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1298 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
4, APRIL 2005
Impedance, Bandwidth, and Q of AntennasArthur D. Yaghjian,
Fellow, IEEE, and Steven R. Best, Senior Member, IEEE
AbstractTo address the need for fundamental universally
validdefinitions of exact bandwidth and quality factor ( ) of tuned
an-tennas, as well as the need for efficient accurate approximate
for-mulas for computing this bandwidth and , exact and approxi-mate
expressions are found for the bandwidth and of a gen-eral
single-feed (one-port) lossy or lossless linear antenna tunedto
resonance or antiresonance. The approximate expression de-rived for
the exact bandwidth of a tuned antenna differs from pre-vious
approximate expressions in that it is inversely proportionalto the
magnitude0
(
0
) of the frequency derivative of the inputimpedance and, for not
too large a bandwidth, it is nearly equalto the exact bandwidth of
the tuned antenna at every frequency0
, that is, throughout antiresonant as well as resonant
frequencybands. It is also shown that an appropriately defined
exact ofa tuned lossy or lossless antenna is approximately
proportional to0
(
0
) and thus this is approximately inversely proportionalto the
bandwidth (for not too large a bandwidth) of a simply tunedantenna
at all frequencies. The exact of a tuned antenna is de-fined in
terms of average internal energies that emerge naturallyfrom
Maxwells equations applied to the tuned antenna. These in-ternal
energies, which are similar but not identical to previouslydefined
quality-factor energies, and the associated are provento increase
without bound as the size of an antenna is decreased.Numerical
solutions to thin straight-wire and wire-loop lossy andlossless
antennas, as well as to a Yagi antenna and a straight-wireantenna
embedded in a lossy dispersive dielectric, confirm the ac-curacy of
the approximate expressions and the inverse relation-ship between
the defined bandwidth and the defined over fre-quency ranges that
cover several resonant and antiresonant fre-quency bands.
Index TermsAntennas, antiresonance, bandwidth, impedance,quality
factor, resonance.
I. INTRODUCTION
THE primary purpose of this paper is twofold: first, to definea
fundamental, universally applicable measure of band-width of a
tuned antenna and to derive a useful approximateexpression for this
bandwidth in terms of the antennas inputimpedance that holds at
every frequency, that is, throughout theentire antiresonant as well
as resonant frequency ranges of theantenna; and second, to define
an exact antenna quality factorindependently of bandwidth, to
derive an approximate expres-sion for this exact , and to show that
this is approximatelyinversely proportional to the defined
bandwidth.
The average internal electric, magnetic, and magnetoelec-tric
energies that we use to define the exact of a linear an-tenna are
similar though not identical to those of previous au-thors [1][8].
The approximate expression for the bandwidth
Manuscript received October 2, 2003; revised September 14, 2004.
This workwas supported by the U.S. Air Force Office of Scientific
Research (AFOSR).
The authors are with the Air Force Research Laboratory, Hanscom
AFB, MA01731 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAP.2005.844443
and its relationship to are both more generally applicable
andmore accurate than previous formulas. As part of the
derivationof the relationship between bandwidth and , exact
expressionsfor the input impedance of the antenna and its
derivative with re-spect to frequency are found in terms of the
fields of the antenna.The exact of a general lossy or lossless
antenna is also re-ex-pressed in terms of two dispersion energies
and the frequencyderivative of the input reactance of the antenna.
The value ofthe total internal energy, as well as one of these
dispersion en-ergies, for an antenna with an asymmetric far-field
magnitudepattern, and thus the value of for such an antenna, is
shownto depend on the chosen position of the origin of the
coordi-nate system to which the fields of the antenna are
referenced. Apractical method is found to emerge naturally from the
deriva-tions that removes this ambiguity from the definition of for
ageneral antenna.1 The validity and accuracy of the expressionsare
confirmed by the numerical solutions to straight-wire andwire-loop,
lossy and lossless tuned antennas, as well as to a Yagiantenna and
a straight-wire antenna embedded in a frequencydependent dielectric
material, over a wide enough range of fre-quencies to cover several
resonant and antiresonant frequencybands. The remainder of the
paper, many of the results of whichwere first presented in [9], is
organized as follows.
Preliminary definitions required for the derivations of the
ex-pressions for impedance, bandwidth, and of an antenna aregiven
in Section II.
In Section III, the fractional conductance bandwidth and
thefractional matched voltage-standing-wave-ratio (VSWR) band-width
are defined and determined approximately for a generaltuned antenna
in terms of the input resistance and magnitude ofthe frequency
derivative of the input impedance of the antenna.It is shown that
the matched VSWR bandwidth is the more fun-damental measure of
bandwidth because, unlike the conduc-tance bandwidth, it exists in
general for all frequencies at whichan antenna is tuned.
(Throughout this paper, we are consideringonly the bandwidth
relative to a change in the accepted powerand not to any additional
loss of bandwidth caused, for example,by a degradation of the
far-field pattern of the antenna.)
In Section IV, the input impedance, its frequency derivative,the
internal energies, and the of a tuned antenna are given interms of
the antenna fields, and the relationship between band-width and is
determined. In particular, the frequency deriva-tive of the input
reactance is expressed in terms of integrals ofthe electric and
magnetic fields of the tuned antenna. These in-tegrals of the
fields are then re-expressed in terms of internal
1This ambiguity in the values of internal energy and Q
engendered by sub-tracting the radiation-field energy of an antenna
with an asymmetric far-fieldmagnitude pattern is not mentioned or
addressed in [1][8], probably becausethese references concentrate
on defining theQ of individual spherical multipoleswhich have
far-field magnitude patterns that are symmetric about the
origin.
0018-926X/$20.00 2005 IEEE
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
1299
energies used to define the of the antenna and two disper-sion
energies: the first dispersion energy determined by an in-tegral
involving the far field and the frequency derivative of thefar
field of the antenna; and the second determined by an in-tegral
involving the fields and the frequency derivative of thefields
within the antenna material. The dependence in the valueof the
far-field dispersion energy on the origin of the coordi-nate
system, and thus the ambiguity in (mentioned above),is removed by
the procedure derived in Section IV-E. An ap-parently new energy
theorem proven in Appendix B is used toderive a number of
inequalities that the constitutive parametersmust satisfy in
lossless antenna material. We find that the Fosterreactance
theorem, which states that the frequency derivative ofthe reactance
of a one-port linear, lossless, passive network is al-ways
positive, does not hold for antennas (whether or not the an-tenna
is lossless because the radiation from the antenna acts as aloss)
[10, