Survey of Small Antenna Theory - McGraw-Hill Professional

106 Pages·2010·3.11 MB·English

June2,2010 12:27 SmallAntennas:MiniaturizationTechniques&Applications/JohnVolakis/162553-4/Ch01
1
CHAPTER
Survey of Small
Antenna Theory
JeffreyChalas,KyoheiFujimoto,JohnL.Volakis,andKubilaySertel
1.1 Introduction
Antennaminiaturizationhaslongbeendiscussedasoneofthemost
significantandinterestingsubjectsinantennaandrelatedfields.Since
thebeginningofradiocommunications,thedesireforsmallandver-
satile antennas has been ever increasing. Today’s needs for more
multifunctionalsystemsfurtherdriverequirementsforsmallmobile
terminals, including cell phones, handheld portable wireless equip-
mentforinternetconnection,short-andlong-rangecommunication
devices, RFIDs (radio frequency identification), etc. Similarly, small
equipment and devices used for data transmission and navigation
(GPSsystems)requiresmallantennas.Theseapplicationsandcontin-
uinggrowthofwirelessdeviceswillcontinuetochallengethecom-
munitytocreatesmallerandmoremultifunctionalantennas.
Thischapterisintendedtoprovideachronologicalreviewofpast
theoreticalworkcrucialtoantennaminiaturization.Throughout,we
shall refer to the small antennas as “electrically small antennas,” or
ESAs,implyingthattheirsizeismuchsmallerthanawavelengthat
theoperationalfrequency.
Wheeler[1]proposedtheESAdefinitionasanantennawhosemax-
imum dimension is less than (cid:2)/2(cid:3), referred to as a “radianlength.”
Anothercommonlyused(andequivalent)definitionofanESAisan
1
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2 Small Antennas
antennathatsatisfiesthecondition
ka <0.5 (1.1)
wherekisthewavenumber2(cid:3)/(cid:2),andaistheradiusoftheminimum
sizespherethatenclosestheantenna(seeFig.1.1).Weshallreferto
thissphericalenclosureasthe“Chusphere.”Smallantennasfittingthe
WheelerdefinitionradiatethefirstordersphericalmodesofaHertzian
dipole (see Fig. 1.2) and have radiation resistances, efficiencies, and
bandwidths. As is well known, these parameters typically decrease
withelectricalsizeka.
Another commonly accepted definition of a small antenna is
ka <1,[2].Thisdefinitioncanbeinterpretedasanantennaenclosed
inside a sphere of radius equal to one radianlength. Such a sphere
is referred to as a “radiansphere” [33], and represents the bound-
ary between the near- and far-field radiation for a Hertzian dipole.
Hansen[2]notesthatforantennasofthissize,higherorderspherical
modes(n>1)areevanescent.
Inthesectionstofollow,thesmallantennaperformancewillbechar-
acterizedbytheirsizeka,qualityfactorQ,fractionalbandwidth B,and
gain G. It is therefore important to have an understanding of these
parameters.Ofparticularinterestishowantennabandwidth(or Q)
is related to the antenna size. As will see, there is an optimum Q
Chu
sphere
z radius = a
Input y
x
FIGURE1.1 Chusphereofradius“a”centeredabouttheorigin.
TheChusphereistheminimumcircumscribingsphereenclosingtheantenna
ofmaximumdimension2a.
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Chapter 1: Survey of Small Antenna Theory 3
z
sin2 (θ)
θ
1.0 1.0
x φ y
FIGURE1.2 TM10orTE10modepowerpatternwiththeregion
(0◦<(cid:4)<90◦,0◦<(cid:5)<90◦)omittedforclarity.
(smallestpossible Q)foragivenantennasize.Followingareviewof
somebasicantennaparameters,ashortdiscussiononlumpedreso-
nant circuits and circuit Q is presented, which lays the foundation
forsmallantennaanalysis.Achronologicalreviewofthesignificant
contributionstosmallantennaswillthenbepresented,withafocus
onthetheoreticaldevelopmentofthefield.
1.2 Small Antenna Parameters
Toestablishafoundationfordiscussingsmallantennas,anoverview
oftheirmostimportantcharacteristicsispresentedbelow.
