Antenna Handbook .
 ....
VOLUME II
ANTENNA THEORY
Edited by
V.T.Lo Electromagnetics Laboratory Department of Elect...
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Antenna Handbook .
 ....
VOLUME II
ANTENNA THEORY
Edited by
V.T.Lo Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois Urbana
s. W. Lee Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois Urbana
InmiI VAN NOSTRAND REINHOLD ~
_______ New York
Copyright e 1993 by Van Nostrand Reinhold Published by Van Nostrand Reinhold
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All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any meansgraphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systemswithout the written permission of the publisher. ARCFF 95 96 97 98 99 ... 8 7 6 5 4 3 2 Ubrary of Congress CataloginginPublication Data
The antenna handbook/edited by Y.T. Lo and S.W. Lee p. cm. Includes bibliographical references and indexes. Contents: v. I. Fundamentals and mathematical techniquesv. 2. Antenna theoryv. 3. Applicationsv. 4. Related topics. ISBN 0442015925 (v. 1).ISBN 0442015933 (v. 2).ISBN 0442015941 (v. 3).ISBN 044201596 (v. 4). 1. Antennas (Electronics) I. Lo, Y.T. II. Lee, S. W. TK7871.6.A496 1993 936502 621.382' 4dc20 elP
Contents ....
Volume II
ANTENNA THEORY
S. Radiation From Apertures
53
E. V. lull 6. Receiving Antennas P. K. Park and C. T. Tai
61
7. Wire and Loop Antennas L. W. Rispin and D. C. Chang
71
8. Horn Antennas Constantine A. Balanis
81
9. FrequencyIndependent Antennas Paul E. Mayes
91
10. Microstrip Antennas William F. Richards
101
ll. A.rray Theory Y. T. Lo
llI
12. The Design of WaveguideFed Slot Arrays Robert S. Elliott
121
13. Periodic Arrays R. l. Mailloux
131
14. Aperiodic Arrays
141
Y. T. Lo IS. Reflector Antennas Y. RahmatSamii
151
16. Lens Antennas 1.1. Lee
161
Appendices A. B. C. D. E. F.
Physical Constants, International Units, Conversion of Units, and Metric Prefixes The Frequency Spectrum Electromagnetic Properties of Materials Vector Analysis VSWR Versus Reflection Coefficient and Mismatch Loss Decibels Versus Voltage and Power Ratios
Index
A3 81 CI DI EI FI
II
Preface .... During thepast decades, new demands for sophisticated spaceage communication and remote sensing systems prompted a surge of R&D activities in the antenna field. There has been an awareness, in the professional community, of the need for a systematic and critical review of the progress made in those activities. This is evidenced by the sudden appearance of many excellent books on the subject after a long dormant period in the sixties and seventies. The goal of this book is to compile a reference to complement those books. We believe that this has been achieved to a great degree. A book of this magnitude cannot be completed without difficulties. We are indebted to many for their dedication and patience and, in particular, to the fortytwo contributing authors. Our first thanks go to Mr. Charlie Dresser and Dr. Edward C. Jordan, who initiated the project and persuaded us to make it a reality. After smooth sailing in the first period, the original sponsoring publisher had some unexpected financial problems which delayed its publication three ·years. In 1988, Van Nostrand Reinhold took over the publication tasks. There were many unsung heroes who devoted their talents to the perfection of the volume. In particular, Mr. Jack Davis spent many arduous hours editing the entire manuscript. Mr. Thomas R. Emrick redrew practically all of the figures with extraordinary precision and professionalism. Ms. Linda Venator, the last publication editor, tied up all of the loose ends at the final stage, including the preparation of the Index. Without their dedication and professionalism, the publication of this book would not have been possible. Finally, we would like to express our appreciation to our teachers, students, and colleagues for their interest and comments. We are particularly indebted to Professor Edward C. Jordan and Professor George A. Deschamps for their encouragement and teaching, which have had a profound influence on our careers and on our ways of thinking about the matured field of electromagnetics and antennas.
This Preface was originally prepared for the first printing in 1988. Unfortunately, it was omitted at that time due to a change in the publication schedule. Since many readers questioned the lack of a Preface, we are pleased to include it here, and in all future printings.
Prefaee to the Second Printing Since the publication of the first printing, we have received many constructive comments from the readers. The foremost was the bulkiness of a single volume for this massive book. The issue of dividing the book into multivolumes had been debated many times. Many users are interested in specific topics and not necessarily the entire book. To meet both needs, the publisher decided to reprint the book in multivolumes. We received this news with great joy, because we now have the opportunity to correct the typos and to insert the original Preface, which includes a heartfelt acknowledgment to all who contributed to this work. We regret to announce the death of Professor Edward C. Jordan on October 18, 1991.
PART B
 . Antenna Theory
Chapter 5
Radiation from Apertures E. V. Jull University of British Columbia
CONTENTS 1. Alternative Formulations for Radiation Fields
2.
3. 4. 5. 6.
