ANTENNA THEORY
`Analysis and Design
`
`CONSTANTINE A. BALANIS
`West Virginia University
`
`1817
`
`HARPER & ROW, PUBLISHERS, New York
`Cambridge, Philadelphia, San Francisco,
`London, Mexico City, Sao Paulo, Sydney
`
`
`
`Ex.1027
`APPLE INC. / Page 1 of 20
`
`

`

`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 embodied in 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 -
`Antenna theory.
`
`(The Harper & Row series in electrical engineering)
`Includes bibliographical references and index.
`I . Antennas (Electronics)
`I. Title.
`II. Series.
`TK7871.6.B353
`621.38'028'3
`8 1-20248
`ISBN 0-06-040458-2
`AACR2
`
`
`
`Ex.1027
`APPLE INC. / Page 2 of 20
`
`

`

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

`viii CONTENTS
`
`25
`
`37
`
`46
`
`2.3 Radiation Power Density
`2.4 Radiation Intensity
`27
`2.5 Directivity
`29
`2.6 Numerical Techniques
`2.7 Gain
`42
`44
`2.8 Antenna Efficiency
`2.9 Half-Power Beamwidth
`2. 10 Beam Efficiency
`46
`2.11 Bandwidth
`47
`2.12 Polarization
`48
`Linear, Circular, and Elliptical Polarizations; Polarization Loss
`Factor
`53
`Input Impedance
`2.13
`57
`2.14 Antenna Radiation Efficiency
`59
`2.15 Antenna as an Aperture: Effective Aperture
`61
`2.16 Directivity and Maximum Effective Aperture
`2.17 Friis Transmission Equation and Radar Range Equation
`Friis Transmission Equation; Radar Range Equation
`2.18 Antenna Temperature
`67
`References
`70
`71
`Problems
`Computer Program- Polar Plot
`Computer Program-Linear Plot
`Computer Program- Directivity
`
`63
`
`75
`78
`80
`
`82
`
`Chapter 3 Radiation Integrals and Auxiliary Potential Functions
`3.1
`Introduction
`82
`3.2 The Vector Potential A for an Electric Current Source J
`83
`85
`3.3
`The Vector Potential F for a Magnetic Current Source M
`3.4 Electric and Magnetic Fields for Electric (J) and Magnetic (M)
`Current Sources
`86
`Solution of the Inhomogeneous Vector Potential Wave
`Equation
`88
`Far-Field Radiation
`3.6
`93
`3.7 Duality Theorem
`3.8 Reciprocity and Reaction Theorems
`Reciprocity for Radiation Patterns
`References
`99
`Problems
`99
`
`3.5
`
`92
`
`94
`
`100
`
`Chapter 4 Linear Wire Antennas
`4.1
`Introduction
`100
`100
`4.2
`Infinitesimal Dipole
`Radiated Fields; Power Density and Radiation Resistance;
`Near-Field (kr « I) Region; Intermediate-Field (kr > I) Region;
`Far-Field (kr » I) Region; Directivity
`
`
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`Ex.1027
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`

