ANTENNA THEORY
`Analysis and Design
`
`CONSTANTINEA. BALANIS
`West Virginia University
`
`1817
`
`HARPER & ROW, PUBLISHERS,New York
`Cambridge, Philadelphia, San Francisco,
`London, Mexico City, So Paulo, Sydney
`
`
`
`SAMSUNG 1048
`
`SAMSUNG 1048
`
`1
`
`

`

`Sponsoring Editor: Carl McNair
`Project Editor: Pamela Landau
`Designer: Michel Craig
`Production Manager: Marion Palen
`Compositor: Science Typographers,Inc.
`‘Printer and Binder: The Murray Printing Company
`Art Studio: Vantage Art Inc.
`
`ANTENNA THEORY
`
`Analysis and Design
`
`Copyright © 1982 by Harper & Row, Publishers, Inc.
`
`All rights reserved. Printed in the United States of America. No part of
`this book may be used or reproduced in any manner whatsoever without written
`permission, except in the case of brief quotations embodiedin critical articles
`and reviews. For information address Harper & Row,Publishers, Inc., 10 East 53d Street, New
`York, NY 10022.
`
`Library of Congress Cataloging in Publication Data
`
`Balanis, Constantine A., 1938—
`Antennatheory.
`
`(The Harper & Rowseriesin electrical engineering)
`Includes bibliographical references and index.
`1. Antennas (Electronics)
`I. Title.
`II. Series.
`TK7871.6. B353
`621.38’028’3
`81-20248
`ISBN 0-06-040458-2
`AACR2
`
`
`
`2
`
`

`

`Contents
`
`Preface
`
`XV
`
`1
`
`1.2
`
`Chapter 1 Antennas
`1
`Introduction
`1.1
`1
`Types of Antennas
`Wire Antennas; Aperture Antennas; Array Antennas; Reflector
`Antennas; Lens Antennas
`7
`Radiation Mechanism
`Current Distribution on a Thin Wire Antenna
`Historical Advancement
`15
`References
`15
`
`1.3
`1.4
`
`1.5
`
`11
`
`Chapter 2 Fundamental Parameters of Antennas
`Introduction
`17
`2.1
`Radiation Pattern
`17
`eds
`Isotropic, Directional, and Omnidirectional Patterns, Principal
`Patterns; Radiation Pattern Lobes; Field Regions; Radian and
`Steradian
`
`17
`
`vii
`
`
`
`3
`
`

`

`viii CONTENTS
`
`25
`
`37
`
`de
`
`46
`
`Radiation Power Density
`Radiation Intensity
`27
`Directivity
`29
`Numerical Techniques
`Gain
`42
`Antenna Efficiency
`Half-Power Beamwidth
`Beam Efficiency
`46
`Bandwidth
`47
`Polarization
`48
`Linear, Circular, and Elliptical Polarizations; Polarization Loss
`Factor
`53
`Input Impedance
`57
`Antenna Radiation Efficiency
`59
`Antenna as an Aperture: Effective Aperture
`61
`Directivity and Maximum Effective Aperture
`Friis Transmission Equation and Radar Range Equation
`Friis Transmission Equation; Radar Range Equation
`Antenna Temperature
`67
`References
`70
`Problems
`71
`Computer Program—Polar Plot
`Computer Program— Linear Plot
`Computer Program— Directivity
`
`63
`
`75
`78
`80
`
`2 2
`
`.4
`en
`2.6
`2.7
`
`2.8
`29
`2.10
`2Al
`
`Dade
`
`2.13
`2.14
`2.15
`2.16
`2.17
`
`2.18
`
`ss
`
`3.6
`Sf
`3.8
`
`Chapter 3 Radiation Integrals and Auxiliary Potential Functions
`3.1
`Introduction
`82
`3.2
`83
`The Vector Potential A for an Electric Current Source J
`85
`The Vector Potential F for a Magnetic Current Source M
`33
`Electric and Magnetic Fields for Electric (J) and Magnetic (M)
`3.4
`Current Sources
`86
`Solution of the InhomogeneousVector Potential Wave
`Equation
`88
`Far-Field Radiation
`93
`Duality Theorem
`Reciprocity and Reaction Theorems
`Reciprocity for Radiation Patterns
`References
`99
`Problems
`99
`
`82
`
`92
`
`94
`
`100
`
`Chapter 4 Linear Wire Antennas
`4.1
`Introduction
`100
`100
`Infinitesimal Dipole
`4.2
`Radiated Fields; Power Density and Radiation Resistance;
`Near-Field (kr < 1) Region; Intermediate-Field (kr > 1) Region;
`Far-Field (kr > 1) Region; Directivity
`
`
`
`4
`
`