PlaneWave Spectra 55 Equivalent Currents 58 Radiation Patterns of Planar Aperture Distributions Approximations 510 Rectangular Apertures 511 Circular Apertures 520 NearField Patterns 524 Aperture Gain Effective Area and Aperture Efficiency NearField Axial Gain and Power Density References
55 510
526 527 529 534
53
Edward V. Jull was born in Calgary, Alberta, Canada. He received a BSc degree in engineering physics from Queen's University, Kingston, Ontario, in 1956, a PhD in electrical engineering in 1960, and a DSc (Eng.) in 1979, both from the University of London, England. In 195657 and 196172 he was a research officer in the microwave section and later the antenna engineering section of the Division of Electrical Engineering of the National Research Council of Canada Laboratories in Ottawa. During 196365 he was a guest worker in the Electromagnetics Institute of the Technical University of Denmark and the Microwave Institute of the Royal Institute of Technology, Stockholm, Sweden. In 1972 he joined the University of British Columbia, Vancouver, Canada, where he is now a professor in the Department of Electrical Engineering. In 1964 Dr. Ju11 was a joint winner of the IEEE Antennas and Propagation Society Best Paper Award. He has been chairman of Canadian Commission VI for the International Union of Radio Science (URSI), chairman of the Canadian National Committee for URSI, an associate editor of Radio Science, and an international director of the Electromagnetics Society. He is currently a vice president of URSI and is the author of Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).
1. Alternative Formulations for Radiation Fields An aperture antenna is an opening in a surface designed to radiate. Examples are radiating slots, horns, and reflectors. It is usually more convenient to calculate aperture radiation patterns from the electromagnetic fields of the aperture rather than from the currents on the antenna. There are now basically two methods for doing this. Traditionally the pattern has been derived from the tangential electric and magnetic fields in the aperture. This aperture field method is an electromagnetic formulation of the HuygensKirchhoff method of optical diffraction. In application it is convenient and accurate for the forward pattern of large apertures. More recently the pattern has also been derived from fields associated with rays which pass through the aperture and rays diffracted by the aperture edges. Its origins can be traced to the early ideas on optical diffraction of Young as more recently formulated by Keller [1] in his geometrical theory of diffraction. It is particularly useful in deriving the radiation pattern in the lateral and rear directions and is described in Chapter 4. This chapter deals only with the derivation of radiation patterns from the tangential fields in the aperture. Two methods of formulating the radiation it:ttegrals are given in this section. They lead to the same result but differ in their concepts of radiation from apertures.
PlaneWave Spectra This approach has the advantages of conceptual simplicity for the radiative fields and completeness in its inclusion of the reactive fields of the aperture [25]. It uses the fact that any radiating field can be represented by a superposition of plane waves in different directions. The amplitude of the plane waves in the various directions of propagation, or the spectrum function, is determined from the tangential fields in the aperture. This spectrum function is the farfield radiation pattern of the aperture for radiation in real direction angles. The reactive aperture fields are represented by the complex directions of propagation in the total spectrum function. The method is most appropriate for planar apertures. If the aperture lies in the z = 0 plane of Fig. 1 and radiates into z > 0, components Ex and Ey of the electric field in z ::: 0 can be written in terms of their corresponding spectrum functions Px and Pyas
If a,/3 are the directions of propagation of each plane wave in Fig. 1, then kx = k sin a cos /3, ky = k sin a sin /3, and k z = k cos a are the components of the propagation vector k for each plane wave. It is necessary to specify 55
Antenna Theory x
y
Fig. 1. Radiation from an aperture in the z = 0 plane: a plane wave radiates in a direction defined by the angles a and p, and the coordinates of a farfield point are (r, 0, t/J).
when k} when k x 2
+ k/ < k 2 + ky 2 > k 2
(2)
so for real angles a,p (with k} + k/ < k 2 ) there is radiation, and for complex angles a,fJ (with k} + k/ > k 2 ), the fields decay exponentially outwards from the aperture. Putting z = 0 in (1) and inverting the transforms shows the relation between the spectrum functions and the aperture fields:
(3)
Each plane wave has a z component associated with its x and y components in (1) and their relative magnitudes follow from k·E = 0 for each plane wave. Thus the total field is
To evaluate (4) at large distances from the aperture it is convenient to first convert to spherical coordinates, then apply stationary phase integration to the two double integrals. The stationary point is a = 8, fJ = f/J, and the final result for kr » 1 is
57
Radiation from Apertures . jkr
E(r,8,q,)
= Je;"r
.
