CHAPTER10 Traveling Wave and Broadband Antennas

 CHAPTER10
TravelingWaveandBroadbandAntennas
目录
10.1 INTRODUCTION .................................................................................................................................................................................................................................... 2 10.2 TRAVELING WAVE ANTENNAS ............................................................................................................................................................................................................. 3 10.2.1 Long Wire ................................................................................................................................................................................................................................ 12 10.2.2 V Antenna ............................................................................................................................................................................................................................... 29 10.2.3 Rhombic Antenna ................................................................................................................................................................................................................... 37 10.3 BROADBAND ANTENNAS ................................................................................................................................................................................................................... 40 10.3.1 Helical Antenna ....................................................................................................................................................................................................................... 40 10.3.3 Yagi‐Uda Array of Linear Elements ......................................................................................................................................................................................... 66 PROBLEMS ................................................................................................................................................................................................................................................. 81 10.1 INTRODUCTION Inthepreviouschapterswehavepresentedthedetailsofclassicalmethods
thatareusedtoanalyzetheradiationcharacteristicsofsomeofthesimplestand
most common forms of antennas (i.e., infinitely thin linear and circular wires,
broadband dipoles, and arrays). In practice there is a myriad of antenna
configurations,anditwouldbealmostimpossibletoconsideralloftheminthis
book.Thegeneralperformancebehaviorofsomeofthemwillbepresentedinthis
chapterwithaminimumofanalyticalformulations.
10.2 TRAVELING WAVE ANTENNAS In Chapter 4, center‐fed linear wire antennas were discussed whose
amplitudecurrentdistributionwas
1. constantforinfinitesimaldipoles( λ/50)
2. linear(triangular)forshortdipoles( /50
3. sinusoidalforlongdipoles( λ/10)
λ/10)
Inallcasesthephasedistributionwasassumedtobeconstant.
Thesinusoidalcurrentdistributionoflongopen‐endedlinearantennasisa
standing wave constructed by two waves of equal amplitude and 180o phase
differenceattheopenendtravelinginoppositedirectionsalongitslength.
The current and voltage distributions on open‐ended wire antennas are
similartothestandingwavepatternsonopen‐endedtransmissionlines. Linear antennas that exhibit current and voltage standing wave
patterns formed by reflections from the open end of the wire are
referredtoasstandingwaveorresonantantennas.
Antennascanbedesignedwhichhavetravelingwave(uniform)patternsin
current and voltage. This can be achieved by properly terminating the antenna
wiresothatthereflectionsareminimizedifnotcompletelyeliminated. An example of such an antenna is a long wire that runs horizontal to the
earth,asshowninFigure10.1. F
Figure10.1
1Beverage(long‐wire))antennaab
boveground
d
The input terrminals cconsist off the gro
ound and one end
d of the wire.
w
This
cconfiguraationiskn
nownasB
Beverage （贝威尔基天线） orwaveantenna.Thereare
m
manyotherconfigu
urationso
oftravelingwavea
antennas.
Ingeneral,allantennaswhosecurrentandvoltagedistributionscan
be represented by one or more traveling waves, usually in the same
direction,arereferredtoastravelingwaveornonresonantantennas.A
progressive phase pattern is usually associated with the current and
voltagedistributions. Standing wave antennas, such as the dipole, can be analyzed as traveling
wave antennas with waves propagating in opposite directions and represented
bytravelingwavecurrents
and
inFigure10.1(a). Besides the long wire antenna there are many examples of traveling wave
antennas such as dielectric rod, helix, and various surface wave antennas.
Apertureantennas,suchasreflectorsandhorns,canalsobetreatedastraveling
waveantennas.Inaddition,arraysofcloselyspacedradiators(usuallylessthan
/2 apart) can also be analyzed as traveling wave antennas by approximating
their current or field distribution by a continuous traveling wave. Yagi‐Uda,
log‐periodic, and slots and holes in a waveguide are some examples of
discrete‐elementtravelingwaveantennas. In general, a traveling wave antenna is usually one that is associated with
radiationfromacontinuoussource.  A traveling wave may be classified as a slow wave if its phase velocity
(
/k, ω: wave angular frequency, k = wave phase constant) is equal or
smallerthanthevelocityoflightcinfree‐space( /c
1).  Afastwaveisonewhosephasevelocityisgreaterthanthespeedoflight
( /c
1).
Ingeneral,therearetwotypesoftravelingwaveantennas. 1
1. Surfacewavea
antenna urface w
wave antenna defin
ned as “aan antenna which
h radiates
One is the su
p
powerflo
owfromd
discontinu
uitiesintthestructturethat interruptabound
dwaveon
n
ttheantennasurfacce.”Asurfacewaveeantenna
ais,ingen
neral,asllowwaveestructure
w
whoseph
haseveloccityofth
hetravelin
ng wave is equal toor
t
lesssthanthee speed of
o
llightinfreee‐space(
c
1).
For slow waave structtures rad
diation ta
akes placce only at
a nonuniiformitiess,
ccurvaturees, and discontin
nuities. D
Discontin
nuities caan be either
e
disscrete or
d
distributeed.
 On
ne type of
o discrette discontinuity on a surfaace
w
wave anttenna is a
a transm
mission lin
ne termin
nated in an
u
unmatcheedload.  A distribu
uted surfface wav
ve antenna can be
aanalyzed interms ofthevaariationo
oftheamplitudeaand
p
phaseoftthecurren
ntalongitsstructu
ure. In geeneral, power flow
ws parallel to the
e structu
ure,
eexceptwh
henlossesarepresent,andforplane
estructurres
tthe fieldss decay exponenti
e
ially away
y from th
he antenn
na.
M
Most of the surfface wav
ve antenn
nas are end‐fire or
n
near‐end‐‐fire radiaators. Praacticalco
onfigurations inclu
ude
lline,planaarsurfacee,curved,,andmod
dulatedsttructures..
2
2. Leaky‐‐wavean
ntenna
Anotther trav
veling waave anten
nna is a leaky‐wav
l
ve anten
nna defineed as “an
n
aantenna that cou
uples po
ower in small in
ncrementss per unit lengtth, either
ccontinuou
usly or discretely
d
y, from a traveliing wavee structu
ure to frree‐spacee”
L
Leaky‐waave anten
nnas conttinuously
y lose ene
ergy due to radiaation, as shown in
n
F
Figure 10
0.2 by a slotted rectangu
ular wave
eguide. The
T
fieldss decay along the
sstructureinthedirrectionoffwavetraavelandiincreaseiinothers. Figu
ure10.2Le
eaky‐wavew
waveguide
eslots;upper(broad)andside((narrow)w
walls.
1
10.2.1Lo
ongWire
An aantenna is
i usually
y classifieed as a long wiree antennaa if it is a straigh
ht
cconductorrwithaleengthfromonetomanywa
avelength
hs. The long wirre of Figgure 10.1(a), in th
he presen
nce of th
he ground, can be
aanalyzed approxim
mately b
by introd
ducing an
n image to take into acccount the
p
presence of the grround. Th
he magnittude and phase off the image are deetermined
d
u
usingthereflection
ncoefficieentforho
orizontalpolarizattionasgiv
venby ,
1
for
(4‐129)
0 , 180 plane
∥

∥,
90 , 270 plane
:thereflectioncoefficientsforparallelandperpendicularpolarization
The angles
and
for
and
are angles of incidence and refraction, respectively.
areintrinsicimpedanceoffree‐spaceandtheground,respectively.
The height of the antenna above the ground must be chosen so that the
reflectedwaveisinphasewiththedirectwaveattheanglesofdesiredmaximum
radiation. The total field can be found by multiplying the field radiated by the
wireinfreespacebythearrayfactorofatwo‐elementarray.
As tthe wavee travels along the wire from the
sourceto
owardtheeload,itccontinuou
uslyleakssenergy. Thiss can bee repressented b
by an atttenuation
n
coefficien
nt. Therefore the current distributiion of thee
forward travelingg wave aalong thee structurre can bee
ntedby
represen
z′

z′
z′
Figure1
10.3Long‐w
wireantennaa
0
(10‐1)
z′ :th
hepropagaationcoeffiicient.  Theatttenuation
nfactor
z′ canalsorepressenttheoh
hmiclosseesofthew
wireaswelll
asgro
oundlossess,whicharreverysm
mallandare
eneglected
d.  Whentheradiattingmediu
umisair,tthelossofenergyinalongwirre dueto
oleakageiis
veryssmall,andiitcanalsobeneglectted.
Thereforethecurrentdistributionof(10‐1)canbeapproximatedby
z′
′
⇒
where
z′
(10‐1)
(10‐1a)
0
isassumedtobeconstant.Inthefarfield
0 (10‐2a); 0
2
sin
10‐2c 10‐2b is used to represent the ratio of the phase constant of the wave along the
transmissionline( )tothatoffree‐space( ),or
wavelengthalongthetransmissionline (10‐3)
Assuming a perfect electric conductor for the ground, the total field for
Figure 10.1(a) is obtained by multiplying each of (10‐2a)–(10‐2c) by the array
factor sin
For
sin
k K
.
1 thetime‐averagepowerdensitycanbewrittenas
| |
⇒
| |
1 (10‐4)
1 (10‐5)
From (10‐5) it is evident that the power distribution of a wire antenna of
length isamultilobepatternwhosenumberoflobesdependsuponitslength. Assumingthat isverylargesuchthatthevariationsinthesinefunctionof
(10‐5) are more rapid than those of the cotangent, the peaks of the lobes occur
approximatelywhen
sin
cos
1
1;
cos
1
m
2
1
2
π, m
0,1,2, …(10‐6)
Theangleswherethepeaksoccuraregivenby
cos
1
2
1 ,
0,1,2, … (10‐7)
Theanglewherethemaximumofthemajorlobeoccursisgivenbym=0. As becomesverylarge( ≫ ),theangleofthemaximumofthe
major lobe approaches zero degrees and the structure becomes a
near‐end‐firearray.
Infindingthevaluesofthemaxima,thevariationsofthecotangenttermin
(10‐5) were negligible. If the effects of the cotangent term were to be included,
thevaluesofthe 2m
2 1 termin(10‐7)shouldbe
1
0.742, 2.93, 4.96, 6.97, 8.99, 11, 13, . .. (10‐8)
Inasimilarmanner,thenullsofthepatterncanbefoundandoccurwhen
sin
cos
1
0 , cos
1
n
Theangleswherethenullsoccuraregivenby
,
1,2,3, … (10‐9)
1
1,2,3,4 … 10‐10 ,
The total radiated power can be found by integrating (10‐5) over a closed
sphereofradius andreducesto
∯
where
∙
2
| | 1.415
10‐11 is the cosine integral of (4‐68a). The radiation resistance is then
foundtobe
| |
2
1.415
(10‐12)
Using(10‐5)and(10‐11)thedirectivitycanbewrittenas
.
.