`

`CONTENTS Ix
`
`4.5
`
`4.3
`Small Dipole
`109
`4.4 Region Separation
`112
`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
`4.6 Half-Wavelength Dipole
`130
`4.7 Linear Elements Near or on Infinite Plane Conductors
`132
`Image Theory; Vertical Electric Dipole; Horizontal Electric Dipole
`4.8 Ground Effects
`148
`Vertical Electric Dipole;
`Curvature
`References
`159
`Problems
`159
`Computer Program-Linear Dipole: Directivity, Radiation
`Resistance, and Input Resistance
`162
`
`Horizontal Electric Dipole; Earth
`
`164
`
`Chapter 5 Loop Antennas
`5.1
`Introduction
`164
`5.2
`Small Circular Loop
`164
`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
`5.3 Circular Loop of Constant Current
`176
`Radiated Fields; Power Density, Radiation Intensity, Radiation
`Resistance, and Directivity
`5.4 Circular Loop with Nonuniform Current
`184
`5.5 Ground and Earth Curvature Effects for Circular Loops
`5.6
`Polygonal Loop Antennas
`191
`Square Loop; Triangular, Rectangular, and Rhombic Loops
`Ferrite Loop
`196
`References
`198
`Problems
`199
`Computer Program-Circular Loop: Directivity and
`Radiation Resistance
`201
`
`5.7
`
`188
`
`Chapter 6 Arrays: Linear, Planar, and Circular
`6.1
`Introduction
`204
`6.2 Two-Element Array
`205
`6.3 N-Element Linear Array: Uniform Amplitude and Spacing
`
`204
`
`212
`
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`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
`N-Element Linear Array: Directivity
`229
`Broadside Array; Ordinary End-Fire Array; Hansen-Woodyard
`End-Fire Array
`N-Element Linear Array: Three-Dimensional Characteristics
`235
`N-E/ements 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
`
`Chapter 7 Self- and Mutual Impedances of Linear Elements and
`Arrays, and Finite Diameter Effects (Moment Method)
`283
`7.1
`Introduction
`283
`7.2 Near-Fields of Dipole
`285
`7.3
`Input Impedance of Dipole
`290
`Induced emf Method; Finite Dipole Input Impedance
`7.4 Mutual Impedance Between Linear Elements
`296
`7.5
`Finite Diameter Wires: The Moment Method
`304
`Integral Equation; Moment Method Solution; 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 Biconical Antenna
`323
`Radiated Fields; Input Impedance
`8.3 Triangular Sheet, Bow-Tie, and Wire Simulation
`8.4 Cylindrical Dipole
`332
`Bandwidth; Input Impedance; Resonance and Ground Plane
`Simulation; Radiation Patterns; Equivalent Radii; Dielectric
`Coating
`
`330
`
`322
`
`
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`Ex.1027
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`

`CONTENTS xi
`
`346
`
`416
`
`372
`
`8.5
`Folded Dipole
`340
`8.6 Discone and Conical Skirt Monopole
`8.7
`Sleeve Dipole
`347
`8.8 Matching Techniques
`349
`Stub-Matching; Quarter-Wavelength Transformer; T-Match;
`Gamma Match; Omega Match; Baluns and Transformers
`References
`368
`Problems
`369
`Chapter 9 Traveling Wave and Broadband Antennas
`9.1
`Introduction
`372
`9.2 Traveling Wave Antennas
`372
`Long Wire; V Antenna; Rhombic Antenna
`9.3 Broadband Antennas
`385
`Helical Antenna; Electric-Magnetic Dipole; Yagi-UdaArray of
`Linear Elements; Yagi-Uda Array of Loops
`References
`409
`Problems
`411
`Chapter 1 O Frequency Independent Antennas and Antenna
`Miniaturization
`413
`10. l
`Introduction
`413
`10.2 Theory
`414
`10.3 Equiangular Spiral Antennas
`Planar Spiral; Conical Spiral
`10.4 Log-Periodic Antennas
`423
`Planar and Wire Surfaces; Dipole Array; Design of Dipole Array
`10.5 Fundamental Limits of Electrically Small Antennas
`439
`References
`444
`Problems
`445
`Chapter 11 Aperture Antennas, and Ground Plane Edge Effects
`(Geometrical Theory of Diffraction)
`446
`11.1
`Introduction
`446
`11.2 Field Equivalence Principle: Huygens' Principle
`11.3 Radiation Equations
`454
`11.4 Directivity
`456
`11.5 Rectangular Apertures
`457
`Uniform Distribution on an Infinite Ground Plane; Uniform
`Distribution in Space; TE10-Mode Distribution on an Infinite
`Ground Plane; Beam Efficiency
`11.6 Circular Apertures
`478
`Uniform Distribution on an Infinite Ground Plane; TEn -Mode
`Distribution on an Infinite Ground Plane; Beam Efficiency
`11.7 Microstrip Antennas
`487
`Radiated Fields; Radiation Conductance; Directivity; Bandwidth;
`Arrays; Circular Polarization
`
`447
`
`
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`Ex.1027
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`