[9(Pxcosq, + Pysinq,)  +cos8(Pxsinq,  Pycosq,)] (5)
where
Px} { Py
=
I
co
co
1
{EiX,Y,O)} co Ey(x,y, O) co
ejk(xsin8cosq,+ysin8sinq,) dx
dy
(6)
The far magnetic field follows from
H(r, 8, q,) = Yor x E(r, 8, q,)
(7)
where Yo = VEo/Jlo is the freespace wave admittance. Equation 5 provides the complete farfield radiation pattern from the Fourier transforms (6) of the tangential electric field in the aperture. It is also possible to use instead the tangential magnetic field in the aperture. Then, in terms of the spectrum functions
Qx} = { Qy
II co
co
co
co
the total electric field for kr
D)}
{HiX,y, Hy(x,y,O)
»
ejk(xsin8cosq,+ysin8sinq,) dxdy
(8)
1 is
where = means asymptotically equal and Zo = VJlo/Eo is the freespace wave impedance. An equivalent result in terms of both electric and magnetic tangential components of the aperture field is half the sum of (5) and (9), i.e., . jkr
E(r,8,q,) = J~;"r {9[Pxcosq, + Pysinq,)  Zocos8(Qxsinq,  Qycosq,)]  +[(Pxsinq,  Pycosq,)cos8 + Zo(Qxcosq, + Qysinq,)]} (10) in which Px, Py and Qx, Qy are defined by (6) and (8), respectively. This superposition of electric and magnetic current sources provides a field which satisfies Huygens' principle in that radiation into z < is suppressed. The three expressions (5), (9), and (10) all yield the exact farfield pattern from the exact aperture field integrated over the entire aperture plane. Usually electric fields are more convenient to measure and calculate than magnetic fields, so (9) is rarely used. The choice between (5) and (10) should depend on how well the true boundary conditions are satisfied by whichever approximations are used. For example, it is convenient to assume that the field vanishes in the aperture plane outside the aperture. Then (5) should be used for apertures moun~ed in a large
°
Antenna Theory conducting plane as the boundary conditions on the conductor are rigorously satisfied. For apertures not in a conducting plane, however, (5) with this assumption generally yields less accuracy near the aperture plane than (10). For apertures which are large in wavelengths the pattern is well predicted away from the aperture plane by all of these expressions. Then the aperture electric and magnetic fields are related by essentially freespace conditions, i.e., Eix, y, 0) = Z"Hy(x,y,O), Ey(x,y,O) = Z"Hix,y,O) and Qx = Z,,Ipy, Qy = Z,,Ipx' Equations 9 and 10 become, respectively, . jkr
E(r, 0, q,) = )e;'r [9(Pxcos q, + Pysin q,)cos 0  +(Pxsin q,  Pycosq,)] (11) and . jkr
E(r,O,q,)
= )~;'r
(1 + cosO)[9(Pxcosq> + Pysinq,)  +(Pxsinq,  Pycosq,)] (12)
For small angles of 0, cosO == 1 and the three expressions (5), (11), and (12) yield essentially identical results in that region of the pattern where their accuracy is highest. All have less precision at angles far off the beam axis, where (12) yields values which are the average of (5) and (11). As (12) is in terms of an aperture electric field which vanishes in the rear (0 = .7l) direction, it is most commonly used for larger apertures, such as horns or reflectors, which are not in a conducting screen. Equivalent Cu"ents
The fields of sources within a surface S of Fig. 2 can be calculated from the tangential electric and magnetic fields Es and Hs of the sources on S. Alternatively, these surface fields may be replaced by equivalent currents [3,6,7]. An electric surface current density Js = it x Hs , where it is a unit vector normally outward from S, represents the tangential magnetic fields. Their contribution to the magnetic vector potential 1 A ='
f
jkR
JsedS 4.71 s R
(13)
gives a magnetic field H = V x A and an electric field E
= .1V )WE"
x V x A
(14)
Similarly, a magnetic surface current density Ks = Es x it represents the tangential electric fields on S and the resulting electric vector potential F =1
f
kR
e/ 4.71 s KdS s R
(15)
Radiation from Apertures
59
Fig. 2. Coordinates for calculation of radiation from a surface S.
provides an electric field E
= v
x F
(16)
and a magnetic field U = V x V x F/(jwp.(I). Contributions from both electric and magnetic current sources give the total electric and magnetic fields outside S:
E
= v x
F + .I_V x V x A JWE(I
U = V x A + .I_V x V x F Jwp.(1
(17)
(18)
With this superposition of the fields of electric and magnetic surface current densities radiation into the region enclosed by S is suppressed and Huygens' principle satisfied. There will also be no fields inside S if that region is a perfect conductor. Then Ks = 0 on S and the total electric field outside S can be calculated from (14) with 2Js in (13). As discussed previously, under "PlaneWave Spectra," the fields can be calculated from the tangential electric or magnetic fields on S or from a superposition of the two sets of fields and all three methods yield the same result if the boundary conditions are rigorously satisfied. Usually Es and Us are approximated by incident fields, and scattered fields on S are neglected. The choice between the three methods should depend on the convenience of accurately approximating the true boundary conditions. At distances r much larger than the maximum dimension of S in Fig. 2, R == r  r· r' and (13) and (15) become
Antenna Theory
510
{FA}
= _eikrf _
4Jlr
{J }
eikt·r' dS
S
(19)
s KS
where r = rtr and r' is the vector distance from the origin to dS. Also (14) and (16) becon1t, in the far field in spherical coordinates, respectively, E(r, 0, f/J) = jw/lo(6A(J = jkF x
+ +Aq,)
(20) (21)
r
Consequently the electric field of electric and magnetic current sources in the far field is (22) and the corresponding magnetic fields are given by (7). Again, if electric or magnetic currents alone are used to calculate the total fields from (20) or (21), a factor of 2 must be included in the right side of (19). With the surface S in the plane z = 0 of Fig. 2 and radiation into z > 0, n = Z in (19), which becomes (23a)
(23b) in which Px , Py and Qx, Qy are defined by (6) and (8). Using (23a) in (20) with a factor of 2 gives (9). Equation 23b in (21) with a factor of 2 gives (5), and (23a) and (23b) in (22) yields (10). Thus the equivalentcurrent and planewave spectrum formulations provide identical results for the radiation fields of an aperture. The equivalent current method is simpler mathematically, but does not account for the reactive fields of the aperture.