`

`CONTENTS ix
`
`4.3.
`4.4
`
`4.5
`
`109
`Small Dipole
`112
`Region Separation
`Far-Field (Fraunhofer) Region; Radiating Near-Field (Fresnel)
`Region; Reactive Near-Field Region
`Finite Length Dipole
`118
`Current Distribution; Radiated Fields: Element Factor, Space
`Factor, and Pattern Multiplication; Power Density, Radiation
`Intensity, and Radiation Resistance; Directivity; Input Resistance;
`Finite Feed Gap
`130
`Half-Wavelength Dipole
`132
`Linear Elements Near or on Infinite Plane Conductors
`Image Theory; Vertical Electric Dipole; HorizontalElectric Dipole
`4.8 Ground Effects
`148
`Vertical Electric Dipole; Horizontal Electric Dipole; Earth
`Curvature
`159
`References
`159
`Problems
`Computer Program—Linear Dipole: Directivity, Radiation
`Resistance, and Input Resistance
`162
`
`4.6
`4.7
`
`164
`
`5.3.
`
`5.7
`
`Chapter 5 Loop Antennas
`164
`5.1
`Introduction
`164
`5.2.
`Small Circular Loop
`Radiated Fields; Small Loop and Infinitesimal Magnetic Dipole;
`Power Density and Radiation Resistance; Near-Field (kr < 1)
`Region; Far-Field (kr > 1) Region; Radiation Intensity and
`Directivity
`176
`Circular Loop of Constant Current
`Radiated Fields; Power Density, Radiation Intensity, Radiation
`Resistance, and Directivity
`184
`Circular Loop with Nonuniform Current
`5.4
`5.5 Ground and Earth Curvature Effects for Circular Loops
`5.6
`Polygonal Loop Antennas
`19]
`Square Loop; Triangular, Rectangular, and Rhombic Loops
`Ferrite Loop
`196
`References
`198
`Problems
`199
`Computer Program—Circular Loop: Directivity and
`Radiation Resistance
`201
`
`188
`
`Chapter6 Arrays: Linear, Planar, and Circular
`6.1.
`Introduction
`204
`205
`6.2
`Two-Element Array
`6.3
`N-Element Linear Array: Uniform Amplitude and Spacing
`
`204
`
`212
`
`
`
`5
`
`

`

`x CONTENTS
`
`6.4
`
`6.5
`
`6.6
`6.7
`
`6.8
`6.9
`
`6.10
`
`Broadside Array; Ordinary End-Fire Array; Phased (Scanning)
`Array; Hansen-Woodyard End-Fire Array
`229
`N-Element Linear Array: Directivity
`Broadside Array; Ordinary End-Fire Array; Hansen-Woodyard
`End-Fire Array
`N-Element Linear Array: Three-Dimensional Characteristics
`235
`N-Elements Along Z-Axis; N-Elements Along X- or Y-Axis
`Rectangular-to-Polar Graphical Solution
`238
`N-Element Linear Array: Uniform Spacing, Nonuniform
`Amplitude
`240
`Array Factor; Binomial Array; Dolph-Tschebyscheff Array
`Superdirectivity
`257
`Planar Array
`260
`Array Factor; Beamwidth; Directivity
`Circular Array
`274
`Array Factor
`References
`Problems
`
`279
`280
`
`del
`12
`ie
`
`7.4
`7
`
`Chapter 7 Self- and Mutual Impedancesof Linear Elements and
`Arrays, and Finite Diameter Effects (Moment Method)
`283
`283
`Introduction
`285
`Near-Fields of Dipole
`290
`Input Impedance of Dipole
`Induced emf Method; Finite Dipole Input Impedance
`Mutual Impedance Between Linear Elements
`296
`Finite Diameter Wires: The Moment Method
`304
`Integral Equation; Moment MethodSolution; Basis Functions;
`Weighting (Testing) Functions; Current Distribution; Input
`Impedance; Radiation Pattern; Source Modeling
`References
`317
`Problems
`318
`Computer Program— Finite Diameter Dipole: Current
`Distribution, Input Impedance, and Radiation Pattern
`
`319
`
`Chapter 8 Broadband Dipoles and Matching Techniques
`8.1
`Introduction
`322
`8.2
`323
`Biconical Antenna
`Radiated Fields; Input Impedance
`Triangular Sheet, Bow-Tie, and Wire Simulation
`Cylindrical Dipole
`332
`Bandwidth; Input Impedance; Resonance and Ground Plane
`Simulation; Radiation Patterns; Equivalent Radii; Dielectric
`Coating
`
`322
`
`330
`
`8.3
`8.4
`
`
`
`6
`
`

