed. 93 Research on printed self-resonant bent antennas is relatively new. .A. printed meander and a printed zigzag antenna are analyzed in [22] and [23]. In both cases the MoM is used and a bent wire is considered printed on a grounded dielectric substrate of infinite extent. .A. study of the input impedance of a meander dipole as a function of antenna length is presented in [22] with the dielectric constant, as a parameter. Computed Eo and radiation patterns are also presented in [22]. The input impedance and radiation pattern of a printed zigzag antenna are presented in [23]. The analyses presented in [22] and [23] are of introductory nature, and hence, do not contain many vital design information, such as resonant resistance as a function of dielectric constant and substrate thickness, radiation pattern as a function of substrate thickness, and directivity as a function of dielectric constant and substrate thickness. .Also antennas printed on dielectric half-spaces are not treated in [22] and [23]. Therefore, an attempt is made to analyze planar printed meander antennas from a design point of view. Some preliminary' results of this study can be found in 147). [48]. We decide to use the finite-difference time-domain (FDTD) technique [49]-[51] because of its versatility and efficiency in modeling a complex heterogeneous geom etry. .A computer program developed in our laboratory is used for the above pur pose. The method proposed by Yee [50] and its current improvements as described in detail in [51] have been the basis of our code. In addition techniques have been developed and implemented to facilitate accurate modeling of three-dimensional ob jects of complex shapes. Berenger's perfectly matched layers (PML) [52] are used as absorbing boundary to ensure reflections below at least 40 dB from the termination of the computational space. There are options to use other absorbing boundary 94 conditions in the code as well, nevertheless, a properly selected PML results in su perior performance at little additional computational cost. The code also provides the means to observe quantities, such as electric and magnetic fields, voltages and currents, reflection coefficients, radiated power etc. The accuracy of the code has been tested and verified previously for a wide range of electromagnetic problems [47]-[48]. [53]-[5o]. 7.1 Antenna Configuration The geometry of the printed meander antenna is shown in Fig. 7.1. A ground met allization under the substrate material is assumed to be present unless otherwise specified. The parameters of the antenna are: antenna length= 21. meander seg ment length= e. number of nieanders= .V. strip width= u\ substrate length= substrate width= Wg. substrate thickness= and dielectric constant= (r- In the FDTD computation scheme the antenna is excited at the gap shown in Fig. 7.1 using gap excitation'. 7.2 FDTD Modeling .A.S mentioned the printed meander antenna of Fig. 7.1 is excited at the gap where two spikes are used for spatial excitation and a frequency shifted Gaussian pulse is used for time excitation. To compute the input impedance of this antenna we compute the voltage across the gap and current at the feed point. To obtain the voltage, the time series for the voltage is computed by multiplying the relevant electric field with the gap size. 95 , y £r Ground metallization 6 Figure 7.1: A printed meander antenna. The time series is then Fourier transformed to give the voltage as a function of frequency. To obtain the current at the feed-point. a time series for the current is computed integrating the relevant magnetic fields along a loop centering the feed- point. The time series for the current is then Fourier transformed to give the current as a function of frequency. Finally, the input impedance is computed by dividing the voltage by the current. To compute the radiation pattern we compute the near-zone fields inside the computational region at the specific frec^uency and then use near to far-zone transformations. 96 Ground plane Gap source Figure 7.2: A printed dipole antenna. 7.3 Results -A.S a simple test, first the input impedance of a printed dipole is computed. The antenna modeled is shown in Fig. 7.2. The folowing typical parameters are consid ered: fr =2.1. 21 = 44 mm, u- = 4 mm. = 244 mm. and = 244 mm. Input impedance of the printed dipole is then computed as a function of frequency with the substrate thickness, as a parameter. FDTD simulations are performed for = S. 16. 24. and 32 mm. For each simulation the input impedance is plotted as a function of frequency. The resonant frequencies determined are 2.37. 2.04. 2.05. and 2.26 GHz. for substrate thicknesses (ri, ) of 8. 16. 24. and 32 mm. respectively. The resonant resistance for the above are 9. 27.5. 58.75. and 88.1 fl. respectively. Next, we express G in terms of the wavelength (A) that corresponds to each of Table 7.1: Resonant resistance, Rres of a printed dipole antenna as a function of substrate thickness: comparison with the results in [34]. Parameters of the antenna are: (r = 2.1. 21 = 44 mm. u’ = 4 m m . = 244 mm. and = 244 mm. ts. A R r e s - D [this work] R res- n [34] 0.0632 9 10 0.1088 27.5 30 0.1640 58.75 60 0.2411 88.1 85 the above resonant frequencies and present our results in Table 7.1. For comparison, we also present similar results obtained from [34]. In [34] the resonant resistance of a printed dipole antenna on a grounded dielectric substrate is given as a function of the substrate thickness, The method of analysis employed is the MoM. In addition the substrate in [34] is of infinite extent and the dielectric constant is 2.1. From Table 7.1 it can be seen that the resonant resistance computed using the FDTD are in good agreement with that obtained from [34]. The small disagreement that can be observed between the results in the 2nd and the 3rd column in Table 7.1 may probably be due to the finite size of the substrate used in our modeling. 7.3.1 Antenna on a Grounded Dielectric Substrate The input impedance and the radiation characteristics of a printed meander dipole are computed. The antenna is printed on a grounded dielectric substrate éis shown in Fig. 7.1. Input Impedance First, the input impedances of a meander dipole in air (antenna .\). and a meander dipole printed on a grounded dielectric substrate (antenna B) are computed as a 98 function of frequency. The parameters of the antenna A are: .V = 2. e = 12 mm. a- = 4 m m . and 21 = 44 mm. whereas, that of the antenna B are: .V = 2. e = 12 mm. 21 = 44 mm. w = A mm. = 244 mm. = 244 mm. = 20 mm. and f.r = 2.1 (P T F E glass). The input impedance for antennas .A. and B vs frequency is shown in Fig. 7.3. .A.ntenna .A. resonates at 2.18 GHz and has a resonant resistance of 50 Q. while antenna B resonates at 1.618 GHz and has a resonant resistance of 18 Q. The shortening ratios for the two antennas are 36 and 52%. respectively. .A.lthough the printed antenna has a larger shortening ratio, its resonant resistance is too small, if it is to he matched to a practical transmission line of characteristic impedance of 50 Q. Next, the input impedances of printed meander antennas on grounded dielectric substrates are computed as function of frequency. Substrates with of 1.0 (air). 2.1 (PTFE glass), and 3.78 (fused quartz) are considered. Other parameters of the antenna are .V = 2. e = 12 m m . 21 = 44 mm. iv = 4 m m . = 244 mm. = 244 mm. 0 = 20 mm. The results of the computations are shown in Fig. 7.4. The resonant frequencies of these antennas are 2.077. 1.618. and 1.258 GHz. while the resonant resistances are 22. 18. and 14 Q for f.r = 1.0. 2.1. and 3.78. respectively. Once again the resonant resistances for the above are small compared to 50 fl it has been described in [34] that the Rres of a printed dipole antenna on a grounded dielectric substrate strongly depends on the dielectric constant, and the substrate thickness, t^- It has also been shown in [34] that the Rres of a printed dipole has a unique functional realtionship with while is constant. Parametric curves are presented in [34] which depict the dependence of the Rres on 0 with as the parameter. 