2. Radiation Patterns of Planar Aperture Distributions The expressions (5), (8), and (10) for the radiating far fields of an aperture in terms of the Fourier transforms of the tangential electric and magnetic aperture fields (6) and (8) are exact, but approximations are required in their application. Approximlltions
In obtaining the approximations the usual assumptions are the following: (a)
The integration limits in (6) and (8) are the antenna aperture dimensions, i.e., fields in the aperture plane outside the aperture are assumed negligible.
Radiation from Apertures
(b) (c)
(d)
511
The aperture field is assumed to be the incident field from the antenna feed, i.e., scattered fields in the aperture are assumed negligible. Aperture electric and magnetic fields are assumed related by freespace conditions, i.e., the aperture is assumed large in wavelengths. This is so in (11) and (12) but not in (5), (9), and (to). Aperture fields are assumed separable in the coordinates of the aperture. Fortunately this assumption applies in many antenna designs; otherwise numerical integration of double integrals is usually required.
Clearly, the accuracy of the final result will depend on the degree to which all of the above assumptions are satisfied.
Recmngular Apertures The Uniform Aperture DistributionIf the rectangular aperture of Fig. 3 is large and not in a conducting plane, its far field is conveniently calculated from (12) (assumption c above). Each component of aperture electric field can be dealt with separately. From (12) with Py = 0, the x component of aperture field produces the far field E(r,8,q,) = A
f
UI2
f
u12
bl2
EAx,y,O)ei(k,x+kzY)dxdy
(24)
b12
where
A
= A(r,8,q,) = j
e jkr 2lr (1
A
+ cos8)(6cosq,
A
 cl»sinq,)
x
~+~~z
y
Fig. 3. Coordinates of a rectangular aperture in the z = 0 plane.
(25)
Antenna Theory
512
and kl = ksin 8 cos lP
(26)
kl = k sin 8 sin lP
Assumption a has been used in (24). For separable aperture fields (assumption d) EAx,y, 0) = EoEI(X)El(Y), where EI(x) and El(y) are the field distributions
normalized to the field Eo. Then (24) becomes (27) where (28)
(29) It is now necessary to choose an aperture field (assumption b). For the ideal case of a field uniform in amplitude and phase across the aperture, EI(x) = El(y) = 1 and
sin(kJa/2) k l a/2
(30)
F (k ) = b sin(kl b/2)
(31)
F (k) = I
l
I
a
k l b/2
l
This specifies the complete threedimensional radiation pattern. For practical reasons it is usually the pattern in the two principal planes which are of major interest. These are the Eplane pattern in the (x, z) or lP = plane,
°
E ( r, 8 , 0)
=
8jeikr (1 2Ar
+ cos
and the Hplane pattern in the (y, z) or
+e2Ar
ikr
E(r, 8,31"/2) =
(1
lP
8) bsin[(31"a/A) sin 8] a (31"a/A) sin (J
(32)
= 31"/2 plane
sin[(31"b/A) sin 8]
+ cos (J) ab (31"b/A) sin (J
(33)
These patterns are of the same form but scaled in (J according to the aperture dimensions in their respective planes. If the aperture is large (a, b »A), the main beam and first side lobes are contained in a small angle (J, so 1 + cos (J ::: 2 and the
513
Radiation from Apertures
function (sin u)/u, where u = (.1lall) sin 0, determines the pattern. This function is plotted in Fig. 4. The first null in the pattern occurs at u = .1l or 0 = sin' (lIa). Hence the full width of the main beam is 2 sin' (lla) :::: 2lla radians for a» l. At u = 1.39 the field is 0.707 its peak value. Thus the halfpower beamwidth in radians is (34)
for a » l. The first side lobes are at u = ± 1.43.1l and are 20 logw(0.217) = 13.3 dB below the peak value of the main beam. Simple DistributionsRadiation patterns are usually characterized by their principal plane halfpower beam widths and first side lobe levels. These parameters are given for several simple symmetrical aperture distributions in Table 1. The pattern functions there are derived from the Fourier cosine transforms of the aperture distributions, i.e., if·E,(x) = E,(x), then (28) becomes (35)
1.0,......~r....r"...
0.8~\~\r+++++tI
0.6 cos" (rx/ 0)
0.4
5
it" ......
:s
it"
0.2
0.027 
0.21++"""""'=~~+_+__++__t
0.217 0.4~
o
_ _'_ _1_ _ _...L...._ _
~
_ _ _..I..._ _~_ _ _l.._ _..,...J
2r u = (ro/).) sin
3r
4r
(J
Fig. 4. Pattern functions of inphase symmetrical field distributions in a rectangular aperture: uniform (n = 0), cosinusoidal (n = 1), and cosinesquared (n = 2).
Antenna Theory
514
The results in Table 1 are arranged in order of increasing beamwidths and decreasing first side lobe levels. For inphase distributions the uniform aperture field has the highest gain but it also has a high side lobe level. The more the distribution decreases toward the aperture edges, the broader is the main beam and the lower the side lobe levels. If the aperture distribution is an odd function, i.e., if E,(x) = E,(x) in (28), its pattern is a Fourier sine transform, (36)
and is itself an odd function [FCk,) = F(k,)]. Several examples are shown in Table 2. These patterns have nulls at (J = and consequently their beamwidths and side lobe levels are unspecified. Instead, the angle at which the first and main lobe of the pattern appears is given, and the examples are arranged in order of increasing values of this angle. Such patterns may arise from the crosspolarized fields of a paraboloidal reflector, or they may be used for tracking on the pattern null, but otherwise they are rarely encountered. The twodimensional patterns of Tables 1 and 2 combine in (27) to give threedimensional patterns of rectangular apertures. For example, an openended rectangular waveguide with the TEO\ mode has an electricfield distribution in the aperture of Fig. 3
°
EAx,y,O) = Eocos(ny/b)
(37)
Table 1. Radiation Patterns of Even Distributions in a Rectangular Aperture Aperture Distribution
N/1 ~.
0/2
I
0/2
0
0/2
1'" "
E(x) (Ixl U
/"
Z
./
L&.I
ii: ~ L&.I
i
0.8
0.7
/
.'
."
cos (7:x/
,
~'./ /'
0
~
./

~?
V
.,,"" ,:7
1 \2x/o)2
.... ~
i"""
~~qM
./
/
I·
/f(X) = cos2 (7:x/ 0)
/
=:.,.;::;;0
.1
0
.x


L
V o
0.4
0.2
0.6
0.8
1.0
c Fig. 12. Onedimensional aperture efficiencies of compound symmetrical distributions in a rectangular aperture.
These are plotted as dashed, broken, and solid curves, respectively, in Fig. 12. The gain of a rectangular aperture with separable distributions is then (88)
Circular apertures with simple distributions have the aperture efficiency or relative gain given in Table 3. For the compound distributions of a parabola and a parabolasquared on a pedestal, the aperture efficiencies are plotted as the dashed and solid curves, respectively, in Fig. 13.
5. NearField Axial Gain and Power Density On the beam axis (() = 0) the gain of an xpolarized aperture distribution is, from (76), G
=
1/2
Yo IEo(r, Prl4nr2
oW
(89)
For a uniform xpolarized distribution in the aperture of Fig, 3 the axial nearfield follows from (68) and (71):
Antenna Theory
530
1.°ITITITIr=::1=..~~:::=:ptJi
 
~~
.. ... ".,.......'"
~
0.9
0_
/
~
i2
,// 0.8
~
~~C+(1C)[1(2r/a)21n
// / n2
n1~·
/
/
;~
c
I·  , .1'
/ '(
1//
/
/
0.7~+l+++++++j.~
o
/
0.2
0.4
c
0.6
0.8
1.0 J
Fig. 13. Aperture efficiencies of compound symmetrical distributions in a circular aperture.
}
J \
)
E(r,O) = 62jEoejkr [C(U)  jS(U)][C(U')  jS(U')]
i
(90)
where C(u) and S(u) are the Fresnel integrals defined by (72), u = alvz;:I, and u' = bIV2rA.. The power radiated is Pr = '/2 Yo Eo2ab and the axial nearfield gain can be written as (91) where (92) is the nearfield gain reduction factor for a onedimensional uniform distribution. This factor is plotted in Fig. 14. For a uniform and cosinusoidal distribution linearly polarized in the aperture of Fig. 4, the axial nearfield is, from (68) with (71) and (74), E(r,O) = 6jEoejkr+j(n'r).f4b2) [C(u)  jS(u)]{ C(v)  C(w)  j[S(v)  S(w)]} (93) where, from (75),
Radiation from Apertures
o
531
"""'"" ~'" ..........
" ~~ .....

0.5
~
"'" "
" '\
'.
......
i'..
.......
I'\.
r\
, ~
\ \.
\
'\
"
RH
"
"
~Rf
~
\
\
1\
\
l~
\
\
\ ~
\
"I\
\
~ ~
2.0
1 \ ~
2.5
o
2
1
3
5
'1 '1
6
7
8
Fig. 14. Nearfield axial gain reduction factors in decibels: R E , (92), for a uniform distribution, and R H , (96), for a cosinusoidal distribution in a rectangular aperture. (After lull (5], © 1981 Peter Peregrinus Ltd.)
V} {w
b
=
± V2r A. +
1
frT
b y' 2"
The power radiated is now P r
(94)
= lj4 Yo El ab and the axial gain can be written as (95)
Antenna Theory
532
in which
_ n 2 [C(U)  c(w)f + [S(v)  s(w)f RH ( v, w )  4 (v _ W)2
(96)
is the gain reduction factor for the cosinusoidal distribution. It is plotted in Fig. 14. The factors RE and RH also appear in expressions for the axial gain of pyramidal and sectoral horns and are tabulated [15]. For the circular aperture of Fig. 8 with an xpolarized uniform distribution the axial field at a distance r is
E(r,O)
= 92jEoei (kr+t)sint
with t = na2/(8lr). As the power radiated is Pr gain is
(97)
= 'h Yo E02 na2/4 the axial nearfield .j
(98) It is evident from (98) that at very close ranges r, this axial gain has maxima at
r = 4(2n
n = 0,1,2, ...