1.2.1 Directivity
It is often stated that small antennas have the familiar doughnut-
shaped(seeFig.1.2)omni-directionalradiationpatternofaHertzian
dipole of directivity D = 1.5. This pattern may be also thought as
theradiationofTE orTM sphericalmodes.However,thisisnot
10 10
the only possible pattern for a small antenna, as seen in the work
presented by Harrington [3], Kwon [4,5], and Pozar [6]. By super-
posingvariouselectricandmagneticHertziandipolearrangements,
unidirectional and bidirectional patterns are theoretically possible,
along with directivities ranging from D = 1.5 to 3 (see Sec. 1.3.11
inthischapter).Wecanstatethatantennashavingsignificantspher-
ical TE and TM mode radiation with n > 1 are generally not
nm nm
of the small type. Small antennas are also classified as superdirec-
tiveantennas,sincefordecreasingsizeka,theirdirectivityDremains
constant[2,7].
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4 Small Antennas
1.2.2 Radiation Efficiency
Radiation efficiency is a critical topic for small antennas but has
notbeenstudiedrigorously.Antennaradiationefficiencyfactor(cid:6)is
simply the ratio of the power radiated by the antenna to the power
deliveredtotheinputterminalsoftheantenna.Oftentheefficiency
factorisseenintheformulaG =(cid:6)(1−|(cid:2)|2)DwhereGistherealized
gainthatincludesthemismatchesbetweenthesourceandmatching
network(seeFig.1.3).WeassumethematchingnetworkofFig.1.3is
lossless.Thelossesintheantennaapartfromradiationarefrequently
modeledthroughaserieslossresistorR ,inwhichcasetheradiation
loss
efficiency(cid:6)canberepresentedas
R R
(cid:6)= rad = rad (1.2)
R +R R
rad loss A
whereR isthetotalantennainputresistanceR +R (seeFig.1.3).
A rad loss
It has been observed that as antenna size ka decreases, R
rad
decreases and the loss resistance R dominates the efficiency ex-
loss
pression of Eq. (1.2). This decrease in efficiency is primarily due to
frequency-dependentconductionanddielectriclosseswithinthean-
tenna.Asmentionedlater,Harrington[3]quantifiedtheefficiencyfor
anidealsphericalantenna,showingthatlossesareextremelypromi-
nentforsmallerkavalues.
A simple method to find the radiation efficiency (cid:6) and separate
R from R istousetheWheelerCap[8]method(seeFig.1.4).The
rad A
Wheeler Cap (shown in Fig. 1.4) is a hollow perfectly electric con-
ducting(PEC),enclosingsphereofthesamesizeastheradiansphere.
WheelernotedthatthesizeandshapeoftheWheelerCapisnotcrit-
ical. However, it must be electrically large enough so that the near-
zone-antennafieldsarenotdisturbedwhilestillpreventingradiation,
andsmallenoughsothatcavityresonancesarenotexcited.Indeed,
Z Antenna
A
R
loss
ZS XA
+ Lossless
passive
V R
− Γ matching rad
network
FIGURE1.3 Losslesspassivematchingnetworkwithantennaloadand
inputreflectioncoefficient(cid:2).
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Chapter 1: Survey of Small Antenna Theory 5
Shielding PEC sphere
equal to 1 radiansphere
+
−
Small antenna
of ka << 0.5
FIGURE1.4 WheelerCapforsmallantennaefficiencymeasurement.(After
Wheeler[8].)
Huang[9]provedrigorouslythatusingtheradiansphere-sizedspher-
icalWheelerCapdoesnotsignificantlyaffectthenearfieldsexcited.
To measure the radiation efficiency (cid:6) using the Wheeler Cap
method,firstacomputationormeasurementisdoneattheantenna
resonantfrequencyintheabsenceoftheWheelerCaptoobtain R .
A
If the antenna is not self-resonant, it must be tuned to resonance
by a reactive element at the input. The tuned antenna is then en-
closed inside the Wheeler Cap, and the measured input resistance
thenyields R .Substitutionof R and R inEq.(1.2)thengives
loss A loss
(cid:6)= R /R .
rad A
1.2.3 Quality Factor
1.2.3.1 AntennaQualityFactor
An intrinsic quantity of interest for a small antenna is the Q factor,
definedin[3]as
2(cid:7) max(W ,W )
Q= 0 E M (1.3)
P
A
W and W are the time averaged stored electric and magnetic
E M
energies,and P istheantennareceivedpower.Theradiatedpower
A
isrelatedtothereceivedpowerthrough P = (cid:6)P ,where(cid:6)isthe
rad A
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6 Small Antennas
antennaefficiency.ItisassumedinEq.(1.3)thatthesmallantennais
tunedtoresonanceatthefrequency(cid:7) ,eitherthroughself-resonance
0
orbyusingalosslessreactivetuningelement.