(10‐13)
A
A. AmplittudePattterns,Ma
axima,an
ndNulls
 Fig10.4
4(a):the3‐Dpatteernofatrravelingw
wireanten
nnawith 5
 Fig10.4
4(b):the3‐Dpatteernofasttandingw
wavewireeantennawith 5 .  Thecorrresponding2‐Dp
patternsaareshown
ninFiguree10.5. 90
0
120
0
60
-10
-20
30
150
-30
-40 180
0
-30
-20
330
210
-10
0
300
240
0
270
Figure10.4
4Three‐dim
mensionalfree‐space
eamplitude
epatterns fortravelin
ngandstan
ndingwave
e
wirean
ntennasof  The paattern fo
ormed by
y the fo
orward
ttraveling wave cu
urrent
has
m
maximum
m radiattion in the fo
orward
d
direction

The p
pattern formed
2
. when
w
by staanding
There
iis maximu
um radiaation in th
he forwarrd and
b
backward
ddirections. Figure10
0.5Two‐dim
mensionalfrree‐space
amplitudep
patternforttravelingan
ndstanding
wavewireanten
nnasof
5 Thellobenearrtheaxisofthewireinthedirectionsoftraveelisthelaargest.  The traaveling wave
w
antenna is ussed
when it is dessired to radiate or
minantly from o
one
receivee predom
directio
on.  As the length off the wire increasses,
nlobeshiifts
themaxximumoffthemain
closer toward the axiis and tthe
numberoflobessincreasee. Figu
ure10.6Frree‐spacep
patternfortraveling
wavewirea
w
antennaof
an
nd
The anglesoffthemaximaofth
hefirstfou
urlobes, computedusing(10‐8),are
p
plotted in
n Figure 10.7(a)
1
fo
or 0.5 10
0 . The correspon
c
nding anggles of the
ffirst fourr nulls, computed
c
d using (10‐10), are sho
own in Figure
F
10
0.7(b) for
0
0.5 10 .Thesecurvescan beusede
effectively
ytodesiggnlongw
wireswhen
n
tthedirecttionoftheemaximu
umornullisdesire
ed.
Figu
ure10.7An
nglesversu
uslengthoffwireante
ennawhere
emaximaa
andnullsocccur
B
B. InputtImpedance
For travelingg wave w
wire anten
nnas the radiation
n in the opposite direction
n
ffromthe maximum
missuppressedby
yreducin
ngthecurrrentrefleectedfrom
mtheend
d
o
ofthewirre.Thisisaccomplishedby
 In
ncreasingthediameterofth
hewire  Orrterminaatingittothegroun
nd,assho
owninFiggure10.1. Ideaally a com
mplete eliimination
n of the
reflection
ns (perfeect match
h) can o
only be
accompliished if the
t
anten
nna is ellevated
only at small heeights (co
ompared to the
ve the grround, an
nd it is
wavelenggth) abov
terminattedbyareesistivelo
oad. Thevalueoftheloadresistorisequaltothecharacteristicimpedanceofthe
wire near the ground (which is found using image theory). For a wire with
diameter and height
above the ground, an approximate value of the
terminationresistanceisobtainedfrom
138 log
(10‐14)
If the antenna is not properly terminated, standing wave pattern would be
created. Therefore the input impedance of the line is not equal to the load
impedance. The transmission line impedance transfer equation can be used to
calculatetheimpedanceattheinputterminals
(10‐15)
C. Polarization
A long‐wire antenna is linearly polarized, and it is always parallel to the
plane formed by the wire and radial vector from the center of the wire to the
observationpoint. The direction of the linear polarization is not the same in all parts of the
pattern, but it is perpendicular to the radial vector (and parallel to the plane
formedbyitandthewire).Thusthewireantennaisnotaneffectiveelementfor
horizontal polarization. Instead it is usually used to transmit or receive waves
thathaveanappreciablevectorcomponentintheverticalplane.Thisiswhatis
known as a Beverage antenna which is used more as a receiving rather than a
transmitting element because of its poor radiation efficiency due to power
absorbedintheloadresistor.
D. ResonantWires
Resonant wire antennas are formed when the load impedance of Figure
10.1(a) is not matched to the characteristic impedance of the line. This causes
reflections which with the incident wave form a standing wave. Resonant
antennas,includingthedipole,wereexaminedinChapter4. Resonant antennas can also be formed by long wires. For resonant long
wireswithlengthsoddmultipleofhalfwavelength( /2,n=1,3,...),the
radiationresistanceisgivenapproximately(within0.5ohms)by
73
69 log
(10‐16)
Forthesameelements,theangleofmaximumradiationisgivenby
(10‐17)
Thisformulaismoreaccurateforsmallvaluesof ,althoughitgivesgood
results even for large values of . It can also be shown that the maximum
directivityisrelatedtotheradiationresistanceby
(10‐18)
10.2.2VA
Antenna
Forssomeapp
plicationsasinglellong‐wire
eantennaisnotpracticalbeecause (1) iitsdirectiivitymay
ybelow
(2) iitssidelo
obesmaybehigh
(3) iitsmainb
beamisin
nclinedattanangle
e,whichisscontrolledbyitslength. One verypracticalarrrayoflongwiresiss
ntenna fo
ormed by
y using ttwo wiress
the V an
eachwith
honeofiitsendscconnected
dtoafeed
d
lineassh
howninF
Figure10..8(a).
In m
most app
plications,, the plan
ne formed
d
bytheleegsofthe Visparaalleltoth
heground
d,
whose p
principal polarizattion is p
parallel to
o
thegroun
ndandth
heplaneo
oftheV.
Becaause of in
ncreased side lobees, the directivity
y of ordin
nary lineaar dipoles
b
beginstodiminish
hforlengtthsgreateerthanab
bout1.25 .Howev
verbyadjustingthe
iincludedaangleofaaV‐dipolee,itsdireectivityca
anbemad
degreaterrandits sidelobes
ssmallerth
hanthoseeofacorrrespondin
nglineard
dipole.