99 200 Printed :-200 X ' Antenna in air 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 Frequency (GHz) 100 60 Printed U 6 0 - 40 Antenna in air 0.5 0.7 0.9 .5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 Frequency (GHz) Figure 7.3: The input impedances of a meander dipole in air and a meander dipole printed on a grounded dielectric substrate. The parameters of the antenna in air are: .V = 2, e = 12 mm, 2/ = 44 mm. and w = 4 mm. whereas that of the printed are: e = 12 mm. 2/ = 44 mm. w = 4 mm. = 244 mm. = 244 mm. = 20 mm. 6r = 2.1. 100 e =i.7S 0.7 0.9 3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 Frequency (GHz) 100 so e =3,78 = 6 0 ' E =1.0 0.5 0.7 0.9 1.5 1.7 2-1 2 .3 2 .5 2 .7 2.93 Frequency (GHz) Figure 7.4; The input impedance of a printed meander antenna as a function of frequency with Cr as a parameter. Other parameters are: .V = 2. e = 12 mm. 21 = 44 mm. ic = 4 m m . = 244 mm. Wg = 244 mm. t, = 20 mm. 1 0 0 100 s -50 o -100 -150 -200 1.2 I 1.4 1.5 16 Frequency (GHz) 30 1.4 1.5 1.6 Frequency (GHz) Figure 7.5: The input impedance of a printed meander antenna as a function of frequency with ri as a parameter. Other parameters .V = 2. e = 12 mm. 2/ = 44 mm. ir = 4 m m . Lg = 244 mm. = 244 mm. €r = 2.1. To examine the effect of the substrate thickness, ri on the Rres of a printed meander antenna, we study its impedance as a function of frequency with 0 as a parameter. We consider an antenna with the following parameters: .V = 2. c = 12 mm. 21 = 44 m m . iv = 4 mm. = 244 mm. = 244 mm. = 2.1. We compute the input impedance of this antenna as a function of frequency with = 8. 20. 30. 40. 50. and 60 mm. The variation of the input impedance as a function of frequency with (<=20. and 40 mm is shown in Fig. 7.5. Increasing (< from 20 to 40 mm changes the resonant frequency from 1.618 to 1.73 GHz and the resonant resistance from 18 to 62 Cl. respectively. The increase in the resonant resistance is substantial. Now. for each (<. we tabulate the resonant frequency and resonant resistance from our computed input impedance data. We express (< in terms of wavelength corresponding to the resonant frequency of the antenna and plot the resonant resis- 1 0 1 SR 50 res Polynomial approx. 0.05 0.15 0.2 0.25 0.3 0.350.1 Substrate thickness, (X) Figure 7.6: Resonant resistance. Rres (O) of a printed meander dipole eerst/.s sub strate thickness, Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. tc = 4 mm. = 244 mm. = 244 m m . Cr = 2.1. tance as a function of t,. Since we do not have an adequate number of data points to plot a smooth curve, we perform a polynomial approximation using two Matlab routines called polyfit and polyval [56]. The result is shown in Fig. 7.6. We also plot the shortening ratio, SR for this antenna as defined by (1.1). .A.ccording to Fig. 7.6. the SR fluctuates between 52.6 and 45.7'/f while varies from 0.04 to 0.36A. On the other hand, the Rres- first increases as increases, and then it decreases as 0 increases. The Rres starts from a low value of 3 Q and keeps increasing up to 62 O while t, increases from 0.04A to 0.24A. The Rres then decreases to 30 f> as t, increases from 0.24A to 0.36A. It is evident that the Rres has a much stronger dependence on than the SR. From Fig. 7.6 it is clear that for the printed meander antenna under consider ation. a thin substrate is not advantageous since it results in a small Rres- As an 1 0 2 SR (%) res Polynomial approx. 0.08 0.12 0.16 0.2 0.24 0.28 Substrate thickness, tg (A.) Figure 7.7; Resonant resistance. Rres (D) of a printed meander dipole versus sub strate thickness, t,,. Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. u- = 4 mm. f ^ = 244 mm. = 244 mm. Cr = 4.0. example, for Rres > 40 H. we have to have 0.16A < < 0.30A. If we want to avoid the polynomial approximation in Fig. 7.6 we would need more data points which would call for more simulations with in between S. 20. 30. 40. 50, and 60 mm. One such simulation takes considerable amount of CPU time. To give an idea, the CPU time taken for the simulations for C, = S. 20. and 30 mm are 734.7. 325.6. 345.8 minutes, respectively on an HP9000/735 workstation. Since a functional relationship can be easily seen once the SR and Rres are plotted as function of U. polynomial approximations of these data points are considered appropriate rather than performing more simulations. Similar results like the one shown in Fig. 7.6 are also obtainerl for fr = 4.0 (quartz). The parameters of the printed meander antenna are the same as before. The Rres and the SR for (r = 4.0 are plotted as function of U in Fig. 7.7. wherefrom 103 270 180 \ X / 0 -30 -20 -10 O(dB) 90 Figure 7.8: xy-plane pattern of a printed meander antenna. Parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. ir = 4 mm. = 244 mm. = 244 mm. = 20 mm. and fr = 1.0. it can he seen that in order to obtain Rres > 400. we have to have 0.125A < < 0.20A. Note that the substrate thickness required for quartz = 4.0) is less than that required for PTFE (fr = 2.1) when the objective is Rrr^ > 400. Also, the SR obtainable for quartz is close to 65%. which was previously .50/( for PTFE glass. Radiation Pattern, and Gain The radiation pattern of the printed meander dipole shown in Fig. 7.1 is computed. First, an antenna with .V = 2. e = 12 mm. 21 = 44 mm. iv = 4 mm. L., = 244 mm. u\s = 244 mm. = 20 mm. and €r = TO (air) is considered. It may be recalled from the input impedance data of this antenna shown in Fig. 7.3 that its resonant frequency and resonant resistance are 2.18 GHz and 50 Q. respectively. The xy- plane pattern at resonance is shown in Fig. 7.8. The pattern shows that the back lobe is suppressed below -30 dB and the half-power beam-width is 80°. The gain of 104 90 180 0 -3 0 -20 -10 G(dB) 9 0 Figure 7.9: yr-plane pattern of a printed meander antenna. Parameters of the antenna are .V = 2. e = 12 m m . 21 = 44 mm. «.• = 4 mm. = 244 mm. u = 244 mm. f., = 20 mm. and = 1.0. the antenna is 8.8 dBi. The yc-plane pattern is shown in Fig. 7.9. It is apparent that the back radiation is small (-28 dB or less). The £"o-component is below -28 (IB. Next, the radiation pattern of a printed meander dipole with = 2.1 is com puted. From the results shown in Fig. 7.6 we select = 0.16A so that the is about 40 Q. Other parameters of the antenna are: .V = 2. c = 12 mm. 21 = 44 mm. tc = 4 mm. £* = 372 mm. and Ws = 372 mm. The radiation pattern is com puted at a frequency of 1.618 GHz which is the resonant frequency of the antenna. Computed .r//-plane pattern is shown in Fig. 7.10. The front to back (F/B) ratio is about 15 (IB. The gain of the antenna is 7.0 dBi. The yr-plane pattern is shown in Fig. 7.11. The cross-polarization is suppressed below -30 dB. The tails in the back of the pattern are most probably due to the finite size of the ground plane and the presence of a dielectric material. 105 90 O(dB) 120 60 -10 150 -20 30 -30 180 210 330 240 300 270 Figure 7.10: x.y-plane pattern of a printed meander antenna at resonance (1.618 GHz). Parameters of the antenna are: .V = 2. e = 12 mm. 21 = 44 mm. ir = 4 m m . = 372 mm. = 372 m m . = 30 mm. and = 2.1. We have seen that for the printed meander dipole with fr = 2.1 the resonant resistance is approx. 50 Q when the thickness of the substrate is about 0.23A. The radiation pattern for this antenna is also computed. The parameters of the antenna are the same as the previous one except that = 0.23A. The resonant frequency of the antenna is 1.73 GHz. Computed .ri/ and yc-plane patterns at resonance are shown in Figs. 7.12 and 7.13. respectively. Fig. 7.12 shows that the main beam is no longer in one direction as was in Fig. 7.10. The pattern shown in Fig. 7.12 is broader than the one shown in Fig. 7.10. The main beams are in the directions of o = 240 and 300 degrees and the gain of the antenna is 6.0 dBi. The i/r-plane pattern [Fig. 7.13] shows that the cross-polarization has slightly increased as compared to Fig. 7.11 and that the direction of maximum radiation is no longer along o = 270°. 