`

`CONTENTS xi
`
`8.5
`8.6
`8.7
`8.8
`
`340
`Folded Dipole
`Discone and Conical Skirt Monopole
`Sleeve Dipole
`347
`349
`Matching Techniques
`Stub-Matching; Quarter-Wavelength Transformer; T-Match;
`Gamma Match; Omega Match; Baluns and Transformers
`References
`368
`Problems
`369
`
`346
`
`372
`
`Chapter 9 Traveling Wave and Broadband Antennas
`9.1
`Introduction
`372
`D2
`aie
`Traveling Wave Antennas
`Long Wire; V Antenna; Rhombic Antenna
`Broadband Antennas
`385
`Helical Antenna; Electric-Magnetic Dipole; Yagi-Uda Array of
`Linear Elements; Yagi-Uda Array of Loops
`References
`409
`Problems
`411
`
`9.3
`
`413
`
`416
`
`Chapter 10 Frequency Independent Antennas and Antenna
`Miniaturization
`413
`Introduction
`Theory
`414
`Equiangular Spiral Antennas
`Planar Spiral; Conical Spiral
`Log-Periodic Antennas
`423
`Planar and Wire Surfaces; Dipole Array; Design of Dipole Array
`Fundamental Limits of Electrically Small Antennas
`439
`References
`dda
`Problems
`445
`
`10.1
`10.2
`10.3
`
`10.4
`
`10.5
`
`447
`
`Chapter 11 Aperture Antennas, and Ground Plane Edge Effects
`(Geometrical Theory of Diffraction)
`446
`Introduction
`446
`Field Equivalence Principle: Huygens’ Principle
`Radiation Equations
`454
`Directivity
`456
`457
`Rectangular Apertures
`Uniform Distribution on an Infinite Ground Plane; Uniform
`Distribution in Space; TE,)-ModeDistribution on an Infinite
`Ground Plane; Beam Efficiency
`Circular Apertures
`478
`Uniform Distribution on an Infinite Ground Plane; TE, , -Mode
`Distribution on an Infinite Ground Plane; Beam Efficiency
`Microstrip Antennas
`487
`Radiated Fields; Radiation Conductance; Directivity; Bandwidth;
`Arrays; Circular Polarization
`
`11.1
`11.2
`1i3
`11.4
`118
`
`11.6
`
`11.7
`
`
`
`7
`
`