Desiggnsform
maximum directivittyusually
yrequire smalleriincluded anglesfor
llongerV’ss.MostVantennassaresymm
metrical(θ
θ
θ and
d
o
).Also
V
V antenn
nas can be
b design
ned to haave unidiirectionall or bidirrectional radiation
n
p
patterns,asshown
ninFigures10.8(b
b)and(c),,respectiv
vely. To aachieve th
he unidirrectional characte
eristics, th
he wires of the V
V antennaa
m
mustben
nonresonaant.Therreflectedw
wavescanberedu
ucedby
 M
MaketheinclinedwiresofttheVrela
ativelythiick  P
ProperlyterminattetheopeenendsofftheV One way to terminatte the V antenna
a is to
aattachalo
oad,usuaallyaresisstorequaalinvalue
etothe
o
open end
d characteristic im
mpedancee of the V‐wire
V
ttransmisssionline,asshown
ninFiguree10.9(a)..
Theterminatiingresisttancecanalsobed
divided
iin half aand each half con
nnected to the ground
g
lleadingto
otheterm
minationo
ofFigure10.9(b).
Figure10.9TermiinatedV
antennas..
 IfthelengthofeachlegoftheVisverylong(typically
5 ),therewill
be sufficient leakage of the field along each leg that when the wave reaches the
endoftheVitwillbesufficientlyreducedthattherewillnotnecessarilybeaneed
foratermination. ThepatternsoftheindividualwiresoftheVantennaareconicalandinclined
atananglefromtheircorrespondingaxes.Theangleofinclinationisdetermined
bythelengthofeachwire. ThepatternsofeachlegofasymmetricalVantennawilladdinthedirection
ofthelinebisectingtheangleoftheVandformonemajorlobe,thetotalincluded
angle 2 oftheVshouldbeequalto 2
,whichistwicetheanglethatthecone
ofmaximumradiationofeachwiremakeswithitsaxis.Whenthisisdone,beams
2and3ofFigure10.8(b)arealignedandaddconstructively. Similarly forr Figure 10.8(c),, beams 2 and 3 are aligned and add
d
cconstructtivelyintheforwarddirectiion,while
ebeams5
5and8arealignedandadd
d
cconstructtivelyinth
hebackw
warddirecction.  If 2θ
2
2θ
theemainlob
beissplitintotwodistinctb
beams.  If (2θ
2 ), the maximum of the single major
2θ
m
lobe is still along the
p
planethatbisects theVbuttitistilteedupwarrdfromth
heplane oftheVd
duetothe
eexistenceofGND.
ForasymmetricalVantennawithlegseachoflength ,thereisanoptimum
included angle which leads to the largest directivity. The polynomials for
optimumincludedanglesandmaximumdirectivitiesaregivenby
2θ
149 /
603.4 /
13.39 /
78.27 /
D
2.94 /
1.15,
809.5 /
443.6 10
0.5
/
1.5
169.77 10
1.5
/
3
1.5
/
19a
19b
3 (10‐20)
The corresponding input impedances of the V’s are slightly smaller than
thoseofstraightdipoles.
Anottherform
mofaVan
ntennaisshownin
nFigure1
10.11(a).T
TheVisfformedby
y
aamonopo
olewire, bentataanangleo
overagro
oundplan
ne,andb
byitsimaageshown
n
d
dashed.T
Theinclud
dedangleaswellastheleng
gthcanbeeusedto tunetheantenna.  For 2θ
θ
p
primarily
y
120 , the an
ntenna eexhibits
verticcal
polaarization
with
rradiation pattern
ns almosst identiical to
tthoseofstraightdiipoles.  As 2θ
120 ,ahorizontallypo
olarized
ffield com
mponent iss excited which teends to
ffill the pattern toward the horrizontal
d
direction,, makingg it a v
very atttractive
ccommunicationan
ntennaforraircraft.
The computted impeedance o
of the ground plane and
a
free‐space V
V
cconfiguraationsobttainedbytheMoMisshown
nplottediinFigure10.11(a).
Anottherpractticalform
mofadipo
oleantenna,particcularlyusefulforairplaneor
ggroundpllaneapplications,isthe 90
0 bentw
wireconfiggurationofFiguree10.11(b)).
T
Thecomp
putedimp
pedanceo
oftheanteennaissh
hownplotttedinFiggure10.1
11(b). This antennaa can b
be tuned by
aadjusting itsperpeendicularrandparallel
llengths
and
.Th
he radiaation
p
pattern in
n the plaane of the antenn
na is
n
nearly om
mnidirecttional fo
or h
F
For h
0.1 thee pattern
n approacches
tthatofvertical λ/2
2 dipole.
0
0.1λ .
1
10.2.3Rh
hombicA
Antenna
A
A.Geome
etryandRadiatio
onCharaccteristicss
TwoVantenn
nascanb
beconnecctedatth
heiropen endstofformadiamondor
rrhombic aantenna, as shown
n in Figurre 10.12((a). To acchieve thee single main
m
lobee,
b
beams2, 3,6,and 7arealignedand
daddcon
nstructively.Theottherend isusedto
o
ffeedtheaantenna.
 Theanttennaisu
usuallyteerminated
datonee
endinareesistoroff600–800
0ohms,in
n
orderttoreduceifnotelim
minatereeflections..  Ifeachlegislon
ngenough
h(>5λ)su
ufficientle
eakageocccursalon
ngeachleegthatthe
hatreach
hesthefarrendofth
herhomb
busissuffficientlyrreducedth
hatitmay
y
waveth
notben
necessary
ytoterminatetherhombuss.
Anotther conffiguration
n of a rh
hombus is that off Figure 10.12(b) which is
fformedby
yaninverrtedVanditsimagge(shown
ndashed)). TheinvertedVisconnectedtothegroundthrougharesistor.Aswiththe“V”
antennas, the pattern of rhombic antennas can be controlled by varying the
element lengths, angles between elements, and the plane of the rhombus.
Rhombic antennas are usually preferred over V’s for nonresonant and
unidirectionalpatternapplicationsbecausetheyarelessdifficulttoterminate. Additional directivity and reduction in side lobes can be obtained by
stacking, vertically or horizontally, a number of rhombic and/or V antennas to
formarrays.