AntennaQisaquantityofinterestandcanbealsoevaluatedusing
equivalentcircuitrepresentationsoftheantenna.Anotherimportant
characteristicofQisthatitisinverselyproportionaltoantennaband-
width(approximately).AcommonlyusedapproximationbetweenQ
andthe3dBfractionalbandwidth Boftheantennais
1
Q≈ for Q(cid:4)1 (1.4)
B
Equation (1.4) is based on resonant circuit analysis and tends to
becomemoreaccurateasQincreases.Anexplicitrelationshipbetween
Qandbandwidthisgivenlater(seeSec.1.3.10orYaghjianandBest
[10]).Forthemomentletusreviewthelumpedresonantcircuitanal-
ysisusedforcomputing Qinthischapter.
ThereadermaywonderwhyQisthequantityofinterestratherthan
bandwidthitself.Onepracticalreasonisthatbandwidthremainsan
ambiguous term. Though it is often implied that bandwidth refers
to the 3 dB bandwidth, this is not always the case for antennas. It
isdesirabletofindanindependentlyderivedquantity Qthatisalso
related to bandwidth. This idea is given in Sec. 1.3.10 by Yaghjian
andBest[10].However,themostimportantreasonthatQremainsof
interestforsmallantennasisthatafundamentallowerlimitonQcan
be found using a number of techniques (and consequently the max
bandwidthofasmallantenna).ThisfundamentallimitationonQ(or
max bandwidth) drives the majority of the work examined later on
smallantennas.
1.2.3.2 QualityFactorforLumpedCircuits
Wheeler[1]recognizedthatasmallantennaradiatingthesinglespher-
icalTE modecanbeaccuratelyrepresentedasaRLCcombination
10
of Fig. 1.5a. We note the series capacitor represents the ideal tuning
elementinEq.(1.3)whichbringstheantennatoresonance.Similarly,a
smallantennaradiatingonlyasphericalTM modecanbeaccurately
10
representedbytheparallelRLCcombinationasinFig.1.5b,wherethe
shunt inductor represents the ideal tuning element in Eq.(1.3) that
brings the antenna to resonance. More complicated, high-Q circuits
can be accurately represented as a series [for X(cid:5) ((cid:7) ) > 0] or paral-
in 0
lel [for X(cid:5) ((cid:7) ) < 0] RLC circuits within the neighborhood of their
in 0
resonantfrequencies.
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Chapter 1: Survey of Small Antenna Theory 7
Z
in
C
L
I
+
V R
−
(a)
Z
in
I
+
V L C R
−
(b)
FIGURE1.5 SeriesandparallelRLCcircuits.(a)SeriesRLC,(b)ParallelRLC.
Series RLC Circuit For the series RLC circuit of Fig. 1.5a, the input
impedanceis
(cid:2) (cid:3)
j (cid:7)2−(cid:7)2 1
Z = R+ j(cid:7)L− = R+ j(cid:7)L 0 with (cid:7) = √ (1.5)
in (cid:7)C (cid:7)2 0 LC
where (cid:7) represents the resonant frequency at which the input
0
impedance is purely real. This resonance occurs when the average
storedelectricenergyisequaltotheaveragestoredmagneticenergy
in the circuit. Using the general definition of Q in Eq. (1.3) and rec-
ognizing the current is the same in all circuit components, we find
that
(cid:2) (cid:3)
1
2(cid:7) LI2
2(cid:7) W 0 4 (cid:7) L 1
Q= 0 H = = 0 = (1.6)
P 1 R (cid:7) RC
a I2R 0
2
where I isthecurrentthroughtheseriesRLCcircuitinFig.1.5a.The
bandwidthoftheseriesRLCcircuitcanbeestimatedafterintroducing
theapproximation
F((cid:7)) =(cid:7)2−(cid:7)2 ≈ F((cid:7) )+((cid:7)−(cid:7) )F(cid:5)((cid:7) ) =2(cid:7)(cid:3)(cid:7) (1.7)
0 0 0 0
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8 Small Antennas
validforsmall(cid:3)(cid:7)=(cid:7)−(cid:7) .WiththisTaylorseries,Eq.(1.5)becomes
0
(cid:2) (cid:3)
2(cid:7)((cid:7)−(cid:7) )
Z ≈ R+ j(cid:7)L 0 = R+ j2L(cid:3)(cid:7) (1.8)
in (cid:7)2
FromEq.(1.8),itisthenevidentthatthe3dBpointsoccurwhen
2L(cid:3)(cid:7) =±R (1.9)
3dB
where(cid:3)(cid:7) isthedifferencebetweenthe3dBfrequencyandreso-
3dB
nantfrequency.UsingEqs.(1.6)and(1.9)wecannowwrite
(cid:3)(cid:7)
2Q 3dB = QB =1 (1.10)
(cid:7)
0
sincebydefinition,2(cid:3)(cid:7) /(cid:7) = B foranantennahaving(cid:7) asits
3dB 0 0
operationalfrequency.Fromthisresult,wethenhavetherelationship
B = 1/QasmentionedinEq.(1.4).Figure1.6depictstheimpedance
asafunctionoffrequencyforatypicalseriesRLCcircuitforvarious
Q(cid:4)1values.