`

`xii CONTENTS
`
`11.8
`
`11.9
`
`496
`Babinet’s Principle
`Ground Plane Edge Effects: The Geometrical Theory of
`Diffraction
`502
`Edge Diffraction Coefficient; Aperture on a Finite-Size Ground
`Plane; Curved-Edge Diffraction; Equivalent Currents in
`Diffraction; Oblique Incidence Edge Diffraction
`References
`522
`Problems
`524
`Computer Program— Diffraction Coefficient
`
`529
`
`Chapter 12 Horns
`12.1
`532
`Introduction
`532
`E-Plane Sectoral Horn
`12.2
`Aperture Fields; Radiated Fields; Directivity
`H-Plane Sectoral Horn
`550
`Aperture Fields; Radiated Fields; Directivity
`Pyramidal Horn
`565
`Aperture Fields, Equivalent, and Radiated Fields; Directivity;
`Design Procedure
`Conical Horn
`Corrugated Horn
`Phase Center
`References
`
`532
`
`577
`
`579
`
`587
`589
`
`12.3
`
`12.4
`
`12.5
`12.6
`12.7
`
`Problems
`
`590
`
`593
`
`13.2
`13.3
`
`13.4
`
`13.5
`13.6
`
`Chapter 13 Reflectors and Lens Antennas
`13.1
`Introduction
`593
`Plane Reflector
`594
`Corner Reflector
`594
`90° Corner Reflector; Other Corner Reflectors
`Parabolic Reflector
`604
`Front-Fed Parabolic Reflector; Cassegrain Reflectors
`Spherical Reflector
`642
`646
`Lens Antennas
`Lenses with n> 1; Lenses withn< 1; Lenses with Variable Index
`of Refraction
`References
`Problems
`
`654
`656
`
`Chapter 14 Antenna Synthesis and Continuous Sources
`14.1
`Introduction
`658
`14.2
`659
`Continuous Sources
`Line-Source; Discretization of Continuous Sources
`Schelkunoff Polynomial Method
`661
`Fourier Transform Method
`666
`Line-Source; Linear Array
`
`14.3
`14.4
`
`658
`
`
`
`8
`
`

`

`CONTENTS xiii
`
`679
`
`673
`14.5. Woodward Method
`Line-Source; Linear Array
`14.6 Taylor Line-Source (Tschebyscheff Error)
`Design Procedure
`684
`14.7. Taylor Line-Source (One-Parameter)
`14.8 Triangular, Cosine, and Cosine-Squared Amplitude
`Distributions
`690
`694
`14.9 Line-Source Phase Distributions
`696
`14.10 Continuous Aperture Sources
`Rectangular Aperture; Circular Aperture
`References
`698
`Problems
`699
`
`703
`
`Chapter 15 Antenna Measurements
`15.1
`Introduction
`703
`704
`15.2 Antenna Ranges
`Reflection Ranges; Free-Space Ranges
`15.3 Radiation Patterns
`710
`Instrumentation; Amplitude Pattern; Phase Measurements
`15.4 Gain Measurements
`716
`Absolute-Gain Measurements; Gain-Transfer(Gain-Comparison)
`Measurements
` Directivity Measurements
`15.5
`15.6 Radiation Efficiency
`725
`15.7
`Impedance Measurements
`15.8 Current Measurements
`15.9 Polarization Measurements
`15.10 Scale Model Measurements
`References
`734
`
`723
`
`725
`727
`
`728
`733
`
`Appendix | f(x) =
`
`sin(x)
`
`737
`
`Appendix Il f(x) =
`
`
`
`sin(Nx)
`Nsin(x)|’
`
`N=1,3,5,10,20
`
`740
`
`Appendix III Cosine and Sine Integrals
`
`743
`
`Appendix IV Fresnel Integrals
`
`Appendix V Bessel Functions
`
`748
`
`755
`
`Appendix VI Identities
`
`768
`
`Appendix VII Vector Analysis
`
`771
`
`Appendix VIII Television and Radio Frequency Spectrum
`
`781
`
`Index
`
`783
`
`
`
`9
`
`