10.3 BROADBAND ANTENNAS In Chapter 9 broadband dipole antennas were discussed. There are
numerous other antenna designs that exhibit greater broadband characteristics
than those of the dipoles. Some of these antennas can also provide circular
polarization,adesiredextrafeatureformanyapplications. 10.3.1HelicalAntenna
Another basic, simple, and practical configuration of an electromagnetic
radiatoristhatofaconductingwirewoundintheformofascrewthreadforming
ahelix. Inmostcasesthehelixisusedwithagroundplane.Thegroundplanecan
takedifferentforms. O
Oneisforthegroun
ndtobefflat,assho
own
in Figuree 10.13. Typically
y the diam
meter of the
groundp
planeshou
uldbeatleast 3 //4.  The groun
nd plane ccan also b
be cuppe
ed in
ndricalcaavityorin
ntheform
mof
theformofacylin
afrustrumcavity. The helix is usually cconnected
d to the
onductor ofacoaxxialtransm
mission
centerco
line at the feed
d point w
with thee outer
or of thee line aattached to the
conducto
groundp
plane.
Thegeometricalconfigurationofahelixconsistsusuallyof

:turns, 

:diameter 
 :spacebetweeneachturn. :Thetotallengthoftheantenna :thetotallengthofthewire
,
√
isthecircumferenceofthehelix.
√
 Anotherimportantparameteristhepitchangle (10‐24)
 The radiation characteristics of the antenna can be varied by controlling the
sizeofitsgeometricalpropertiescomparedtothewavelength.  The input impedance is critically dependent upon
and the size of the
conductingwire,anditcanbeadjustedbycontrollingtheirvalues.  The geeneral po
olarization
n of the antenna is elliptiical. How
wever circular and
d
linearp
polarizatiionscanb
beachieveedoverd
differentfr
frequency
yranges.
The helicalan
ntennacaanoperatteinmanymodes;;howeverrthetwo
oprincipaal
o
onesaretthenormal(broad
dside)and
dtheaxiall(end‐firee)modess. Figure10
0.14Three‐d
dimensionalnormalizedamplitudelinearpowerrpatternsfornormaland
dend‐fire
mode
eshelicaldesigns.
 Figure 10.14(a), representing the normal mode, has its maximum in a
planenormaltotheaxisandisnearlynullalongtheaxis.Thepatternissimilarin
shapetothatofasmalldipoleorcircularloop.  Figure10.14(b),representativeoftheaxialmode,hasitsmaximumalong
the axis of the helix, and it is similar to that of an end‐fire array. The axial
(end‐fire) mode is usually the most practical because it can achieve circular
polarizationoverawiderbandwidth(usually2:1)anditismoreefficient.
A helix can always receive a signal transmitted from a rotating linearly
polarized antenna. Therefore helices are usually positioned on the ground for
space telemetry applications of satellites, space probes, and ballistic missiles to
transmitorreceivesignals.
A.NormalMode
To achieve the normal mode of operation, the dimensions of the helix are
usuallysmallcomparedtothewavelength(i.e.,

toaloopofdiameter

toalinearwireoflength when
≪
whenthepitchangle
).Thehelixreduces
⟹0 ⟹ 90 . Sincethelimitinggeometriesofthehelixarealoopandadipole,thefarfield
radiatedbyasmallhelixinthenormalmodecanbedescribedintermsof
componentsofthedipoleandloop,respectively. and
In the normal mode, the helix of Figure 10.15(a) can be simulated
approximately by
small loops and
short dipoles connected together in
series as shown in Figure 10.15(b). The fields are obtained by superposition of
thefieldsfromtheseelementalradiators. Sincee in the normal mode tthe helix
d
dimension
nsaresm
mall,  the currrent throughout its length
h can be
assumeedtobeconstant
 its reelative far‐field
f
pattern to be
independent of the num
mber of lo
oops and
dipoles.
shortd
Figure1
10.15Norma
al(broadside
e)modefor
helicalantennaanditsequivalent.
Thuss its operration can
n be desccribed by
y the sum
m of the fiields radiiated by a
a
ssmallloop
pofradiu
us
and
dashortd
dipoleof length ,withits
,
axisperp
pendiculaar
ttotheplaneoftheloop,and
deachwitththesam
meconstaantcurren
ntdistribution.
The far‐zone electric field
constantcurrent
radiated by a short dipole of length and
is
(4‐26a/10‐25)
Theelectricfield
radiatedbyaloopis
/
(10-26)
A comparison of (10‐25) and (10‐26) indicates that the two components are in
time‐phase quadrature, a necessary but not sufficient condition for circular or
ellipticalpolarization.
Theratioofthemagnitudesofthe
axialratio(AR),anditisgivenby
and
componentsisdefinedasthe
Byvaryingthe
|
|
|
|
(10‐27)
and/or theaxialratioattainsvaluesof 0
 AR=0occurswhen
AR
∞. 0 leadingtoalinearlypolarizedwaveofhorizontal
polarization(thehelixisaloop).  AR
∞,
0 and the radiated wave is linearly polarized with vertical
polarization(thehelixisaverticaldipole).  AR = 1, the radiated field is circularly polarized in all directions other than
θ
0 wherethefieldsvanish.
2
,
/
/ 2 To achieve the normal mode of operation, it has been assumed that the
current throughout the length of the helix is of constant magnitude and phase.
Thisissatisfiedtoalargeextentprovided  The total length of the helix wire
wavelength(
≪
is very small compared to the
)  Itsendisterminatedproperlytoreducemultiplereflections. Because of the critical dependence of its radiation characteristics on its
geometricaldimensions,whichmustbeverysmallcomparedtothewavelength,
thismodeofoperationisverynarrowinbandwidthanditsradiationefficiencyis
verysmall.Practicallythismodeofoperationislimited,anditisseldomutilized.