106 O(dB) 120. 60 -10 150 -20 30 -30 180 210 330 240 300 270 Figure 7.11: .i/c-plane pattern of a printed meander antenna at resonance ( 1.618 GHz). Parameters of the antenna are: .V = 2. e = 12 mm. 2/ = 44 mm, w = 4 mm. I., = .372 mm. = 372 mm. = 30 mm. and = 2.1. 90 O(dB) 120 60 ■10 150.QO 30 -30 180 210 330 240 300 270 Figure 7.12: x.y-plane pattern of a printed meander antenna at resonance (1.73 GHz). Parameters of the antenna are .V = 2. e = 12 mm. 21 — 44 mm. iv = 4 mm. Lg = 372 mm. = 372 mm. f., = 40 mm. and = 2.1. lOi 90 O(dB) 120 60 -10 150. -20 .30 180 210 330 240 300 270 Figure 7.13: /yc-plane pattern of a printed meander antenna antenna at resonance (1.73 GHz). Parameters of the antenna are .V = 2, e = 12 mm. 21 = 44 mm. tc = 4 mm. Ls = 372 m m . ii-g = 372 mm. G = 40 mm. and €r = 2.1. 7.3.2 Antenna on a Dielectric Half-Space We use the term dielectric half-space to indicate that there is no metallic ground plane beneath the substrate material [34]. This configuration may be of interest when an omnidirectional radiation pattern is desired. The dielectric under the meander antenna can increase the SR. Input Impedance We have seen that important design information for a printed meander antenna can be found from a study of its Rres and SR both as function of G [Figs. 7.6 and 7.7]. For the antenna on a dielectric half-space we conduct similar investigation. We consider an antenna with the following constant parameters: .V = 2. e = 12 mm. 21 = 44 mm. «,• = 4 mm. Ls = 244 mm. = 244 mm. Cr = 2.1. We compute 108 SR tL 44 C % 42 Polynomial approx. 1 40 res 1136“ 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Substrate thickness, r^ (A.) Figure 7.14: Resonant resistance. Rreg (O) of a printed meander dipole on a dielec tric half-space versus substrate thickness, Other parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. tc = 4 mm. = 244 mm. tc, = 244 mm. (r = 2.1. the input impedance of this antenna as a function of frequency with = 0. 4. 10. 20. 30. 40. 50. and 60 mm. .A. zero substrate thickness indicates that there is no dielectric material present. Similarly as before, we plot both the Rres and SR as function of f, in Fig. 7.14 from which it can be seen that the SR sharply increases from 36 to 47'/ as t, increases from 0 to 0.06A. The increase in the SR is less sharper (from 47 to 40.5%) when ts increases from 0.06 to 0.36A. The Rres on the other hand, decreases from 46 to 34 Q as t, increases from 0 to 0.24A. It increases again from 34 to 38 Q as f, increases from 0.24 to 0.36A. Radiation Pattern and Gain For radiation pattern computation we consider the same antenna as above with = 20 mm. We know from our input impedance data that the resonant frequency 109 90 CKdB) 120 150, -20 30 -30 180 210 330 240 300 270 Figure 7.15: z«/-plane pattern of a printed meander dipole on a dielectric half-space. Parameters of the antenna are .V = 2, c = 12 mm. 21 = 44 m m . u- = 4 mm. = 372 mm. - 372 mm. = 20 mm. and = 2.1. of this antenna is 1.77 GHz which makes G = 0.12A. We may observe from Fig. 7.14 that the SR and Rres for this antenna are 48% and 36.5 Q. respectively. Computed T//-plane pattern for this antenna at resonance is shown in Fig. 7.15. It can be seen that although there is no metallic ground plane present, the pattern is not omnidirectional. The main beams are in the directions of o = 30 and 150 degrees. The gain of this antenna is 3.9 dBi. The presence of a directional pattern is probably due to the thickness of the substrate. The ,(/:-plane pattern is shown in Fig. 7.16. The fg-pattern is broad with no nulls. The pattern shows that the cross-polarization is below -20 dB. The radiation pattern for this antenna is also computed for = 4 mm. Since the resonant frequency for this case is 1.907 GHz. this gives us t, = 0.04A. It may be recalled that the Rres and SR for this situation are 39.5 Q and 44%. respectively. Computed .ri/-plane pattern for the antenna is shown in Fig. 7.17. The pattern is 1 1 0 90 D(dB) 120 60 -20 150 30 -40 !✓ 180 210 330 240 300 270 Figure 7.16; yr-plane pattern of a printed meander dipole on a dielectric half-space. Parameters of the antenna are .V = 2. c = 12 mm, 21 = 44 mm. ir = 4 mm. Is = 372 mm. = 372 mm. = 20 mm. and = 2.1. 90 120 60 O(dB) 150 -10 -20 -30 180 210 330 240 300 270 Figure 7.17: .r//-plane pattern of a printed meander dipole on a dielectric half-space. Parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. w = 4 mm. Is = 372 mm. Ws = 372 mm. ^ = 4 mm. and = 2.1. I l l O(dB) 120 6 0 -10 150 30 180 210 330 240 300 270 Figure 7.18: yz-plane pattern of a printed meander dipole on a dielectric lialf-space. Parameters of the antenna are .V = 2. e = 12 mm. 21 = 44 mm. a- = 4 m m . Ls = 372 mm. = 372 mm. t* = 4 mm. and €r = 2.1. nearly uniform. Also computed yr-plane pattern is shown in Fig. 7.18 from which it can he seen that the cross-polarization is below -22 dB. The gain of the antenna is 2.5 dBi. 7.4 Discussion .A. study of the printed meander antenna is conducted. Two cases are considered: (1) antenna printed on a grounded dielectric sushtrate. and (2) antenna printed on a dielectric half-space (no ground metallization present). The results of the analysis of the printed meander dipole on a grounded dielectric substrate provides some vital design information, such as: • the dependence of the resonant resistance. Rres and the shortening ratio. SR 1 1 2 on the substrate thickness, while the dielectric constant, is constant • the dependence of the radiation pattern and the gain on the substrate thickness (fj) while the dielectric constant, is constant Graphs are presented which depict the dependence of the Rres and the S R on G of a printed meander antenna with = 2.1. It is observed that the SR obtainable is close to approximately 50% and that the SR is not a strongly varying function of t,. In contrast, the Rres is a strongly varying function of It is also observed that a thin substrate gives a small resonant resistance (less than 3 (1 for < 0.04A). In a situation where Rres > 40 Q is required, the substrate thickness, should be w ithin 0.16A and 0.30A. Similar results are also presented for = 4.0. The SR obtainable for €r = 4 0 is approximately 60%. Once again, the SR is not a strongly varying function of t,. The dependence of the Rres on 0 shows that in a situation where Rres > 40 0 is reciuired. the substrate thickness. 0 should be within 0.125A and 0.20A. Thus, a relatively thin substrate can be used when a larger is utilized. Radiation pattern and gain results for a printed meander antenna with = 2.1 show that both of the above depend on Results for 0 = 0.16A show that the gain of the antenna is 7.0 dBi and the front to back (F/B) ratio is about 15 dB. Note that the resonant resistance for this case is 40 O. Radiation patterns for = 0.23A show that there is no longer a single main beam in the pattern as was the case for tg = 0.16A. The gain for this case is 6.0 dBi. Note that the resonant resistance of the antenna is 50 Q. Thus, although increasing substrate thickness from 0.16A to 0.23A results in an increase in the resonant resistance from 40 Q to 50 Q. the radiation pattern and gain of the antenna deteriorates. 