`

`2.13
`
`INPUTIMPEDANCE 53
`
`|
`
`|
`
`|
`|
`
` coc 7
`
`|
`Vp |
`|
`|
`|
`|
`|
`Lo
`
`| |
`
`PLF = [Ay * Ba*l? = 1
`(aligned)
`
`PLF = [iy * bg*|? = cos” Wp
`(rotated)
`
`PLF = Idw * ba*|? = 0
`(orthogonal)
`
`(a) PLFfor transmitting and receiving
`aperture antennas
`
`on
`
`lI!
`
`1I!| |
`
`2
`! Wo i)
`\\
`\\
`
`\
`
`\
`
`PLE = |dy * Oa*l? = 1
`(aligned)
`
`PLF = [bw * ba*|* = cos” vp
`(rotated)
`
`PLF = [By * by*|? =0
`(orthogonal)
`
`(b) PLF for transmitting and receiving
`linear antennas
`
`Figure 2.17 Polarization loss factors (PLF) for aperture and wire antennas.
`
`The polarization loss must always be taken into account in the link
`calculations design of a communication system because in some cases it may
`be a very critical factor. Link calculations of communication systems for
`outer space explorations are very stringent because of
`limitations in
`spacecraft weight. In such cases, power is a limiting consideration. The
`design must properly take into accountall loss factors to ensure a successful
`operation of the system.
`
`2.13 INPUT IMPEDANCE
`
`Input impedanceis defined as “the impedance presented by an antennaatits
`terminals or the ratio of the voltage to current at a pair of terminals or the
`ratio of the appropriate components of the electric to magnetic fields at a
`point.” In this section weare primarily interested in the input impedanceat
`a pair of terminals which are the input terminals of the antenna. In Figure
`2.18(a) these terminals are designated as a—b. Theratio of the voltage to
`current at these terminals, with no load attached, defines the impedance of
`
`
`
`10
`
`10
`
`

`

`54
`
`FUNDAMENTAL PARAMETERS OF ANTENNAS
`
`ACRES
`
`Generator
`iZg)
`
`Antenna
`a
`
`b
`
`7
`
`Radiated
`wave
`
`(a) Antenna in transmitting mode
`
`a
`
`Tr Ris
`VeI fe,
`
`Rg
`
`xe
`
`|
`
`R
`
`|
`eS|wae:
`
`b
`
`(b) Thevenin equivalent
`
`(c) Norton equivalent
`Figure 2.18 Transmitting antenna and its equivalentcircuits.
`
`the antenna as
`
`Z4=Ryat JX
`
`(2-72)
`
`where
`(ohms)
`Z, =antenna impedanceat terminals a—b
`(ohms)
`R,=antennaresistance at terminalsa—b
`(ohms)
`X,=antennareactance at terminals a—b
`In general the resistive part of (2-72) consists of two components; thatis
`
`
`
`11
`
`11
`
`

`

`2.13 INPUTIMPEDANCE 55
`
`where
`
`R,= radiation resistance of the antenna
`R,=lossresistance of the antenna
`
`The radiation resistance will be considered in more detail in later chapters,
`and it will be illustrated with examples.
`If we assumethat the antennais attached to a generator with internal
`impedance
`ZR, tik,
`
`(2-74)
`
`where
`(ohms)
`R,=resistance of generator impedance
`(ohms)
`X,=reactance of generator impedance
`and the antenna is used in the transmitting mode, we can represent the
`antenna and generator by an equivalentcircuit* shown in Figure 2.18(b). To
`find the amount of power delivered to R, for radiation and the amount
`dissipated in R, as heat (J?R,/2), we first find the current developed
`within the loop which is given by
`
`V,
`V,
`V,
`
`we ee's~ 7, Z,+Z, (R,+R,+R,) +i(X,+X,)
`and its magnitude by
`yeea (2-75a)
`[(R,+R,+R,) +(Xi+X,)|
`where V, is the peak generator voltage. The powerdelivered to the antenna
`for radiation is given by
`
`
`R,
`1
`\V,.°
`
`P= fr =—|oT (W) (2-76)
`
`2 le 2|(R,+R,+R,)+(X,+X,)
`and that dissipated as heat by
`R
`l
`a
`
`
`P,==|1,?R,= —|—_—_—___+_|; W) (2-77)
`E
`5 el
`LB
`9)
`(R,+R,+R,)+(X,+X,)°
`(
`)
`The remaining power is dissipated as heat on the internal resistance R, of
`the generator, and it is given by
`
`I,|
`
`V,
`
`
`
`(4)
`
`(@-75)
`
`
`
`
`
`[HA
`=
`
`
`
`R,
`(R,+R,+R,) +(X,+X,)°
`
`
`
`(Ww)
`
`(2-78)
`
`*This circuit can be used to represent small and simple antennas. It cannot be used for
`antennas with lossy dielectric or antennas over lossy ground because their loss resistance
`cannotbe represented in series with the radiation resistance.
`
`
`
`12
`
`12
`
`