B.AxialMode
Amorepracticalmodeofoperation,whichcanbegeneratedwithgreatease,
istheaxialorend‐firemode.Inthismodeofoperation,
 Thereisonlyonemajorlobeanditsmaximumradiationintensityisalong
theaxisofthehelix.  Theminorlobesareatobliqueanglestotheaxis.
Toexcitethismode,thediameter
andspacing mustbelargefractions
of the wavelength. To achieve circular polarization, primarily in the major lobe,
the circumference of the helix must be in the
near optimum), and the spacing about S
12
α
14 .
range (with / = 1
/4. The pitch angle is usually
Most often the antenna is used in conjunction with a ground plane, whose
diameterisatleast
/2,anditisfedbyacoaxialline.However,othertypesof
feeds (such as waveguides and dielectric rods) are possible, especially at
microwavefrequencies. Thedimensionsofthehelixforthismodeofoperationarenotas
critical,thusresultinginagreaterbandwidth. C.DesignProcedure
The terminal impedance of a helix radiating in the axial mode is nearly
resistive with values between 100 and 200 ohms. Smaller values, even near 50
ohms, can be obtained by properly designing the feed. Empirical expressions,
based on a large number of measurements, have been derived. The input
impedance(purelyresistive)isobtainedby
140
(10‐30)
whichisaccuratetoabout ±20%,thehalf‐powerbeamwidthby
HPBW degree
thebeamwidthbetweennullsby
∙
√
/
(10‐31)
∙
FNBW degree
√
dimensionless
15
/
(10‐32)
(10‐33)
theaxialratio(fortheconditionofincreaseddirectivity)by
(10‐34)
andthenormalizedfar‐fieldpatternby
/
/
cos
10‐35 / (10‐35a)
/
For ordinary end‐fire radiation (10‐35b)
/
ForHansen‐Woodyardend‐fireradiation (10‐35c)
All these relations are approximately valid provided 12
/
α
14 , 3/4
4/3 andN>3.
Thefar‐fieldpatternofthehelix,asgivenby(10‐35),hasbeendevelopedby
assuming that the helix consists of an array of
identical turns, a uniform
spacing betweenthem,andtheelementsareplacedalongthez‐axis.  The cos termin(10‐35)representsthefieldpatternofasingleturn,
 Thelastterm
/
/
in(10‐35)isthearrayfactorofauniformarrayof
elements.  Thetotalfieldisobtainedbymultiplyingthefieldfromoneturnwiththearray
factor.

Thevalueofpistheratioofthevelocitywithwhichthewavetravelsalong
the helix wire, and it is selected according to (10‐35b) for ordinary end‐fire
radiationor(10‐35c)forHansen‐Woodyardend‐fireradiation.
(1) Forordinaryend‐fire Therelativephase amongthevariousturnsofthehelix(elementsofthe
array)isgivenby(6‐7a),or
(10‐36) where isthespacingbetweentheturns. Foranend‐firedesign,theradiationfromeachoneoftheturnsalong θ
0 must be in phase. Since the wave along the helix wire between turns travels a
distance
0
withawavevelocity 0
( <1where
0
isthewavevelocity
infreespace)andthedesiredmaximumradiationisalong θ
0 ,then(10‐36)
forordinaryend‐fireradiationisequalto cos
2
,
0,1,2, … (10‐37) Solving(10‐37)for leadsto
/
(10‐38) For
0 and
1, 0
√
. This corresponds to a straight
1, and it
wire (α 90 ), and not a helix. Therefore the next value is
corresponds to the first transmission mode for a helix. Substituting
1 in
(10‐38)leadsto
/
(10‐38a) (2) forHansen‐Woodyardend‐fireradiation
In a similar manner, it can be shown that for Hansen‐Woodyard end‐fire
radiation(10‐37)isequalto
cos
,
2
0,1,2, … (10‐39) whichwhensolvedfor leadsto
/
/
(10‐40) Example10.1
Designa10‐turnhelixtooperateintheaxialmode.Foranoptimumdesign, 1.Determinethe:
a.Circumference(in
),pitchangle(indegrees),andseparationbetweenturns(in
b.Relative(tofreespace)wavevelocityalongthewireofthehelixfor:
i.Ordinaryend‐firedesign
ii.Hansen‐Woodyardend‐firedesign
c.Half‐powerbeamwidthofthemainlobe(indegrees)
d.Directivity(indB)using:
i.Aformula
)
ii.ThecomputerprogramDirectivityofChapter2
e.Axialratio(dimensionlessandindB)
2. Plot the normalized three‐dimensional linear power pattern for the ordinary and
Hansen‐Woodyarddesigns.
Solution:
1. a.Foranoptimumdesign
Condition: 12
⟹
,α
α
14 , 3/4
13 ⟹ S
/
Ctanα
tan13
b.Thelengthofasingleturnis
Thereforetherelativewavevelocityis:
i.
Ordinaryend‐fire:
4/3 andN>3
1.0263 0.231 ii.
1.0263
0.231 1
/
/ 1
Hansen‐Woodyardend‐fire:
/
.
.
0.8337
=0.8012
c.Thehalf‐powerbeamwidthaccordingto(10‐31)is
52 ∙
HPBW degree
√
/
52
√10 ∙ 0.231
34.21o d.Thedirectivityis:
i.