`xii CONTENTS
`
`496
`11.8 Babinet's Principle
`11.9 Ground Plane Edge Effects: The Geometrical Theory of
`502
`Diffraction
`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
`
`532
`Chapter 12 Horns
`12.1
`Introduction
`532
`532
`12.2 £-Plane Sectoral Horn
`Aperture Fields; Radiated Fields; Directivity
`12.3 H-Plane Sectoral Horn
`550
`Aperture Fields; Radiated Fields; Directivity
`12.4 Pyramidal Horn
`565
`Aperture Fields, Equivalent, and Radiated Fields; Directivity;
`Design Procedure
`12.5 Conical Horn
`12.6 Corrugated Horn
`12.7 Phase Center
`587
`References
`589
`Problems
`590
`
`577
`579
`
`593
`
`Chapter 13 Reflectors and Lens Antennas
`13.1
`Introduction
`593
`13.2 Plane Reflector
`594
`13.3 Corner Reflector
`594
`90° Corner Reflector; Other Corner Reflectors
`13.4 Parabolic Reflector
`604
`Front-Fed Parabolic Reflector; Cassegrain Reflectors
`13.5 Spherical Reflector
`642
`13.6 Lens Antennas
`646
`Lenses with n > I; Lenses with n < I; Lenses with Variable Index
`of Refraction
`References
`Problems
`
`654
`656
`
`Chapter 14 Antenna Synthesis and Continuous Sources
`14.1
`Introduction
`658
`659
`14.2 Continuous Sources
`Line-Source; Discretization of Continuous Sources
`14.3 Schelkunoff Polynomial Method
`661
`14.4 Fourier Transform Method
`666
`Line-Source; Linear Array
`
`658
`
`
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`Ex.1027
`APPLE INC. / Page 8 of 20
`
`

`

`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
`15.2 Antenna Ranges
`704
`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
`725
`Radiation Efficiency
`Impedance Measurements
`725
`Current Measurements
`727
`Polarization Measurements
`Scale Model Measurements
`References
`734
`
`15.5
`15.6
`15.7
`15.8
`15.9
`15.10
`
`723
`
`728
`733
`
`sin(x)
`.
`Appendix I f(x) = -~
`X
`-1 sin(Nx) I _
`
`Appendix II fN(x) - Nsin(x), N - 1,3,5,10,20
`
`737
`
`740
`
`Appendix Ill Cosine and Sine Integrals
`
`743
`
`Appendix IV Fresnel Integrals
`
`748
`
`Appendix V Bessel Functions
`
`755
`
`Appendix VI Identities
`
`768
`
`Appendix VII Vector Analysis
`
`771
`
`Appendix VIII Television and Radio Frequency Spectrum
`
`781
`
`Index
`
`783
`
`
`
`Ex.1027
`APPLE INC. / Page 9 of 20
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`

`

`2.13
`
`INPUT IMPEDANCE 53
`
`r
`I
`I
`I
`I
`L
`
`7
`I
`I
`I
`I
`_..J
`
`PLF = I.Ow • Pa * I 2 = I
`(aligned)
`
`PLF = I.Ow • Pa *12 = cos 2 I/Ip
`(rotated)
`
`(a) PLF for transmitting and receiving
`aperture antennas
`
`4 I
`
`I
`
`I
`I
`L ____ _j
`PLF = I.Ow • Pa*1 2 = 0
`(orthogonal)
`
`\
`\
`\
`
`\
`
`\
`
`\
`
`\
`
`\
`\
`
`\
`
`PLF = IPw • Pa*l 2 = I
`(aligned)
`
`PLF = IPw • Pa *12 = cos2 I/Ip
`(rotated)
`
`PLF = I.Ow • Pa*1 2 = 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 account all loss factors to ensure a successful
`operation of the system.
`
`2.13
`
`INPUT IMPEDANCE
`
`Input impedance is defined as "the impedance presented by an antenna at its
`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 we are primarily interested in the input impedance at
`a pair of terminals which are the input terminals of the antenna. In Figure
`2.18(a) these terminals are designated as a - b. The ratio of the voltage to
`current at these terminals, with no load attached, defines the impedance of
`
`
`
`Ex.1027
`APPLE INC. / Page 10 of 20
`
`