`

`56 FUNDAMENTAL PARAMETERS OF ANTENNAS
`
`The maximum power delivered to the antenna occurs when we have
`conjugate matching; that is when
`R,+R,=R,
`X,=—X,
`For this case
`
`
`R,
`\V,\°
`P= 5 —— |=
`4(R,+R,)
`
`
`IVa?|oR
`= : te
`(2-82)
`(R,+R,)
`
`
`
`pull|_Re f_WPf_ a )_ Ir (2-83)
`
`
`
`
`
`
`
`
`nn) 8|R,+R,|(R,+R,)° 8R,
`From (2-81)—(2-83), it is clear that
`
`
`
`
`R,+R
`VP.
`VP}
`oR
`(2-84)
`al 14. ,
`P =P.+P,= |
`al ——__, = |
`
`
`
`° 8|(R,+R;) 8|(R-+Rz)
`
`
`(2-79)
`(2-80)
`
`(2-81)
`
`
`R,
`\V,P
`
`5 nn
`(R,+R,)
`
`
`
`The power supplied by the generator during conjugate matching is
`V,*
`_ \V,l°
`1
`
`2(R,+R,)
`R,+R,
`
`1
`l
`P= 5,1= 5 V5
`
`
`
`
`
`4
`
`|
`
`(W)
`
`(2-85)
`
`Ofthe powerthat is provided by the generator, half is dissipated as heat in
`the internal resistance (R,) of the generator and the otherhalf is delivered
`to the antenna. This only happens when we have conjugate matching. Of the
`powerthatis delivered to the antenna,partis radiated through the mecha-
`nism provided by the radiation resistance and the otheris dissipated as heat
`which influencespartof the overall efficiency of the antenna. If the antenna
`is lossless (¢,;= 1), then half of the total power supplied by the generatoris
`radiated by the antenna during conjugate matching. In this section we have
`assumed a perfect match between the antenna and the interconnecting
`transmission line (e,=1). Any mismatch losses will reduce the overall
`efficiency. Figure 2.18(c) illustrates the Norton equivalent of the antenna
`and its source in the transmitting mode.
`The use of the antenna in the receiving mode is shown in Figure
`2.19(a). The incident wave impinges upon the antenna, and it induces a
`voltage V; which is analogousto V, of the transmitting mode. The Thévenin
`equivalentcircuit of the antennaandits load is shownin Figure 2.19(b) and
`the Norton equivalent in Figure 2.19(c). The discussion for the antenna and
`its load in the receiving modeparallels that for the transmitting mode.
`The input impedanceof an antennais generally a function of frequency.
`Thus the antenna will be matched to the interconnecting transmission line
`
`
`
`13
`
`13
`
`

`

`2.14 ANTENNA RADIATION EFFICIENCY 57
`
`Antenna
`ad
`
`Load
`(Zr)
`
`js
`
`Incident
`wave
`fi
`
`(a) Antenna in receiving mode
`
`Ry |
`AN Vr
`
`
`k,
`
`Rr
`|
`AT
`Lo Xa
`
`(b) Thevenin equivalent
`
`a
`
`rToTTT
`Gr
`Br
`eS
`eo
`Ba
`fa
`|
`|. {|
`[|
`|
`|
`
`(c) Norton equivalent
`Figure 2.19 Antenna and its equivalent circuits in the receiving mode.
`
`and other associated equipment only within a bandwidth. In addition, the
`input impedance of the antenna depends on many factors including its
`geometry, its method of excitation, and its proximity to surrounding objects.
`Because of their complex geometries, only a limited numberof practical
`antennas have been investigated analytically. For many others, the input
`impedance has been determined experimentally.
`
`2.14 ANTENNA RADIATION EFFICIENCY
`
`The antennaefficiency that takes into account the reflection, conduction,
`and dielectric losses was discussed in Section 2.8. The conduction and
`dielectric losses of an antenna are very difficult to compute and in most
`
`
`
`14
`
`14
`
`