Using(10‐33): dimensionless
15
15 ∙ 10 ∙ 0.231
34.65
15.397
e.Theaxialratioaccordingto(10‐34)is:
AR
2N
1 /2N
21/20
1.05 dimensionless 0.21dB
2. Thethreee‐dimensionallineaarpowerp
patternsfo
orthetwo end‐fired
designs,orrdinaryand
d
Hansen‐‐Woodyard
d,areshow
wninFigure10.16
Figure10
0.16Three‐d
dimensionalnormalizedamplitudeliinearpowerrpatternsforrhelicalord
dinary(p=
Hansen‐Wo
0
0.8337)and
oodyard(p=0.8012)end
d‐firedesign
ns
D.FeedDesign
The nominal impedance of a helical antenna operating in the axial mode,
computedusing(10‐30)
140
/ is 100~200. However, many practical transmission lines have characteristic
impedance of about 50. The input impedance of the helix must be reduced to
near that value. There may be a number of ways by which this can be
accomplished.Onewaytoproperlydesignthefirst1/4turnofthehelixwhichis
nexttothefeed. Tobringtheinputimpedanceofthehelixfromnearly150downto50,
tthewireo
ofthefirst1/4turn
nshouldbeflatintheformofastrip
pandthetransition
n
iintoaheliixshouldbeveryggradual.
留出
出螺旋最底部的 1／4 圈，制
制成平行
行于接地面
面的锥削过
过渡段，将
将 140～
150  的螺旋阻
阻抗变换为
为 50 的同
同轴线阻抗。其结
结构细节如
如图所示
This isaccom
mplishedb
bymakin
ngthewirrefromth
hefeed,aatthebegginningo
of
the formation of the helix, in the form of a strip of width
by flattening it and
nearlytouchingthegroundplanewhichiscoveredwithadielectricslabofheight
(10‐41) √
where
=widthofstripconductorofthehelixstartingatthefeed
=dielectricconstantofthedielectricslabcoveringthegroundplane
=characteristicimpedanceoftheinputtransmissionline
Typicallythestripconfigurationofthehelixtransitionsfromthestriptothe
regular circular wire and the designed pitch angle of the helix very gradually
withinthefirst1/4–1/2turn. This modification decreases the characteristic impedance of the
conductor‐ground plane effective transmission line, and it provides a lower
impedanceoverasubstantialbutreducedbandwidth.  For example, a 50 helix has a VSWR of less than 2:1 over a 40%
bandwidthcomparedtoa70%bandwidthfora140helix.  In addition, the 50 helix has a VSWR of less than 1.2:1 over a 12%
bandwidthascontrastedtoa20%bandwidthforoneof140.
Asimpleandeffectivewayofincreasingthethicknessoftheconductornearthe
feedpointwillbetobondathinmetalstriptothehelixconductor.Forexample,a
metalstrip70‐mmwidewasusedtoprovidea50impedanceinahelixwhose
conductingwirewas13‐mmindiameteranditwasoperatingat230.77MHz. 1
10.3.3Ya
agi‐UdaA
ArrayofL
LinearEllements
Anotther very
y practical radiatorr in the HF
H (3–30 MHz), VHF (30–3
300 MHz)),
aandUHF((300–3,000MHz)rangesistheYagi‐‐Udaanteenna. This antennaa consistts of a n
number of
o
llinear dip
pole elem
ments, ass shown in Figurre
1
10.19, oneof whicchisenerrgizeddirrectlyby a
ffeed transsmission line whille the oth
hers act as
a
p
parasitic radiato
ors or reflectorr, whosse
ccurrentsaareinduccedbymu
utualcoup
pling.
This radiator isanend
d‐firearrray,Yagid
designateedtherow
wofdireectorsasaa
““wavecan
nal.” T
Toachiev
vetheend
d‐firebeam
mformattion,  The parasiticc elementts in the direction
n of the beam
b
aree smaller in length
h
tthantheffeedelem
ment.  Thed
driveneleementisresonantwithitsllengthsligghtlylesssthan λ//2  The lengths of
o the dirrectors sh
hould be about 0.4~0.45λ. The direectors are
n
notnecessarilyoftthesamelengthan
nd/ordia
ameter.  The separation between the directors is 0.3~0.4 , and it is not necessarily
uniform.
 Thegainwasindependentoftheradiiofthedirectorsupto~0.024 .  The length of the reflector is greater than that of the feed. In addition, the
separation is smaller than the spacing between the driven element and the
nearestdirector,anditis~0.25 .
 Sincethelengthofeachdirectorissmallerthanitscorrespondingresonant
length,theimpedanceofeachiscapacitiveanditscurrentleadstheinducedemf.
 Similarly the impedance of the reflector is inductive and the phases of the
currentslagthoseoftheinducedemfs.  The phase of the currents in the directors and reflectors is not determined
solelybytheirlengthsbutalsobytheirspacingtotheadjacentelements.
 Thus, properly spaced elements with lengths slightly less than their
corresponding resonant lengths (less than /2) act as directors because they
form an array with currents approximately equal in magnitude and with equal
progressive phase shifts which will reinforce the field of the energized element
towardthedirectors.  Similarly, a properly spaced element with a length of /2 or slightly
greaterwillactasareflector.
ThusaYagi‐Udaarraymayberegardedasastructuresupportingatraveling
wave. Higher resonances are available near lengths of , 3 /2,and so forth, but
areseldomused.
Figure10.23Directivity
yandfront‐tto‐backratio
o,asa Figu
ure10.24Directivityand
dfront‐to‐ba
ackratio,asa
functionofdirectorspa
ffunctionofreflectorspaccing,ofa15‐‐elementYag
gi‐Uda
acing,for15‐‐element
a
array.
Yagi‐Udaarray.
1. Thetotalfieldrepresented
4
∑
10‐65 10‐65a 2. InputImpedanceandMatchingTechniques
The input impedance of a Yagi‐Uda array, measured at the center of the
drivenelement,  Usuallysmall  Strongly influenced by the spacing between the reflector and feed
element. Some of these values are low for matching to a 50‐, 78‐, or 300‐ohm
transmissionlines.