+ l)l'
(99)
and is zero at
n
= 1,2, ...
(100)
These correspond to ranges at which the aperture contains even and odd numbers of Fresnel zones, respectively. The situation is illustrated in Fig. 15a, where the axial nearfield power density of a uniform circular aperture of diameter a, (101)
is plotted. Fig. 15b shows corresponding results for a uniform square aperture of side a. The axial power density
(102) is not completely canceled at any range because the Fresnel zones are annular and incomplete in a square aperture.
.~
£
Radiation from Apertures
533
28
"
"
(\ 1/
~,
1\~ a
J
lJ
o
0,02
0,01
0,05
I" t"' i""
0,5
0.2
0.10
f'. r....
1.0
r/2tJ2/).
r
/ ~\
"
f1 1\
\I
b
v
rJ
('~
f\
V
I
~
J
\
~
~
.
\
I
~ fiV
1\
" ......
o 0.01
002
0.05
"
0.2
0.5
i' 1.0
Fig. 15. Axial power density in the near field of uniform distributions, (a) Circular aperture of diameter a; (b) Square aperture of side a,(After lull/51, © 1981 Peter Peregrinus Ltd,)
Antenna Theory
534
6. References [1] J. B. Keller, "Diffraction by an aperture," J. Appl. Phys., vol. 28, pp. 426444, 1957. [2] [3] [4] [5] [6] [7] [8] [9]
[to] [11] [12] [13] [14] [15]
P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, New York: Pergamon Press, 1966, pp. 1137. R. E. Collin and F. J. Zucker, Antenna Theory, Part I, New York: McGrawHill Book Co., 1969, pp. 6274. D. R. Rhodes, Synthesis of Planar Antenna Sources, Oxford: Oxford University Press, 1974, pp. 942. E. V. Jull, Aperture Antennas and Diffraction Theory, Stevenage, Herts., England: Peter Peregrinus, 1981, pp. 717. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, New York: John Wiley & Sons, 1981, pp. 375384. C. A. Balanis, Antenna Theory: Analysis and Design, New York: Harper and Row, 1982, pp. 446456. S. Silver, ed., Microwave Antenna Theory and Design, New York: McGrawHill Book Co., 1946, pp. 159168. T. T. Taylor, "Design of linesource antennas for narrow beamwidth and low side lobe levels," IRE Trans. Antennas Propag., vol. AP3, pp. 1628, January 1955. R. C. Hansen, ed., Microwave Scanning Antennas, Volume I, New York: Academic Press, 1964, pp. 419, 427. T. T. Taylor, "Design of circular apertures for narrow beamwidth and low side lobes," IRE Trans. Antennas Propag., vol. AP8, pp. 1722, January 1960. R. C. Hansen, "Table of Taylor distributions for circular aperture antennas," IRE Trans. Antennas Propag., vol. AP8, pp. 2326, January 1960. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS, US Dept. of Commerce, 1964, pp. 321322 (reprinted by Dover Publications, 1965). R. C. Hansen and L. L. Baillin, "A new method of nearfield analysis," IRE Trans. Antennas Propag., vol. AP7, pp. 458467, 1959. E. V. Jull, "Finite range gain of sectoral and pyramidal horns," Electron. Lett., vol. 6, pp. 680681, 1970.
Chapter 6
Recei·ving Antennas P. K. Park Hughes Aircraft Company
c. T. Tai University of Michigan
CONTENTS 1. Equivalent Circuit of a Receiving Antenna 2. Vector Effective Height of an Antenna 3. Receiving Cross Section, ImpedanceMatching Factor, and PolarizationMatching Factor 4. Generalized Friis Transmission Formula 5. Mutual Impedance between Distant Antennas 6. Small Antennas The Short Dipole 610 The Small Loop 613 7. Ferrite Loop Antennas 8. Bandwidth and Efficiency 9. Noise 10. Satellite TV Earth Station Receiving Antenna 11. References
63 63 66 68 69 69
618 622 624 625
632
61
Pyong Kiel Park was born in PyongAnDo, Korea. He received his BSEE degree from InHa University, Korea, in 1962, the MSEE degree from Yon Sei University, Korea, in 1965, and the PhD in electrical engineering from UCLA in 1979. He is a senior staff engineer at Hughes Aircraft Company, Missile Systems Division, which he joined in 1979. Currently he is responsible for designing a monopulse concurrent low sidelobe antenna, a coherent side lobe canceller antenna of the AIM54, and their monopulse feed networks. His prior experience includes lecturer at Yon Sei University, assistant professor at Kwang Woon Univer~ity, member of the technical staff at TRW, and senior engineer at Ford Aeronatronics. His main interests are in electromagnetic theory, antennas, and microwave circuits. He is a Senior Member of IEEE.
ChenTo Tai received his BS degree in physics from TsingHua University in 1937 and his DSc degree in communication engineering from Harvard University in 1947. He has been a professor of electrical engineering at the University of Michigan since 1964. He has been a visiting professor at several universities in the United States. Europe, and the Far East. and held an honorary professorship from the Shanghai Normal University. A Life Fellow of IEEE and a past chairman of the Antennas and Propagation Society (1971). he received a Distinguished Faculty Award from the University of Michigan in 1975. and several awards from the Department of Electrical and Computer Engineering and the College of Engineering at that university.