113 Similarly, the result of the study of the printed meander dipole on a dielectric half-space also gives some important design information, such as • the relationships of the Rres and the SR with • the dependence of the radiation pattern and the gain on ts Graphs are presented which show the dependence of the Rres and the SR on ts for a printed meander antenna on a dielectric half-space with Cp = 2.1. It is observed th at the SR sharply rises and the Rres sharply decreases as G is increased. This relationship holds for G value of up to 0.06A. The SR varies less strongly with ts for ts > 0.06A. Since by increasing ts over 0.06A does not improve the SR significantly. ts < 0.06A is a good choice. Results of radiation pattern and gain computation for Cp = 2.1 and t, = 0.12A are presented. It is observed that the T(/-plane pattern of the antenna is not om nidirectional. This is why the antenna has a gain of 3.9 dBi. Note that the gain of a half-wave dipole with an omnidirectional pattern is 2.1 dBi [16]. However, for the printed meander antenna on a dielectric half-space, an omnidirectional pattern results when G = 0.04A is utilized. The gain for this case is 2.5 dBi. Thus, if an omnidirectional radiation pattern is required, the substrate thickness should be sm aller than 0.04A. 114 C hapter 8 Applications Potential applications of self-resonant bent antennas are discussed. A wide-hand dual meander-sleeve antenna is described. The antenna can be used as a vehicular antenna operating simultaneously in both bands (824-894 MHz and 1850-1990 MHz) of the personal communication services (PCS). Novel plane sheet reflector antennas are described for application in base stations. The novelty of these antennas lies in their feed design. MLBT dipoles and monopoles are used as feeds of plane sheet re flectors instead of conventional half-wave dipoles. This helps in reducing the overall sizes of such reflectors without deteriorating their performance. 115 8.1 Dual meander-Sleeve Antenna 8.1.1 Introduction Recently, we have proposed a wide-hand dual meander-sleeve antenna [57]-[33] for application as a vehicular antenna in the personal communication ser\ices (PCS). The PCS system in North America relies on the microcellular concept of communica tion in order to ensure efficient utilization of the frequency spectrum. The frequency hands allocated for the PCS systems in North America are: 824-894 MHz and 1850- 1990 MHz [58]. The microcellular concept involves radio paths ranging from 200 m to 1000 m and hase station antennas of ahout the height of a lamp post with low transmitted power (of the order of tens of milliwatts). In the PCS system vehicle mounted or hand held radio receivers used hy suhscrihers establish communication with other suhscrihers via the hase station unit. Depending on the subsystem, the requirements for antennas are different. For example, a base station antenna may require a gain of 7 to 15 dB d'. while lower gain is sufficient for a handset. -\ntenna characteristics required for vehicular applications are: (i) bandwidth of 8.2% in the 824-894 MHz hand and 7.3% in the 1850-1990 MHz hand for \'S \\'R < 2. (ii) omnidirectional radiation pattern, (iii) 3 dB gain with reference to a half-wave dipole antenna (gain required in the .lapanese system (NTT) is 0 dBd). and (iv) vertical polarization [2]. \ehicular antennas currently in use are mainly of the whip or sleeve type. These antennas are mounted either on the tnmk-lid or roof-top of a car. Since there are two different frequency hands in the PCS system, suhscrihers who MBd - dB with reference to a half-wave dipole 116 travel over serv ice areas employing different frequency bands, may need two separate antennas unless a dual-frequency antenna is used. A dual-frequency trunk-lid-type antenna described by [-59] was mentioned in [2] where by adding parasitic elements to the antenna its dual-frequency operation was made possible. This antenna operated in the 900 and 1500 MHz bands. Kagoshima et al. [60] proposed a dual frequency low profile printed dipole yagi-array for a cabin mounted vehicular antenna. This antenna could be attached at the back of a rear view mirror within a vehicle interior. The dimensions of the antenna were approximately 90 mm in width. 70 mm in depth, and 60 mm in height. This antenna also operated in the 900 and 1500 MHz bands. In the following we describe a novel dual-frequency vehicidar antenna which is considerably shorter than the antennas presented in [2] and [60]. 8.1.2 Design Considerations Inspired by the wide-band characteristic of the dual meander antenna [see for in stance. Figs. 6.6 and 6.7]. we decide to use it for the above application. We recall that the parameters of the dual meander antenna are: c,. e,. u\ L. and a [Fig. 6.1]. From the analysis presented in chapter 6 we select c, = e> = 0.8 cm. w = 0.5 cm. L = 6.4 cm. and a = 1.25 mm to make the antenna resonate within the 824-894 MHz band. ir 0.5 0.8 — Sleeve Sleeve 0.8 Ground plane Figure 8.1: .A. dual meander-sleeve antenna. To make dual-frequency operation possible, we decide to use a sleeve. The reason for selecting the sleeve configuration is its broadband characteristic. Such a characteristic is a result of the presence of a sleeve that provides smooth impedance- freciuency response [61]. .Another advantage of the sleeve antenna is its capability to operate over a frequency range of 4 : 1 with little change in the radiation pattern. Considering the open-sleeve dipole configuration described by King and Wong [62] we decide to use two posts as a sleeve. The geometn.' of the antenna with the sleeve is shown in Fig. 8.1. Now we design the sleeve such that the antenna also resonates within the 1850- 1990 MHz band. To determine the sleeve parameters. I and 25. we compute the input impedance of the antenna using the Numerical Electromagnetic Code (NEC) [1 1 - lis — 1=2.4 cm - - -1=2.8 cm 1=3.2 c 900 1100 1300 1500 1700 1900 2100 Frequency, MHz 200 1=2.4 cm =2.8 cm - 1=3.2 cm TOO 900 1100 1300 1500 1700 1900 2100 Frequency, MHz Figure 8.2: Computed input impedance of the dual meander-sleeve antenna vs fre quency with the sleeve length I as a parameter; sleeve spacing 2S = 3.2 cm. 8.1.3 Numerical Results Initially. 25 is kept constant (3.2 cm) while I is varied. The results are shown in Fig. 8.2. from which it can be seen that for I = 3.2 cm. the 3rd resonance of the antenna occurs at 1730 MHz. For I = 2.8 cm. the 3rd resonance occurs at 1970 MHz. and for / = 2.4 cm it occurs at 2177 MHz. Based on Fig. 8.2 we consider the antenna to have its 3rd resonance around 1930 MHz. One important reason for selecting the 3rd resonance for the operation of the antenna is because its resistance at this resonance is close to 30 Q [Fig. 8.2]. It is also worth noting that the resistance varies little over a wide frequency range around this resonance, resulting in low \'S \\’R. In the next step. / is varied while 25 is kept constant in order to obtain the 3rd resonance around 1930 MHz. For three different sleeve spacings (25) the input impedance of the dual meander-sleeve antenna is computed over the frequency range of 700-2100 MHz. The result is shown in Fig. 8.3. wherefrom. it is apparent that 119 200 2S=3.2 cm; 1=2.9 cm 150 — 2S=2.8 cm; 1=2.7 cm - 2S=3.6 cm; 1=3.0 cm c 100 50 - 1000 1200 1400800 1600 1800 2000 Frequency, MHz 100 2S=3.2 cm; 1=2.9 cm 50 — 2S=2.8 cm; 1=2.7 cm - 2S=3.6 cm; 1=3.0 cm -50 -100 800 1000 1200 1400 1600 1800 2000 Frequency. MHz Figure 8.3: Computed input impedance of the dual meander-sleeve antenna vs fre quency for 3rd resonance around 1930 MHz. 120 wdth 2S = 2.8 cm and I = 2.7 cm. the resistance around the 3rd resonance is too small compared to 50 Q. This would result in a larger compared to the other two cases (25 = 3.6 cm and I = 3.0 cm or 25 = 3.2 cm and I = 2.9 cm) since for the other two cases the resistances around the 3rd resonance are closer to 50 Cl. Thus, either the arrangement of 25 = 3.6 cm and I = 3.0 cm or the arrangement of 25 = 3.2 cm and I = 2.9 cm can be used. To limit the size of the antenna we select 25 = 3.2 cm. 8.1.4 Experimental Procedure A dual meander antenna with the previously mentioned parameters is then manufac tured and tested. Based on the impedance frequency response of Figs. 8.2 and 8.3. experiments are conducted by connecting two posts (length 4.0 cm) to the ground plane at a distance of 1.6 cm from the axis of the antenna. The \'S\\'R-frequency response is measured after successively trimmimg the posts. To measure the \'SW R frequency response experiments are conducted using an HP 8720C vector network analyzer (\'NA). Each antenna is soldered to the center pin of an N type female connector. The outer conductor of the connector is connected to a ground plane of dimension 90 cm x90 cm. The \"N.A. is calibrated using standard short, open and 50 n load. The radiation pattern, and gain are also measured using the HP 8720C vector network analyzer. .A. large ground plane of dimension 122 cm x 120 cm is used for this purpose. For the H-plane pattern, a straight-wire quarter-wave monopole is used as a transmitting antenna which is connected to port 1 of the \'NA. The receiving dual meander-sleeve antenna is connected to port 2. The receiving antenna 121 is rotated 360° and the S-parameters are recorded in a computer connected to the \'NA by the GPIB bus. To measure the E-plane pattern a straight-wire dipole is used as a transmitting antenna. The dipole is connected to a semi-circular structure that is rotated 90° and the S-parameters recorded in a computer. .A.ntenna gain is measured by using the gain comparison method where it is mea sured by comparing the received power of the test antenna with that of a standard antenna gain of which is known. .A. straight quarter-wave monopole antenna with a gain of 5.1 dBi’- [30] is used as the standard gain antenna. 8.1.5 Experimental Results Measured \'S\VR frequency response is shown in Fig. 8.4. .According to Fig. 8.4. the dual meander-sleeve antenna can operate in both PCS bands when the sleeve length is about 2.9 cm. Measured H-plane radiation patterns are shown in Fig. 8.5 and Fig. 8.6. The H-plane pattern at 1890 MHz [Fig. 8.6] shows some non-uniformity, wiiich is less than 1.5 dB. The E-plane patterns shown in Fig. 8.7 and 7.8 are very close to the pattern of a straight-wire quarter-wave monopole. For operation in the lower PCS band (824-894 MHz), the sleeve is not needed. The sleeve (I = 2.9 cm and 25 = 3.2 cm) gives a 3rd resonance around 1930 MHz which falls in the upper PCS band (1850-1990 MHz) [Figs. 8.3 and 8.4]. The sleeve also provides a smooth impedance frequency response that results in a relatively low \'SW R throughout the frequency range of 700-2100 MHz. The dual meander-sleeve antenna has negligible cross polarization in both the *ciBi - dB with reference to an isotropic radiator 122 DC I > 2.3 2.9 3.9 900 1100 1300 1500 1700 1900 2100 Frequency, MHz Figure 8.4: Measured \ ’S\VR frequency characteristics of the dual meander-sleeve antenna with sleeve length I (cm) as the parameter. 90 'X'\ 1.0 \0.75 ^ 0.5 ,0.25 180 I 270 Figure 8.5: Measured H-plane pattern of the dual meander-sleeve antenna at 890 MHz. Scale: linear. 123 90 180 0.25 270 Figure 8.6: Measured H-plane pattern of the dual meander-sleeve antenna at 1890 MHz. Scale: linear. 60 90 90 0 0.25 0.5 0.75 1 0 Figure 8.7: Measured E-plane pattern of the dual meander-sleeve antenna at 890 MHz. Scale: linear. 124 60 9 0 L — ______— 9 0 0 0.25 0.5 0.75 '0 Figure 8.8: Measured E-plane pattern of the dual meander-sleeve antenna at 1890 MHz. Scale: linear. 824-894 MHz and 1850-1990 MHz bands. The gain of the antenna at 890 MHz is 2.4 dBd. The maximum gain at 1890 MHz is 4.4 dBd and the minimum is 2.9 dBd. For PCS application the antenna can be mounted on the roof-top or trunk-lid of a vehicle. The approximate overall volume of the antenna is 64 mm in height, 34 mm in width and 3 mm in depth, which is small compared to the antennas described in [2] and [60]. Since the dual meander-sleeve antenna operates over a 2.6:1 bandwidth within \'SW R< 3. it can also be used in other broad-band applications. 8.2 Plane Sheet Reflector Antennas 8.2.1 Introduction In mobile telephone systems, sector beam antennas with various beamwdiths [60°. 90°. 120° etc.] are widely used in the base stations in order to increase the com munication capabilities. To achieve an omnidirectional horizontal pattern several sectors are usually combined. For example, for antenna systems with 90°. and 120° beam widths, the number of sectors required would be four, and three, respectively. 125 H sj \ L - Length of the comer S - antenna to comer spacing 21 - length of feed antenna H - height of the comer y - comer angle G - spacing between the elements of the wire grid Figure 8.9: .A. comer reflector antenna. .A corner reflector antenna is generally a good choice for the above application because the beamwidth required for the base station can be easily obtained by adjusting the parameters of the antenna e.g. aperture angle (" ). comer length (I) etc [Fig. 8.9]. The required gain is obtained by decreasing the beamwidth in the vertical plane rather than in the horizontal plane [2|. Therefore, a base station antenna system may be visualized as an arrangement of several sectors of collinear arrays each element of which is a corner reflector. However, if the service area is not a circle, but rather a sector, a single sector of collinear array with the appropriate beamwidth can provide the necessar}' coverage. Earlier study of the comer reflector antenna is due to Kraus [19]. The interested reader is referred to one of his finest description that contains the study of comer reflectors with various different aperture angles. Recent work related to corner reflectors include the work of Kamata et al. [63], Suzuki and Kagoshima [64]. and 126 Maruyama and Kagoshima [65]. The study presented in [63] proposed a rocket-borne antenna consisting of a monopole and a major angle comer reflector. The antenna operated in the S band (2290 MHz). The primary radiator was a monopole buried in teflon (Cr = 2) so that the structure was small. Suzuki and Kagoshima [64] described a corner reflector antenna with the same beamwidth in two frequency bands. The sector beamwidth for this antenna was the same at both 900 and 1500 MHz. The dual-frequency operation was realized by placing a parasitic rod near the primary radiating dipole. Mamyama and Kagoshima [65] introduced dual-frequency coraer- reflector antennas fed by elements connected to parallel feed lines. The antenna consisting of two radiating elements with an in-phase connection, had equal beams at two frequencies. The antenna worked as a sector beam antenna with difference of 30° in the 3 dB beam. It is well known that the commonly used feed of a comer reflector is either a straight-wire dipole [19] or a triangular dipole [ 66 ]. .A.s Kraus [19] suggests that the height [H] of the corner reflector has to be at least 1.2 times that of the feed dipole for its proper operation, it is apparent that in a base station where the antenna system consists of several sectors of collinear arrays of comer reflectors, the size of each collinear array can become sufficiently large. Thus, if it is possible to reduce the overall dimension of the array by designing reduced sized corner reflectors, certainly without deteriorating the performance of the array (gain, bandwidth, beamwidth etc.). that should come as an adavntage. The study that follows now describes novel MLBT antennas that can be used as feeds for plane sheet metallic refelctors instead of straight-wire half-wave dipoles to achieve the above objectives. Note that a plane sheet reflector is a special case where the aperture angle. 7 = 180°. The antennas described in this context are 127 modeled using the Numerical Electromagnetic Code (NEC) [1]. 