`

`120 LINEAR WIRE ANTENNAS
`
`The pattern multiplication for continuous sources is analogous to the
`pattern multiplication of (6-5) for discrete-element antennas(arrays).
`For the currentdistribution of (4-56), (4-58a) can be written as
`
`E,=
`Jn
`4
`sinafsn 7 +z’}le
`dz
`ke.|po | ( 1 ke
`
`+f
`sink( 72 Je
`dz
`(4-60)
`
`mw
`
`ff
`
`
`Tr
`
`k\|— ’
`
`+jkz'cos@ Jot
`
`H1/2
`
`ile ,
`
`+jkz'cos@ 3,7
`
`-
`
`Each oneof the integrals in (4-60) can beintegrated using
`
`fes*sin(Bx+y) dx= ao [asin(Bx+y)—Beos(Bx+y)]
`a=jkcosé
`B=+k
`y=kl/2
`After some mathematical manipulations, (4-60) takes the form of
`
`
`[Feos0} ~cos( 5]
`
`
`| Tye cos|= Cos cos| >
`
`
`(4-62a)
`y= 2ar
`sin 8
`
`
`In a similar manner, or by using the established relationship between
`the E, and H, in the far-field as given by (3-58b) or (4-27), the total H,
`component can be written as
`E,
`[pew cos 7c0s 8 — cos|
`Hy") =I"dar
`sind
`
`(4-61)
`(4-61a)
`(4-61b)
`(4-61c)
`
`(4-62b)
`
`kl
`kl
`
`
`4.5.3 Power Density, Radiation Intensity, and Radiation
`Resistance
`For the dipole, the average Poynting vector can be written as
`Xa,
`1
`}= 1
`Rela,
`EyXa,H,*)=1Re|
`aE
`Wav= 7 Re[EXH*] = 5 Re[dpBy XGMy"]=7Re| da koX ay
`cos( Sco ) (+) 2
`Lol?
`2
`s
`cos
`2
`1
`
`av &,Way—4 7 al Veep? sin@ ( )
`W..=4W.,=4,—|E,|"=
`=
`4-63
`
`
`
`
`
`and the radiation intensity as
`
`TAG cos{ cos) —cos( 5] °
`
`ol}_keGer May =M 8a sin@ (4-64)
`
`kl
`
`—,2
`
`—,»l
`
`
`
`
`
`15
`
`15
`
`

`

`4.5 FINITELENGTH DIPOLE 121
`
`The normalized (to 0 dB) elevation powerpatterns, as given by (4-64),
`for /=A/4, 4/2, 34/4, and A are shown plotted in Figure 4.5. The current
`distribution of each is given by (4-56). The power patterns for an infinitesi-
`mal dipole /<\ (U~sin’ 0) is also included for comparison. Asthe length of
`the antenna increases, the beam becomes narrower. Because of that, the
`directivity should also increase with length.
`It
`is found that
`the 3-dB
`
`
`power
`
` (dBdown) *Relative
`
`
`
`a ae 1<<h
`
`tipieecem PALS
`
`1=h2
`
`a i= 3/4
`
`URieoemnnsscve 1=r
`
`Figure 4.5 Elevation plane amplitude patterns for a thin dipole with sinusoidal current
`distribution (/=d _/4,4 /2,3A /4, A).
`
`
`
`16
`
`16
`
`

`

`122 LINEAR WIRE ANTENNAS
`
`beamwidth of each is equal to
`
`<r
`
`3-dB beamwidth=90°
`
`—-3-dB beamwidth=87°
`I=\/4
`(4-65)
`[=\/2—-3-dB beamwidth=78°
`1=3\/4—3-dB beamwidth=64°
`fox
`3-dB beamwidth =47.8°
`
`
`
`As the length of the dipole increases beyond one wavelength (/>A), the
`number of lobes begin to increase. The normalized power pattern for a
`dipole with /=1.25A is shown in Figure 4.6. The currentdistribution for the
`dipoles with /=A/4, 4/2, A, 3A/2, and 2A, as given by (4-56), is shown in
`Figure 4.7.
`
`
`
`Figure 4.6 Elevation plane amplitude pattern for a thin dipole of /=1.25X and
`sinusoidal current distribution.
`
`
`
`17
`
`17
`
`