5
|Z | = |R(1 + j2QΔω/ω)|
4.5 in 0
4
3.5 Q = 500
3
R Q = 200
/
|n 2.5
Zi
|
2
Q = 100
1.5
1
0.5 3 dB above |Zin(ω0)|
0
0.994 0.996 0.998 1 1.002 1.004 1.006
ω/ω
0
FIGURE1.6 NormalizedimpedancemagnitudeforaseriesRLCcircuitnear
resonance.
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Chapter 1: Survey of Small Antenna Theory 9
ParallelRLCCircuit FortheparallelRLCcircuitofFig.1.5b,theinput
admittanceis
(cid:2) (cid:3)
j (cid:7)2−(cid:7)2 1
Y =G+ j(cid:7)C− =G+ j(cid:7)C 0 with (cid:7) = √ (1.11)
in (cid:7)L (cid:7)2 0 LC
where(cid:7) isagaintheresonantfrequencyforwhichtheinputadmit-
0
tanceispurelyreal.UsingthegeneraldefinitionofQandrecognizing
fortheparallelRLCcircuitthevoltageVacrosseachcomponentisthe
same,the QfortheparallelRLCcircuitatresonanceisfoundtobe
(cid:2) (cid:3)
1
2(cid:7) CV2
2(cid:7) W 0 4 (cid:7) C 1
Q= 0 E = = 0 = (1.12)
P 1 G (cid:7) GL
a V2G 0
2
FromthedualnatureoftheseriesandparallelRLCcircuits,thesame
bandwidthrelationsobtainedinEq.(1.10)holdfortheparallelRLC
circuit. Figure 1.7 depicts the impedance as a function of frequency
foratypicalparallelRLCcircuithaving Q(cid:4)1.
Arbitrary Lumped Networks In many cases, tuning the antenna impe-
dancetoresonanceusingasinglelosslessreactiveelementdoesnot
giveasuitablevaluefortheinputresistancetomatchthetransmis-
sionline.Tominimizemismatches(reflections)anddelivermaximum
powertotheantenna,twodegreesoffreedomareneededtoprovide
1.2
|Z | = |R(1 + jQΔω/ω)|−1
in 0
1
0.8
R
/
|
Zin 0.6 Q = 100
|
0.4
Q = 200
0.2
3 dB below |Z (ω)|
in 0 Q = 500
0
0.994 0.996 0.998 1 1.002 1.004 1.006
ω/ω
0
FIGURE1.7 NormalizedimpedancemagnitudeforaparallelRLCcircuit
nearresonance.
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10 Small Antennas
LC tuner Antenna / Load Circuit
L
I 1 IL
+ +
C1 CL VL RL
V
Zin − −
FIGURE1.8 AntennacircuitwithLCtuner.
animpedancematchtoatransmissionline.Figure1.8showsanexam-
pleofalumpedmatchingnetworkwithtwodegreesoffreedom—one
seriesandanothershuntelement.Withthesetwodegreesoffreedom,
anarbitraryloadimpedancecanbetransformedtoarealimpedance
value,andmatchedtothetransmissionline.
Tofindthe QforthecircuitconfigurationinFig.1.8,wenotethat
atresonance
Z =Re(Z ) = R
in in in
1
P = P = |I |2R
in L L L
2
where Z istheinputimpedance, P istheinputpower,and P is
in in L
the power at the load. Using these conditions, we can find Q from
Eq.(1.3).ToemployEq.(1.3),wenote
1|V|2 1|V |2
P = = P = L (1.13)
in L
2 R 2 R
in L
1
W = |V|2C (1.14)
E1 1
4
1 1 R
W = |V |2C = C L |V|2 (1.15)
EL L L L
4 4 R
in
SubstitutingthesequantitiesintoEq.(1.3),weget
2(cid:7) W 2(cid:7) (W +W )
Q= 0 E = 0 E1 EL =(cid:7) C R +(cid:7) C R (1.16)
0 1 in 0 L L
P P
L L
FindingQforarbitrarycircuittopologiescanbecomeacumbersome
procedure.FormulatingthetopologyusingapproximateRLCcircuits
canthereforebebeneficial.OnemethodusedbyChu[11]istoequate
theinputresistance,reactance,andreactancefrequencyderivativeof