62
1. Equivalent Circuit of a Receiving Antenna By applying either Thevenin's theorem or Norton's theorem to a receiving antenna placed in a harmonically oscillating incident electromagnetic field, the load current can be calculated from the corresponding equivalent circuit (Fig. 1), i.e., (1) and (2) where
Voc = the opencircuit voltage measured at the receiving terminals Zin
= the input impedance of the antenna when it is operating in its transmitting
mode
ZL = the load impedance /sc
= the shortcircuit current through
the receiving terminals
The opencircuit voltage and the shortcircuit current are related by (3)
2. Vector Effective Height of an Antenna When an antenna is operating in its transmitting mode the farzone electric and magnetic fields can be written in the form E =
jkZoN e jkr
~'
4.71r
(4)
and
rx E H=Zo
(5)
where 63
Antenna Theory
+ lie
a
c
b
Fig. I. Illustration for calculating load current with Thevenin's and Norton's theorems. (a) Receiving antenna in incident electromagnetic field. (b) Thevenin's equivalent circuit for load current. (c) Norton's equivalent circuit for load current.
k = 2n/).. = w(,uO€O)ll2
Zo = Receiving Antennas
613
from the antenna remains constant. In practice the available power cannot be obtained because the antenna and its matching network are generally lossy, and also the high antenna reactance generally limits the bandwidth over which an effective match can be made with a simple network. To induce resonance an inductance may be added in series with the antenna and the generator. The loading inductor keeps the current distribution nearly constant from the feed to the load point, with a linear decrease from the load to the end as shown in row 2 of Table 2. The inductive loading is advantageous since the loading increases the current moment, and the receiving parametereffective lengthvaries as the current moment. The location of the inductor in the radiating structure affects the value of the radiation resistance in the sense that the current distribution along the antenna is modified. The optimum location for this inductance with regard to efficiency is about fourtenths of the length of the antenna above the ground plane [7]. The radiation efficiency of this inductively loaded monopole is still small (in practice, on the order of 10 percent at the lower end of the band). Top loading as shown in row 3 of Table 2 is another traditional approach toward achieving a fairly high efficiency in a relatively small size antenna. Note that the current distribution in these. structures is almost constant. Fig. 4 shows plots of theoretical radiation resistance Rr of a toploaded monopole antenna for an assumed linear current distribution. Further improvement of radiation efficiency can be achieved by combining the inductive loading and the top loading as shown in row 4 of Table 2. The Snwll Loop
Radiation Pattern and GainThe radiation pattern of a small loop is identical with that of a short dipole oriented normal to the plane of the loop with E and H fields interchanged. If the normal direction to the plane of the loop is the z axis (see Fig. 5a), the radiated electromagnetic fields are given by E = 120n nN A Isin e
x [vWb(h + zo;.rc  8) + Wa(h  zo;8)]  RaQI[ejkllh cos (;I Wa(2h; 8)
+ ejkllh cos (;I Wa(2h;.rc  8)]
(56)
Fig~. 16 and 17 show the current distributions on two quite different monopole sleeve antennas. In Fig. 16 the currents on the larger, b conductor are near resonance while those on the smaller, a conductor are not. In Fig. 17 the opposite situation is the case. The current distributions on the antennas depend not only on the overall length (including image) but on the position of the coaxial junction(s) as well, the currents emanating from the junction(s) being excited by the internal source or by the effect of the junction on the external currents. The data for these figures were calculated by using (52) and assuming an incident current of 1/Zo = 1/90 = 11.1 mAo Such a current could be generated by a matched (source impedance Ze = Zo) source having an open circuit voltage of 2 V. Furthermore, these figures are consistent with the experimentally determined distributions obtained by Taylor [30] and readily available in the book The Theory of Linear Antennas ([2], Section 111.30) by King. The input impedance to a monopole sleeve antenna for several different overall lengths is shown in Fig. 18 as a function of the position of the coaxial junction. The same conductor radii (ka = 0.02 and kb = 0.09) were chosen here as they were used in the preceding figures. Note that the input resistance becomes very large in all cases when kzo = 2.rczo/lo :: .rc/4 and 3.rc/4, corresponding to situations in which the junctions are At,/2 and 3At,/2 apart. At these positions the source and its image are opposing one another. For the present theory to yield results comparable to the experimental data measured by Taylor (see [2], Section 111.30) we found that it was necessary to include a shunt susceptance of j2 x 10 3 siemens in Fig. 18 in order to account for the finite thickness of the outer conductor in the experiment.
The Coaxial Sleeve Antenna with a Decoupling Choke
In most practical applications, sleeve antennas are fed via the inherent coaxial line within them with the feed line exciting the larger conductor in the manner illustrated in Fig. 19. Normally of coaxial construction itself, the outer sheath of the
Antenna Theory
732
180
10
1:0
I
8
koa 0 .02 kob O.09
1:0
h
90
/roro0.4...
koh 0.9...