`

`54 FUNDAMENTAL PARAMETERS OF ANTENNAS
`
`Generator
`(Zg)
`
`Antenna
`
`a
`
`b
`
`Radiated
`wave
`
`(a) Antenna in transmitting mode
`
`Vg ~
`
`a
`
`R,
`
`X
`l
`
`b •
`
`R1, l
`J
`
`R,
`
`x,,
`
`(b) Thevenin equivalent
`
`a •
`
`b •
`
`c,
`
`(c) Norton equivalent
`Figure 2.18 Transmitting antenna and its equivalent circuits.
`
`the antenna as
`
`(2-72)
`
`where
`ZA =antenna impedance at terminals a-b
`RA =antenna resistance at terminals a-b
`XA =antenna reactance at terminals a - b
`In general the resistive part of (2-72) consists of two components; that is
`
`(ohms)
`(ohms)
`
`(ohms)
`
`(2-73)
`
`
`
`Ex.1027
`APPLE INC. / Page 11 of 20
`
`

`

`2.13
`
`INPUT IMPEDANCE 55
`
`(2-74)
`
`(ohms)
`
`(ohms)
`
`(A)
`
`<2-75>
`
`where
`R,= radiation resistance of the antenna
`R L = loss resistance of the antenna
`The radiation resistance will be considered in more detail in later chapters,
`and it will be illustrated with examples.
`If we assume that the antenna is attached to a generator with internal
`impedance
`Zg=Rg+)Xg
`where
`R g = resistance of generator impedance
`Xg = reactance of generator impedance
`and the antenna is used in the transmitting mode, we can represent the
`antenna and generator by an equivalent circuit* shown in Figure 2.18(b). To
`find the amount of power delivered to R, for radiation and the amount
`dissipated in R L as heat (/2 R Lf2), we first find the current developed
`within the loop which is given by
`vg
`vg
`vg
`Jg= z1 = zA+zg - (R,+RL+Rg)+J(xA+xg)
`and its magnitude by
`III=
`g
`
`(2-75a)
`
`IVgl
`[(R,+RL+Rg)2+(xA+xg)2]'12
`where Vg is the peak generator voltage. The power delivered to the antenna
`for radiation is given by
`
`2
`Iv 1
`p-1-12R _ _ g_
`,- 2 1 gl ,-
`[
`2
`
`·
`R
`r
`(R,+RL+Rg) 2+(xA+xg) 2
`
`l
`
`(W) (2-76)
`
`(W) (2-77)
`
`The remaining power is dissipated as heat on the internal resistance R g of
`the generator, and it is given by
`
`(W)
`
`(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 Joss resistance
`cannot be represented in series with the radiation resistance.
`
`
`
`Ex.1027
`APPLE INC. / Page 12 of 20
`
`