8.2.2 MLBT Dipole Feed Initially, the MLBT dipole showm in Fig. 8.10a is considered as a feed for a plane sheet reflector. The antenna is so designed that it radiates predominantly an E q- component of field with the cross-polarization (E^ component) being below -25 dB. One typical design is Lu,.,>e = 0.61A. 21 = 0.29A. a = 24°. h = 0.06A. For comparison, a straight-wire half-wave dipole oriented along the r-axis is also modeled as a feed for comparison. A plane sheet reflector (■ = 180°) is modeled using the wire-grid concept shown in Fig. 8.9. The reflector is placed such that its center has the coordinates ( —5.0.0). Since the center of the feed is at ( 0 .0.0 ). the distance between the feed and the reflector is 5. The spacing between the elements of the wire grid {G in Fig. 8.9) is approximately 0.05A. Now the parameters of the reflector (5. L and H) are varied to examine its characteristics. Given these circumstances, the MLBT feed in Fig. 8.10a did not prove to l)e advantageous over a straight-wire half-wave dipole feed. Next we consider the MLBT dipole (Type B) shown in Fig. 8.10b. The antenna is so designed that it also radiates predominantly an Eo component of field. The parameters of this antenna as normalized to the wavelength at 835 MHz are: wire length. Itt-.re = 2.95A. a = 120°. 21 = 0.28A. h = 0.48A. wire radius, a = 0.0035A. and A = 4. The input impedance of this antenna is computed over the frequency range of 200-1200 MHz. The results are shown in Fig. 8.11. wherefrom it can be seen that 128 21 a a V - y 21 (a) (b) Figure 8.10: (a) MLBT dipole (Tvpe A), and (b) MLBT dipole (Type B). tlie antenna has resonances ^ at 232. 520. 835 MHz [Fig. 8.11]. The resonant resistances at these frequencies are 3.7. 4.8. and 77 ft respectively. .A.t 835 MHz the antenna radiates predominantly Eo component of field with the cross-polarization being below -13 (IB [Fig. 8.12]. .Assuming a 75fl coaxial line feed, this antenna has a bandwidth of 8.3% within \'SW R< 2. Its elevation-plane. Eo pattern is shown in Fig. 8.13. The pattern is similar to the pattern of a straight-wire dipole. The MLBT dipole just described is then selected as a trial feed element for a plane sheet reflector. The input impedance and radiation fields are computed. It is observed that depending on S there is a shift in the resonant frequency. This may be explained using image theory. .According to which we assume that there is no reflector, but there is an exact image of the primaiw" feed at a distance S from the plane of the reflector. Thus, the distance between the primary' feed and ^\Ve are considering series type resonances for which the reactance changes from capacitive to inductive 129 1000 750 500 250 -250 -500 Resistance -750 Reactance !00 400 600 800 1000 1200 Frequency (MHz)(MHz) Figure 8.11: Computed input impedance of the MLBT dipole ( Type B) as a func tion of frequency (computed using NEC): wire length. = 2.95A, a = 120°. 21 = 0.2SA. h = 0.48A. a = 0.0035A. and .V = 4. -30 J-20 -10 G(dB) 270 90 180 Figure 8.12: Horizontal-plane pattern of the MLBT dipole (Type B) at 835 MHz (computed using NEC). 130 -30 - z o y i o 90 90 180 Figure 8.13: Elevation-plane pattern of the MLBT dipole (Type B) at 835 MHz (computed using NEC). Eo(MLBT feed) -20 270 90 180 Figure 8.14: Horizontal-plane pattern of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire 0.44A dipole and an MLBT dipole, respectively: S = 0.21A. H = 0.65A. L = 0.21 A. and G = 0.05A. 131 -5 Straight-wire dipole feed -1 0 -1 5 -2 5 MLBT dipole feed -3 0 ~^â)0 825 850 875 900 925 950 Frequency (MHz) Figure 8.15; Return loss vs frequency characteristic for a plane sheet reflector. Reflector parameters are: L = 0.21A, H = 0.65A. 5 = 0.21A. and G = 0.05A. M LBT feed param eters are - o = 120°. 21 = 0.28A. h = 0.48A. .V = 4. and n = 0.0036A. Dipole feed parameters are: h = 0.44A. and a = 0.0036A. the image is 25. It is the mutual impedance between the primar>' feed and its image that changes the resonant frequency of the reflector. It also changes the magnitude of the resonant resistance. For instance, a plane sheet reflector that uses the MLBT dipole feed shown in Fig. 8.10b has a resonant frequency of 873 MHz when 5 = 7.2 cm, whereas, the MLBT dipole operating without a reflector has a resonant frequency of 835 MHz. We now examine the characteristics of a plane sheet reflector that uses the MLBT dipole shown in Fig. 8.10b as a feed. The parameters of the reflector are: 5 = 0.21 A. L = 0.21A, H = 0.65A, and G = 0.05A. For comparison we also consider a 0.44A long straight-wire dipole as a feed for the same reflector. Computed horizontal-plane patterns for both feeds are shown in Fig. 8.14. It is apparent that the MLBT feed results in superior £’ The gain for the MLBT feed is 7.6 dBi compared to the gain for the dipole feed of 132 6.9 clBi. The only limitation of the MLBT feed is its cross-polarization, which is however, suppressed below - IS (IB. -A. comparison of the return loss versus frequency characteristic for both feeds is depicted in Fig. 8.15. To calculate the return loss for the MLBT feed a coaxial line with a characteristic impedance of Toft is considered as the feeding transmission line. The reason for this is because the resonant resistance of the reflector with the MLBT feed is close to Toft. For the straight-wire dipole feed a coaxial line with a characteristic impedance of 50Q is assumed since the resonant resistance for the reflector with the straight-wire dipole feed is close to 60Q. Considering a return loss upper limit of -9.5 (IB (\'SW R< 2) the bandwidths for the dipole and the MLBT feeds are 7.5%. and 8.5% respectively. Thus the reflector fed by the straight-wire dipole is slightly more wideband than the reflector fed by the MLBT antenna. Since a bandwidth of about 8% is normally adequate for present day PCS systems, the MLBT feed is certainly a potential candidate for such applications. Next we design a plane sheet reflector such that it has the same front to back {F/D) ratio for both the MLBT and straight-wire dipole feeds. To achieve our objective we consider a reflector with L = 0.21A. S = 0.21A. and G = 0.Ü5A. and vary H. For the results shown in Fig. 8.16. we find that the reflector height. H for the MLBT feed is 0.65A. whereas that for the straight-wire dipole feed is 0.98A. Thus, for similar F/D ratio, the MLBT feed can reduce the height of the reflector by 34%. The beamwidth for the MLBT feed in this case is 100° and that for the straight-wire dipole feed 112°. Finally, we design a plane sheet reflector such that it has the same front to back rato (F/D). gain, and beamwidth when it uses the MLBT and straight-wire dipole feeds. The horizontal-plane patterns for both feeds are shown in Fig. 8.17. 133 ICO 112 270 90 -30 -20 Dipole feed MLBT feed -10 O(dB) 180 Figure 8.16: Horizontal-plane patterns of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire 0.44A dipole and an .MLBT dipole, respectively. Reflector height (H) for the dipole feed is 0.98A and that for the M LBT feed is 0.65A. 270 90 -30 -20 E„(MLBT feed) -10 Eg (Dipole feed) O(dB) 180 Figure 8.17: Horizontal-plane patterns of plane sheet reflectors with MLBT and straight-wire dipole feeds (computed using NEC). Reflector parameters for the dipole feed are: L = 0.31A. S = 0.21A. G = 0.05A. and H = 0.81 A. Reflector parameters for the MLBT feed are: L = 0.21A. S = 0.21A. G = 0.05A. and H = 0.65A. 134 0 (dB ) 3 0 -1 0 1ST Dipole -20 60 -3 0 120 150 180 Figure 8.18: Elevation-plane pattern of a plane sheet reflector (computed using NEC). The feed elements are a straight-wire dipole and an MLBT. respectively. S = 0.20A. G = 0.05A We find that the F/D ratio, gain and beamwidth are 1Ô.5 dB. 7.6 dBi. and 100°. respectively. The length (2£) and height {H) of the reflector for the MLBT feed are 0.21A and 0.65A whereas for the staright-wire dipole feed these are 0.31 A and O.SIA respectively. Thus, for similar radiation characteristics the MLBT feed can reduce the dimensions of the plane sheet reflector by about 46%. Note that the cross-polarization for the MLBT feed is < —18 dB. The elevation-plane pattern for both feeds shown in Fig. 8.18 are similar. 