`

`4.5 FINITELENGTH DIPOLE 123
`
`
`
` io
`
`Current [,
`
`oe sere ce: som. see 1=r/4
`
`[= r/2
`eosccseneee soeei=}R
`
`------ 1=3n/2
`
`1=2K
`
`Figure 4.7 Current distributions along the length of a linear wire antenna.
`
`To find the total power radiated, the average Poynting vector of (4-63)
`is integrated over a sphere of radius r. Thus
`
`Using (4-63), we can write (4-66) as
`
`= ffWor-ds=f"[°a,Way-a,r?sinddOdp
`=sfpsfw.W,,r*sin0dbdo
`Pat =f"[Wovr2sin6dddé
`ess Zoove) —cosl 5)
`_ ol? cos|
`— cos
`cos|
`5
`
`sin@
`"An 0
`
`d@
`
`18
`
`(4-66)
`
`(4-67)
`
`
`
`18
`
`

`

`124 LINEAR WIRE ANTENNAS
`
`After some extensive mathematical manipulations, it can be shown that
`(4-67) reduces to
`
`P= ql {C +In(kl) — C,(kL) + $sin(KI)[S,(2k1)—258,(k1)]
`(4-68)
`+ 4eos( k1)[C +In(kI/2)+ C,(2k1) —2C,(k1)]}
`where C =0.5772 (Euler’s constant) and C,(x) and S;(x) are the cosine and
`sine integrals (see Appendix III) given by
`
`(4-68a)
`C(x)=— f oe has
`CC
`x COS
`(4-68)
`(x)= [4
`4-68b
`S(x)=
`d
`The derivation of (4-68) from (4-67) is assigned as a problem at the end of
`the chapter (Prob. 4.10). C,(x) is related to C;,(x) by
`Cia(x) = In(yx)— C(x) = In(y) +In(x)— C(x)
`= 0.5772 +In(x)— C(x)
`
`
`x sin y
`
`(4-69)
`
`where
`
`Calx)=f(=Jay
`C(x), S,(x) and C,,(x) are tabulated in Appendix IIT.
`The radiation resistance can be obtained using (4-18) and (4-68) and
`can be written as
`
`(4-69)
`
`
`R,= —84 = SE (C+In( kl) — C,(kl) + Zink!)
`
`(4-70)
`x [S,(2k1) —25,(k1)] + 5e0s(k/)
`
`
`x[C +In(kI/2) + C,(2kl1)—2C,(k1)]}
`Shownin Figure 4.8 is a plot of R, as a function of / (in wavelengths) when
`the antennais radiating into free-space (7 ~ 1207).
`
`4.5.4 Directivity
`As wasillustrated in Figure 4.5, the radiation pattern of a dipole becomes
`more directional as its length increases. When the overall length is greater
`than about one wavelength, the numberof lobes increases and the antenna
`
`
`
`19
`
`19
`
`

`

`
`
`Radiation resistance
`
`4.5
`
`FINITELENGTH DIPOLE 125
`
`—_— am ee Directivity
`
` Directivity
`(dimensionless)
`
`0.50
`
`1.00
`
`1.50
`
`2.00
`
`2.50
`
`
`
`
`
`Radiationresistance(ohms)
`
`400
`
`twweSooOo
`
`_ °Oo
`
`0
`
`Figure 4.8 Radiation resistance and directivity of a thin dipole with sinusoidal current
`distribution.
`
`Dipole length (wavelengths)
`
`loses its directional properties. The parameterthat is used as a “figure-of-
`merit” for the directional properties of the antennais the directivity which
`was defined in Section 2.5.
`The directivity was defined mathematically by (2-22), or
`D.=
`F(9,9)| max
`[" [ F(@,9)sineaaae
`
`0 tT
`
`35
`
`pa
`
`0
`
`0
`
`(4-71)
`
`where F(0,$) is related to the radiation intensity U by (2-18), or
`U=UF(6,¢)
`
`(4-72)
`
`From (4-64), the dipole antenna of length / has
`
`cos| —cos @}]—cos| —
`_
`2
`2
`
`(4
`
`(=) 2
`
`(4-73)
`
`and
`
`ol?
`
`Gay° T Sq?
`
`Because the pattern is not a function of , (4-71) reduces to
`___2F(4)|max
`['F(0)sinodo
`
`0
`
`(4-73a)
`
`(4-74)
`
`
`
`20
`
`20
`
`