`

`56 FUNDAMENTAL PARAMETERS OF ANTENNAS
`
`The maximum power delivered to the antenna occurs when we have
`conjugate matching; that is when
`Rr+ RL= R g
`XA=- Xg
`
`(2-79)
`
`(2-80)
`
`For th.is case
`
`P = Wl
`8
`g
`
`[
`
`R g
`( R r + R L) 2
`
`]=IV/ [ 1 ]=IV/
`
`8
`
`R r + R L
`
`8R g
`
`From (2-81)- (2-83), it is clear that
`
`p - p + p - Wl
`8
`L-
`g- r
`
`[
`
`R g
`(Rr + RL)2 -
`
`]·
`(Rr + RL)2
`
`8
`
`The power supplied by the generator during conjugate matching is
`
`(2-81)
`
`(2-82)
`
`(2-83)
`
`(2-84)
`
`]-IV/ [ Rr+ RL
`V * l IV 12
`Ps= 2Vgl /= 2Vg 2(Rr : RL) =+ Rr+ RL
`
`1
`
`1
`
`[
`
`[
`
`1
`
`]
`
`{W)
`
`(2-85)
`
`Of the power that is provided by the generator, half is dissipated as heat in
`the internal resistance (R g) of the generator and the other half is delivered
`to the antenna. Th.is only happens when we have conjugate matching. Of the
`power that is delivered to the antenna, part is radiated through the mecha(cid:173)
`nism provided by the radiation resistance and the other is dissipated as heat
`which influences part of the overall efficiency of the antenna. If the antenna
`is lossless ( e cd= 1 ), then half of the total power supplied by the generator is
`radiated by the antenna during conjugate matching. In th.is section we have
`assumed a perfect match between the antenna and the interconnecting
`transmission line (er= 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 Vr which is analogous to Vg of the transmitting mode. The Thevenin
`equivalent circuit of the antenna and its load is shown in Figure 2.19(b) and
`the Norton equivalent in Figure 2.19(c). The discussion for the antenna and
`its load in the receiving mode parallels that for the transmitting mode.
`The input impedance of an antenna is generally a function of frequency.
`Thus the antenna will be matched to the interconnecting transmission line
`
`
`
`Ex.1027
`APPLE INC. / Page 13 of 20
`
`

`

`2.14 ANTENNA RADIATION EFFICIENCY 57
`
`Load
`(Zr)
`
`Antenna
`a
`
`b
`
`(a) Antenna in receiving mode
`
`a
`
`Incident
`wave
`
`Rr
`
`R,
`
`(b) Thevenin equivalent
`
`Gr
`
`Br
`
`a
`I
`
`b
`
`G,
`
`(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 number of practical
`antennas have been investigated analytically. For many others, the input
`impedance has been determined experimentally.
`
`2.14 ANTENNA RADIATION EFFICIENCY
`
`The antenna efficiency 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
`
`
`
`Ex.1027
`APPLE INC. / Page 14 of 20
`
`

`

`120 LINEAR WI RE ANTENNAS
`
`The pattern multiplication for continuous sources is analogous to the
`pattern multiplication of (6-5) for discrete-element antennas (arrays).
`For the current distribution of (4-56), (4-58a) can be written as
`Eo '.::e )TJ kloe-jkr sin o[Jo sin[k ( .!._ + z')] e+jkz'cosO dz'
`_ 112
`11\ in[ k ( i- z')] e+jkz'cos O dz']
`
`477r
`
`2
`
`+ fa +
`Each one of the integrals in ( 4-60) can be integrated using
`eax
`f
`[ a sin{,Bx+y ) - ,Bcos{ ,Bx + y))
`eax sin{,Bx+y) dx=
`2
`a + ,B
`2
`
`where
`a =jkcos O
`,B= ±k
`y = kl/ 2
`After some mathematical manipulations, (4-60) takes the form of
`
`(4-60)
`
`(4-61)
`
`(4-61 a)
`(4-61b)
`(4-61c)
`
`(4-62a)
`
`In a similar manner, or by using the established relationship between
`the E0 and Hq, in the far-field as given by (3-58b) or (4-27), the total Hq,
`component can be written as
`
`~- Eo ~- .loe -ikr [cos(~/ cosO ) -cos (~) l
`
`H
`q,
`
`T/
`
`1 - - _...;..._ _ _ _ _ .;.._.;_
`sin0
`277r
`
`(4-62b)
`
`4.5.3 Power Density, Radiation Intensity, and Radiation
`Resistance
`For the dipole, the average Poynting vector can be written as
`
`Wav= l Re[E X H*] = l Re [ a0 E0 X aq,Hq, *] = l Re[ a 0 E0 Xaq, E;*]
`_ ~ [ cos ( ;' cos O ) - cos ( ; ' ) l 2
`I I ol2 cos ( ;' cos o ) - cos ( ;' ) l 2
`
`_ , _I
`_ ,
`w av-a,Jtv,,v-a, 2 IEol - 11
`T/
`and the radiation intensity as
`
`2
`
`2 2
`877 r
`
`_
`_
`U-r Wav_T/ __ 2
`2
`877
`
`. 0
`sm
`
`. 0
`sm
`
`(4-63)
`
`(4-64)
`
`
`
`Ex.1027
`APPLE INC. / Page 15 of 20
`
`