8.2.3 MLBT Monopole Feed Observing the superior radiation characteristics of the MLBT dipole feed we decide to examine the characteristics of an MLBT monopole feed. A plane sheet reflector with an MLBT monopole feed is. therefore, designed. The configuration is shown in 135 Reflector g y - - V 500 % coax feed y y l Figure 8.19: A compact plane sheet reflector fed by an MLBT monopole. Parameters of the antenna are: h = 0.5A, a = 114°. I = O.ISA. h = O.OISA. and e = 0.074A. Radius of wire is 0.0005A. Fig. 8.19. The antenna is fed by a 50 (1 coaxial line from behind the reflector.The MLBT monopole radiates predominantly Eo component of field. Note that if the MLBT monopole in Fig. 8.19 is replaced by a straight-wire monopole, the result would be radiation of component of field. However, as we know that in a base station vertical polarization is required, such an antenna would not he useful. In contrast a properly designed MLBT monopole [Fig. 8.19] can provide predominantly vertical polarization and eliminate the necessity for a balun that is required for a dipole feed that utilizes a coaxial line. To compute the input impedance of this antenna we consider a perfectly con ducting plane sheet reflector of infinite size. Input impedance computed using NEC is shown in Fig. 8.20 as a function of frequency. .Although the antenna has sev eral resonances, the one at 940 MHz is the most important one because (1) the impedance-frequency response in this frequency region is smoother than from any 136 6 0 0 400 Resistance œ 200 Reactance -400 -600 700 800 900 1000 1100 1200 1300 Frequency (MHz) Figure 8.20: Input impedance of the compact plane sheet reflector antenna (com puted using NEC). other resonance region. (2) the real part of the input impedance at 940 MHz is close to 50 Q which is the characteristic impedance of the feeding coaxial line, and (3) for resonances at 720 and 805 MHz the antenna radiates large amount of horizontal component of fields. The antenna is then manufactured and its return loss versus frequency character istic measured using an HP 8720C vector network analyzer. The results are shown in Fig. 8.21. The antenna has a bandwidth of 5.3% within return loss< —9.5dB. Computed and measured radiation pattern for this antenna is shown in Fig. 8.22. The antenna has a gain of 8.4 dBi and half-power beamwidth of 94°. The cross polarization is suppressed below -17 dB within the beamwidth. The MLBT monopole feed described above has several advantages: ( 1 ) it is compact because the feed antenna is not placed at a distance S from the reflector. 131 -5 -15 too 850 900 950 1000 1050 1100 Frequency (MHz) Figure 8.21; Measured Return loss versus frequency characteristics of the compact plane sheet reflector. Q(dB) ■10 -20 -30 270 94 180 Figure 8.22: Horizontal-plane pattern of the compact plane sheet reflector: solid lines-computed. crosses-measured. 138 (2) since it is a monopole feed, a balun is no longer required which is always a necessity for a dipole feed that uses a coaxial line, and that (3) it has adequate gain, beamwidth. and bandwidth. 8.3 Discussion A number of applications of self-resonant bent antennas are described. This includes • a dual meander-sleeve antenna for application as a vehicular antenna in both bands of the personal communication sen,ices (PCS). • a novel MLBT dipole designed using NEC for application as a feed for a plane sheet reflector antenna. The new feed results in superior radiation character istics as compared to a conventional half-wave dipole feed. • a compact plane sheet reflector antenna designed using NEC. fed by an MLBT monopole for application in a base station. The new antenna eliminates the need for a balun which is required when a dipole is fed by a coaxial line. The dual meander-sleeve antenna described is designed using the Numerical Elec tromagnetic Code (NEC) [l|. The antenna is smaller than a quarter-wave monopole and has the attractive property of operating simultaneously in both bands (824-894 and 1850-1990 MHz) of the PCS system within \'S\VR< 2.0. The bandwidth of the antenna is 38% in the lower PCS band and 14% in the upper PCS band. The gain at 890 MHz is 2.4 dBd. The maximum gain at 1890 MHz is 4.4 dBd and the minimum is 2.9 dBd. .A. novel MLBT dipole antenna is introduced as a potential feed for a plane sheet reflector antenna. The reflector is expected to be used in a base station. The 139 new feed has the feature of reducing the overall dimension of the reflector without deteriorating its performance. For instance, for the same front to hack ( F/D) ratio, the MLBT dipole can reduce the height of a plane sheet reflector by 34% as compared to a half-wave dipole feed. In addition, for a constant F/D ratio and half-power beamwidth (HPBW). the MLBT dipole can reduce the overall dimension of a plane sheet reflector by 46% as opposed to a half-wave dipole feed. The only limitation of the MLBT dipole is its cross-polarization which is usually suppressed below - IS dB. The compact plane sheet reflector antenna described consists of an MLBT monopole and a plane sheet metallic reflector. The antenna has a bandwidth of 5.3% within return loss< —9.5 dB. The gain and half-power beamwidth of the antenna are 8.4 dBi and 94°. respectively. The antenna is compact because it utilizes an MLBT monopole as its feed. 140 C hapter 9 Conclusions and Future Work 9.1 Conclusions A self-resonant bent antenna need not to he an XMHA or an ILA or an MLD. Any antenna manufactured by l)ending a wire or metal ribbon would attain self resonance itself and be smaller than 0.25A (in the monopole case) or 0.5A (in the dipole case). However, the shortening ratio, SR. the resonant resistance. and the cross-polarization strongly depend on the geometr}' of the antenna. Thus, a new antenna may provide improvement in any or all of the above three parameters. This indicates that an optimization is the solution when there are a particular set of design constraints and a specific set of design goals [67]. For instance, the geometn," of a curv ilinear dipole is optimized in [ 68 ] to obtain maximum directivity. However, it should be stressed that the primary focus of this dissertation is not on optimization, but on analysis. Thus instead of optimizing self-resonant bent antennas for a specific set of design goals, we focused on analyzing several existing 141 self-resonant bent antennas and on examining their potential applications. The original contributions of this study can be summarized as follows: An expression for the input impedance of an inverted-L antenna (ILA) is derived using the induced EMF method. .A. sinusoidal distribution of current that drops to zero at the antenna end is assumed and closed-form expressions for the near-fields of the antenna are derived. The near-fields are then used to derive an expression for the input impedance. The final equation contains integrals that can be evaluated using a suitable numerical integration routine. L'nlike the transmission line model of King and Harrison [5] and the point matching scheme of W'unsch and Hu [20] our model is not restricted to antennas that are electrically small. The model proposed can be extended to treat other antennas of similar geometries. The input impedance and radiation characteristics of a meander-line dipole (MLD) and a meander-line bow-tie (MLBT) antenna are investigated. Similarly to the inverted-L antenna analytical solutions for the input impedance of the MLD and the MLBT antennas are also obtainable. However, because of the geometries of these antennas such solutions would involve large number of parameters and there fore. be complicated in nature. Because of this the input impedance of the MLD and the MLBT antenna are computed using the Numerical Electromagnetics Code (NEC) [I]. The dependency of the shortening ratio. SR. and the resonant resis tance. Rres of these antennas on their parameters is examined. The impedance as a function of wire radius is also studied. It has been observed that for the same SR the MLBT antenna has larger Rres than the MLD antenna. To study the radiation characteristics of the MLD and the MLBT antennas sim plified analytical models are proposed. Closed-form expressions for the vertically and horizontally polarized components of fields are derived. Results obtained using the 142 analytical models are verified by comparing them with results obtained from NEC computation. It is noted that both of these antennas have some cross-polarization which for a constant antenna length can be minimized by increasing the number of meander sections. The gain of these antennas is calculated as a function of antenna length. The gain is greater than for a straight-wire dipole of the same length. .A. dual meander antenna is analyzed using NEC. The results of this study indicate that the dual meander generates extremely low cross-polarization, than the MLD and the MLBT antennas. In addition the dual meander is wideband compared with the MLD. .A. planar printed meander antenna is analyzed using the Finite-Difference Time- Domain Technique. The characteristics of this antenna are studied iis a function of the dielectric constant of the substrate. Cr. and substrate thickness. The de pendence of the resonant resistance of a printed meander dipole on the dielectric constant of the substrate. 