~ 
I
o
"~
2
... r .... ___
 ....  .... _  .... , PHASE
Q
I
~ 90
~ MAGNITUDE
~ o o
I
0.2
0.6
0.4
~
0.8
180
1.0
zlh
Fig. 16. Current distribution on a monopole sleeve antenna due to an incident coaxial current of ZoIexp(jkoz).
feed line and the hollow larger conductor form another sleevetype junction. With the proper choice of terminating impedance Z, = lIY, for the bc coaxial line indicated in the figure, the sleeve antenna itself may be virtually isolated from its feed line at a single frequency. In the most practical situation, Z, is chosen so that the equivalent of an open circuit appears at the lower truncation of conductor b. If physically permissible, this can be accomplished by shorting the bc coaxial line at a distance of Ao/4 from the opening. Obviously this type of decoupling has a strong dependence on the operating frequency. In the forthcoming analysis, current waves traveling up the c conductor toward the b and a conductors, which form the intended antenna structure, are not considered. Such waves could result if the antenna is not mounted sufficiently high enough above the surr a). The only difference is the complex exponential term which is used here to represent the quadratic phase variations of the fields over the aperture of the horn. The necessity of the quadratic phase term in (lb)(ld) can be illustrated geometrically. Referring to Fig. 2b, let us assume that at the imaginary apex of the horn (shown dashed) there exists a line source radiating cylindrical waves. As the waves travel in the outward radial direction the constant phase fronts are cylindrical. At any point y' at the mouth of the horn the phase of the field will not be the same as that at the origin (y' = 0). The phase is different because the wave has traveled different distances from the apex to the aperture. The difference in travel path, designated as o(y'), can be obtained by referring to Fig~ 2b. For any point y'
Antenna Theory y'
1 X'
p;;+:I~
zI
a
y'
E
Z'
PI
I I
I I
b
bl /2
~
Fig. 2. Eplane horn and coordinate system. (a) Eplane sectoral horn. (b) Eplane view. (After Balanis /6/, © 1982; reprinted by permission of Harper & Row Publishers, Inc.)
(2a) or
Using the binomial expansion and retaining only the first two terms of (2b) reduces it to
Horn Antennas
87
(2c)
When (2c) is multiplied by the phase factor k, the result is identical with the quadratic phase term in (lb)(ld).
Radiated Fields To find the fields radiated by the horn, only the tangential components of the E and/or H fields over a closed surface need to be known [6]. The closed surface is chosen to coincide with an infinite plane passing through the aperture of the horn. To solve for the fields the approximate equivalent of Section 11.5.2 of [6] is used. That is, we assume that the equivalent current densities J.\. and Ms exist over the horn aperture, and they are zero elsewhere. Doing this yields
Js
=0 x
Ha
M s = 0 x
x') = 8 E ~os(:: x')
= 8yy J = 8 EI cos(:: Y", a
Ea
= axMx A
xl
a
ejkO(y')
'
a/2 < x' < a/2 (3)
ejkO(y')
'
where Ea and Ha represent, respectively, the electric and magnetic fields at the horn aperture, as given by (la)(le), and 0 = 8z is a unit vector normal to the horn aperture. Using (3) and the equations of Section 11.3 of Reference 6 it can be shown that the farzone radiated fields are given by [6] (4a)
(4b)
(4c) where (4d) (4e)
Antenna Theory
88
(4f)
t = I
~Jrkel 1
, (_ kb  k el) 2 y
(4g)
(4h) k x = k sin 0 cos 1>
(4i)
ky = k sin 0 sin 1>
(4j)
C(x) and S(x) are known as the cQsine and sine Fresnel integrals and are well tabulated [6]. Computer subroutines are also available for efficient numerical evaluation of each [10,11]. In the principal E and Hplanes the electric field reduces to EPlane (1) = Jr/2) Er = Eq, = 0
Eo =
(5a)
_ja~~,ejkT _e j(k e, sin 0)/2(2/.n/(1 2
+ COSO)F(t"t2)]
(5b)
(5c)
(5d) HPlane (0 = 0)
E,. = Eo = 0 E = q,
11
.a~Elejkr{ 8r
]
=
(6a)
_~ ~k 2
Jr(!1
" +bl/!; 
t2 =
2
Jrel
(1
1
[cos [(112) ka sin 0] ] "" sin of _ (Jr/2)2 F(t(,t 2 )J(6b)
+ cosO) [(1I2)ka
(6c) (6d)
To better understand the performance of an Eplane sectoral horn and gain some insight into its performance as an effiCient radiator, a threedimensional
Hom Antennas
89
normalized field pattern has b~n''pJ,qtted in Fig. 3 utilizing (4a)(4j). As expected, the Eplane pattern is much rni~rr than the Hplane because of the flaring and larger dimensions of the horn in that direction. Fig. 3 provides an excellent visual view of the overall radiation performance of the horn. To display additional details the corresponding normalized E and Hplane patterns (in decibels) are illustrated in Fig. 4. These patterns also illustrate the narrowness of the Eplane and provide information on the relative levels of the pattern in those two planes. To examine the behavior of the pattern as a function of flaring, the Eplane patterns for a horn antenna with (>1 = 15l and with flare angles of 20° < 21/Je < 35° are plotted in Fig. Sa and for 40° < 21/Je < 55° in Fig. 5b. In each figure a total of four patterns are illustrated. Since each pattern is symmetrical, only half of each pattern is displayed. For small included angl~s ~he M!!