`

`4.5 FINITE LENGTH DIPOLE 121
`
`The normalized (to O dB) elevation power patterns, as given by (4-64),
`for l = t../ 4, t../ 2, 3t../ 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(cid:173)
`mal dipole /« A ( U ~ sin2 0) is also included for comparison. As the 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
`
`...
`" ;!:
`8. '2
`"
`;!:
`
`180°
`
`- - - - - l << X
`
`- - - - / =X/4
`
`- - - - / = X/2
`
`I = 3A/4
`-
`-
`-
`- -
`.............. / = X
`
`Figure 4.5 Elevation plane amplitude patterns for a thin dipole with sinusoidal current
`distribution(/ = A/ 4, A/ 2,3A / 4, A).
`
`
`
`Ex.1027
`APPLE INC. / Page 16 of 20
`
`

`

`122 LINEAR WIRE ANTENNAS
`
`beamwidth of each is equal to
`!«>..
`3-dB beamwidth = 90°
`3-dB beamwidth= 87°
`
`!= >../ 4
`
`/ = >../2
`
`3-dB beamwidth=78°
`
`(4-65)
`
`3-dB beamwidth=64°
`/=3>../4
`3-dB beamwidth=47.8°
`!= >..
`As the length of the dipole increases beyond one wavelength (I>>..), the
`number of lobes begin to increase. The normalized power pattern for a
`dipole with /= 1.25>.. is shown in Figure 4.6. The current distribution for the
`dipoles with /= >../ 4, >../ 2, >.. , 3>../ 2, and 2>.., as given by (4-56), is shown in
`Figure 4.7.
`
`180°
`Figure 4.6 Elevation plane amplitude pattern for a thin dipole of / = 1.25A and
`sinusoidal current distribution.
`
`
`
`Ex.1027
`APPLE INC. / Page 17 of 20
`
`

`

`4.5 FINITE LENGTH DIPOLE 123
`
`/
`
`\
`.. ,
`\
`\.·
`.. ·· \
`..
`
`\
`
`Io
`
`Current I ,
`
`1/2
`
`1/2
`
`- - - - - - 1 = X/4
`- - - - / = X/ 2
`•••••• •• •••• ••• l = A
`
`- -
`
`-
`
`-
`
`-
`
`- I= 3X/2
`
`- - - -1 =2X
`
`Figure 4.7 Current distributions along the length of a linear wire antenna.
`
`S
`
`O
`
`0
`
`To find the total power radiated, the average Poynting vector of (4-63)
`is integrated over a sphere of radius r. Thus
`Prad = 11 Wav· ds= 12
`.,, 1.,, a,w.iv· a,r 2 sin0 dOdcp
`= L2.,,1.,,w.ivr2 sin0d0dcp
`
`0
`0
`Using (4-63), we can write (4-66) as
`
`(4-66)
`
`(4-67)
`
`
`
`Ex.1027
`APPLE INC. / Page 18 of 20
`
`