6 ^- and the substrate thickness is described. The analysis of the printed meander may be considered as another important contribution of this investigation. .A. wide-band dual meander-sleeve antenna is proposed for dual-frequency vehic ular application in the personal communication services (PCS). The antenna can be used as a vehicular antenna operating simultaneously in the 824-894 MHz and 1850-1990 MHz bands of the PCS. The antenna we propose has bandwidths of 38% in the 824-894 MHz band and 14% in the 1850-1990 MHz band within \ ’SWR< 2. The radiation patterns are omnidirectional in the horizontal plane in both frequency bands. The gain of the antenna is 2.4 dB at 890 MHz with reference to a half-wave dipole. The maximum gain at 1890 MHz is 4.4 dB and the minimum is 2.9 dB with reference to a half-wave dipole. The polarization of the antenna is vertical. 143 An MLBT dipole can be used as a feed to plane sheet reflectors for application in base stations. For similar radiation characteristics (front to back ratio, half-power beamwidth. and gain) the MLBT dipole feed can reduce the reflector dimensions by 46% compared to a straight-wire dipole feed. A compact plane sheet reflector antenna is described that uses an MLBT monopole as a feed. The antenna is fed by a 50 Q coaxial line from behind the reflector. It ra diates predominantly vertical polarization and has a gain of 8.4 dBi and bandwidth of 0.3%. .A plane sheet reflector which has a dipole as its primar\' radiator needs a balun if a coaxial line is used to feed the dipole. In contrast the new antenna we propose does not need any balun since the primary radiator is an MLBT monopole. Also, if the primary’ radiator of a plane sheet reflector is a dipole, then the dipole must be placed at a distance S from the plane of the reflector. The bandwidth, and radiation pattern of the reflector strongly depends on S [19]. T he new antenna we describe is not affected by this problem as because its primary radiator is an MLBT monopole. In addition since 5 = 0. the antenna is relatively more compact than a conventinal plane sheet reflector. 9.2 Future Work In the future the analysis presented for the inverted-L antenna can be extended to treat other antennas, such as the T-antenna. the folded unipole antenna, and the M-antenna. The analysis of the printed meander antenna can also be extended considering various other dielectric materials, such as quartz (Cr = 4.0). alumina (fr = 9.8) etc. The effect of a dielectric cover on the antenna characteristics can be investigated as well. Another important topic could be to incorporate feeding trans- 144 mission lines in the models and study their effect on the antenna characteristics. The bandwidth and efficiency of printed meander antennas can also be examined. From antenna application perspective, the potentials of the MLBT type feeds can further be explored considering comer reflectors with aperture angles< 180°. Such feeds can also be tested for trough [ 66 ] and superior angle reflectors [2|. The ap plication prospects of the printed meander antenna, such as in a collinear array for base station application can also be explored. BlBLIOGRAPWi' 145 Bibliography [1] G. .1. Burke, and A. .1. Poggio. 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J Integrating the first integral in (A.3) by parts twice APPEXDLX A. SOLUTION OF INTEGR.\LS IN EQNS. (3.15)-(3.22) 155 Substituting (A.4) into (A.3) lo El = ;A.5) Sim plifying (.A..5). p-A-fl p - jk R , . ^ ^] v'=o E\ — —730/0 cos k{ / — (/ —/;) ——------h (.7 — // ) sin k{I — i/ — h ) ——— j — + J J (A.6) The integrals in (3.18), (3.20). and (3.21) are solved using the above procedure which can be followed when the obser\ation and the source wire elements are in parallel. The other case is when the source and observation elements are perpendicaular to each other. For instance. 1 /■--'=« r 0 0 E-, = ( -A.. 1 or 1 E-. = A.8) 4,. 7^.60 IZ where I(:') = /osin(Ad + k :'). Integrating (.A.. 8 ) 1 r'=0 di{z’)d The integral in (.A.9) can be calculated as follows [28] f = r " = H„ r " - ° cosiu + , A .10) J:’=-h d:' Of! Jz'=-h at/ Since cos{kl + k:') can be expressed as _ and also since 'P = ‘ (.A. 1 0 ) can be rewritten as = 0 Q r R--') 1 ^ r:'=Q Q fç-jk{R+z') 0„ I R j T A ~ R ------APPENDIX A. SOL UTION OF INTEGFLALS IN EQNS. (3.15)-(3.22 ) 156 Considering the 1st integral in (A. 11] r'= 0 ^ f The term inside the braces in (A. 12) can he expressed as a total differential. Thus /•r'=0 R - z ' ) Il =0] - y) / dz (A. 13) J = R(R- :'+ =] Integrating (A. 13) , - j k ( R - : ’) r'= 0 A = (.V - .'/) [A.14) R {R -z' A Similarly, the 2nd integral in (A .ll) gives z'=0 c'=0 Q (ç-jk\R+z'\ ^ - j k i R+z' I /■> (Iz = iy - y] A.15) = f T Jz'=-h ay R RiR + z' - z] z' = —h S ubstituting (.A.14) and (A.15) into (A .ll) and then (.A .ll) into (A.9) gives E , = + + ,~ jkl A .16) 2 Rx + h where the parameters in (A.16) are listed in Table 3.1. The integrals in (3.17). (3.19). and (3.22) are solved using the above procedure which can be followed when the observation and the source wire elements are per pendicular to each other. APPEXDLX D. ADDREVIATIOXS IX EQXS. (5.24)-(ô.27) 15< Appendix B Abbreviations in Eqns (5.24)-(5.27) — sin/9 „ P'2 = ( A/| sin( —s ; r/, + LL — Ci ) + -S| cos( —s | + f' L — Ci ) | ( B.2) P3 = p sin( —s 1 zi + k'L — Cl) + s 1 cos( — si zi + LL —C i)) ( B.3 ) COS f) sin o _ “ 4 + m:; ' ° p.", = {A/o sin( —■'’■21 /h + LL — Ci ) + .so cos( — sgp/, 4- k L — Ci ) ) ( B.5) Pg = {A/o sin( — s'-j/p + LL — Ci ) + -so cos( —■s-2 f/i + LL — C\)} ( B.6) 'G.) Ps — {A/3 sin( —-51 c/i + kL — C3 ) + .s 1 cos( — s 1 z/, + kL — C>) ) ( B.S) pg = { A/3 sin( —.s 1 z; + k'L — C )) + si cos{ —s 1 z/ + kL — Co) ) ( B.9) APPEN D IX D. ADDREVIATIOXS IX EQXS. (5.24)-(5.27) 158 cos 0 sin o „ Pio = •> , , (B.IO) ■s.j + Mj Pi I = {.\/i sin( s-2 !Jh + — Co) — so cos( .so.i//, + k'L — Co)} ( B .ll pio = (.\/| sin(.5->,fy/ + P I — Co) — socos(.soiji 4- P I — C> ) [ ( B.I2 Pi:\ = { A/;j sin( + P I — C, ) — s, cos( .si c/, 4- P I — C’l )} ( B. 13 pt 1 = e'^^’''{.V/3 sin(sir; 4- P I — C, ) — s, cos( .s, :/ 4- P I — C, )} ( B.I4 Pis = (All sin( —■'io}/h + P I ~ 1^1 ) — ■^■2 cos( —soijh + P I — (At )} {B.15 Pi6 = {AI| sin( — sop/ 4- P I — C\) — .‘to cos( —•‘>•>,7/ 4- P I — I'l )} ( BIG PIT = sin(.sir/, 4- PI — C->) — S| cosisic/, 4- PI — C>)} (B.I7 Pis = e'^‘''{A /i sin(.siC; 4- P I — Co)— .si cos(.s|4- PI — )} (B.lS P i9 = {AA sin(.s'-2p/, 4- P I — Co) — ■‘^■2 i:o s (.s o 4- PI — (Tj)} ( B.19 P’2o — p {AI» sin(.sop/ 4- P I — Co) — .'iocosi.soiji 4- P I — I'-_>)} ( B.20 p._, = f'sW" {.Ç-, sin( Ppu 4- s ,3 ) - P cos( Pp„ 4- s,; )} ( B.22 73 == p''*'{S. 3sin(Pp/ 4- -S6) - Pcos(Pp/ 4- s g ( B.23 7t = { s.T sin( -Ppu + -5(5 ) + P cos( -Ppu 4- -sg)} ( B.24 q-> = f*'*' { 55 sin( -P p / 4- .9(1 ) 4- Pcos( -P p / 4- s,i)} ( B.25 s’3 = COS ^ sin o (B.28 APPEXDLX D. AD BREVIATIO XS IX EQXS. (5.24)-(.5.27) 159 (B.29) .-fr, = jksinÛsino (B.30) ■sg = LL — L ire sin — — nke ( B.31) ,s- = coso iB.32)