TherearemanytechniquesthatcanbeusedtomatchaYagi‐Udaarraytoa
transmission line and eventually to the receiver, which in many cases is a
television set which has a large impedance (on the order of 300 ohms). Two
commonmatchingtechniquesaretheuseofthefoldeddipole,ofSection9.5,asa
driven element and simultaneously as an impedance transformer, and the
Gamma‐matchofSection9.7.4.
3. DesignProcedure
A step‐by‐step design procedure has been established in determining the
geometricalparametersofaYagi‐Udaarrayforadesireddirectivity.Theincluded
graphs can only be used to design arrays with overall lengths (from reflector
element to last director) of 0.4, 0.8, 1.2, 2.2, 3.2, and 4.2 with corresponding
directivities of 7.1, 9.2, 10.2, 12.25, 13.4, and 14.2 dB, respectively, and with a
diameter‐to‐wavelengthratioof 0.001
/
0.04.
Assumethedrivenelementa /2 foldeddipole.Theprocedureisidentical
for all other designs at frequencies where included data can accommodate the
specifications.
Thebasisofthedesignisthedataincludedin
1. Table 10.6 which represents optimized antenna parameters for six
differentlengthsandfora /
0.0085
2. Figure 10.27 which represents uncompensated director and reflector
lengthsfor 0.001
/
0.04
Example10.3
DesignaYagi‐Udaarraywithadirectivity(relativetoa /2 dipoleat
the same height above ground) of 9.2 dB at 50.1
. The desired
diameter of the parasitic elements is 2.54 cm and of the metal supporting
boom5.1cm.Findtheelementspacings,lengths,andtotalarraylength.
Solution:
a. At
/
50.1
thewavelengthis
2.54/598.8
4.24
10 ;
598.8
/
. 5.1/598.8
8.52
10 b. From Table 10.6, the array with desired gain would have five elements.
Fora / =0.0085ratiotheoptimumuncompensatedlengthswouldbe 0.428 ,
0.424 ,
0.482 . Thespacingbetweendirectors=0.2 .Thereflectorspacing 0.2 .Theoverall
antennalength=0.8 .
Itis nowdessiredtofiindtheop
ptimumllengthso
oftheparasiticeleements
f
fora
/ =0.004
424.
c Plot th
c.
he optim
mized len
ngths fro
om Table 10.6 (
0
0.424
,
0.428 ,
0.482 )onFigurre10.27aandmark
kthemby
yadot(·).
d. InFigu
d
ure10.27
7drawa verticalllinethrou
ugh / =0.0042
24interssecting
c
curves
(
(B)
at director
d
uncomp
pensated lengths
0.442 and
r
reflector
length
0.485
5 .Markthesepointsbyan
n“x”.
ee. With a
a dividerr, measu
ure the distance
d
(∆ ) alo
ong direcctor curv
ve (B)
b
between
points
0.428 and
0.4
424 . Transpose
T
e this
d
distance
ownward along
from thee point
0.442 on curvee (B), do
t
thecurve
eanddeteermineth
heuncom
mpensatedlength
0.43
38 .
Thus the boom
u
uncompe
ensated lengths
l
o
of
thee array
y at
50..1
are
a
0.44
42 0.438 0.485 f. Correccttheelem
mentlengthstocompensaatefortheboomd
diameter..From
Figure10.28,aboomdiaameter‐to‐waveleengthratiioof0.00
0852requ
uiresa
fractional lengtth increaase in eaach elemeent of ab
bout 0.00
05λ. Thu
us the
finalleengthsoftheelementsshou
uldbe
0.442
5
0.005
0.4
447 ,
0.485
0.005
0.438
8
0.005
5
0.490
0 0.443 PROBLEMS 10.6.Itisdesiredtoplacethefirstmaximumofalongwiretravelingwave
antenna at an angle of 25 from the axis of the wire. For the wire
antenna,findthe (a)exactrequiredlength
(b)radiationresistance
(c)directivity(indB)
Thewireisradiatingintofreespace.
10.7.Computethedirectivityofalongwirewithlengthsof
2 and 3 .
10.8.Alongwireofdiameterdisplaced(intheair)ataheighthabovethe
ground.
(a)Finditscharacteristicimpedanceassuming
(b)Comparethisvaluewith(10‐14).
≫ .
10.12.DesignasymmetricalVantennasothatitsoptimumdirectivityis8
dB.Findthelengthsofeachleg(in )andthetotalincludedangleoftheV
(indegrees).
10.17.Designafive‐turnhelicalantennawhichat400MHzoperatesinthe
normal mode. The spacing between turns is /50. It is desired that the
antennapossessescircularpolarization.Determinethe
(a)circumferenceofthehelix(in
(b)lengthofasingleturn(in
andinmeters)
andinmeters)
(c)overalllengthoftheentirehelix(in
andinmeters)
(d)pitchangle(indegrees)
10.20. Design a nine‐turn helical antenna operating in the axial mode so
thattheinputimpedanceisabout110ohms.Therequireddirectivityis10
dB(aboveisotropic).Forthehelix,determinetheapproximate:
(a)circumference(in
).
(b)spacingbetweentheturns(in
).
(c)half‐powerbeamwidth(indegrees).
10.36. Design a Yagi‐Uda array of linear dipoles to cover all the VHF TV
channels. Perform the design at 216MHz. Since the gain is not
affectedappreciablyat
,asFigure10.26indicates,thisdesignshould
accommodate all frequencies below 216 MHz. The gain of the antenna
shouldbe14.4dB(aboveisotropic).Theelementsandthesupportingboom
shouldbemadeofaluminumtubingwithoutsidediametersof38in.( 0.95
cm) and 34 in.(1.90 cm), respectively. Find the number of elements, their
lengths and spacings, and the total length of the array (in , meters, and
feet).