`

`124 LINEAR WIRE ANTENNAS
`After some extensive mathematical manipulations, it can be shown that
`( 4-67) reduces to
`2
`
`{ C + In( kl)- c;( kl)+ ½sin( kl) [ si(2kl)-2S;( kl)]
`
`prad = r, I :0
`+ ½cos( kl) [ C + In( kl / 2) + Ci(2kl)-2Ci( kl)]}
`(4-68)
`where C =0.5772 (Euler's constant) and Clx) and S;(x) are the cosine and
`sine integrals (see Appendix III) given by
`C;(x)=- Joo cosy dy= f xcosy dy
`00 y
`Y
`Si ( x) = lax si~ y dy
`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 Ci/ x) by
`Ciix) = In( yx )-C;(x) = In( y )+ ln(x )-C;(x)
`= 0.5772 + ln(x )- C;(x)
`
`;
`
`X
`
`(4-68a)
`
`( 4-68b)
`
`(4-69)
`
`where
`
`Clx), S;(x) and Cin(x) are tabulated in Appendix III.
`The radiation resistance can be obtained using ( 4-18) and ( 4-68) and
`can be written as
`
`(4-69a)
`
`{ C + ln(kl) - Ci( kl) + 15in(kl)
`
`r,
`2P
`R, = ~ = -
`,,,
`I 1ol
`2
`X [ Si(2kl) - 2S;( kl)] + ½cos( kl)
`
`X [ C + ln(k/ / 2) + Ci(2kl)-2Ci(kl)]}
`
`(4-70)
`
`Shown in Figure 4.8 is a plot of R, as a function of 1 (in wavelengths) when
`the antenna is radiating into free-space (r, ~ 1201T).
`
`4.5.4 Directivity
`As was illustrated 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 number of lobes increases and the antenna
`
`
`
`Ex.1027
`APPLE INC. / Page 19 of 20
`
`

`

`- - - Radiation resistance
`-
`-
`-
`
`.., Directivity -I \
`
`I
`I
`
`\
`\
`
`400
`
`,;;;-
`] 300
`-3.
`8
`C
`"'
`-~ 200
`e
`C
`.!2
`'io
`~ 100
`c:,:;
`
`4.5 FINITE LENGTH DIPOLE 125
`
`4.0
`
`3.5
`
`~
`
`~ .,
`3.0 c
`0 ·;;;
`C .,
`E
`~
`~
`2.0 ]
`u e
`0
`
`2.5
`
`1.5
`
`0
`
`0.50
`
`I.SO
`1.00
`2.00
`Dipole length (wavelengths)
`Figure 4.8 Radiation resistance and directivity of a thin dipole with sinusoidal current
`distribution.
`
`2.50
`
`1.0
`3.00
`
`loses its directional properties. The parameter that is used as a "figure-of(cid:173)
`merit" for the directional properties of the antenna is the directivity which
`was defined in Section 2.5.
`The directivity was defined mathematically by (2-22), or
`F( t}' q>) I max
`1 1 F( 0, q>) sin 0 d 0 d q>
`Do= 4~-2-w-n-----=-='---
`
`0
`
`0
`
`(4-71)
`
`where F( 0, q>) is related to the radiation intensity U by (2-18), or
`U = U0 F( 0, q>)
`From (4-64), the dipole antenna of length/ has
`- [ cos ( ~cost} ) - cos ( ~ ) ]
`_
`F( 0, q>) - F( 0) -
`
`2
`
`. O
`sm
`
`and
`
`Because the pattern is not a function of q>, ( 4-71) reduces to
`2F( O)I max
`0 = _n __ ....:....;;;."-----
`D
`1 F( 0) sin 0 d 0
`
`0
`
`(4-72)
`
`(4-73)
`
`(4-73a)
`
`(4-74)
`
`
`
`Ex.1027
`APPLE INC. / Page 20 of 20
`
`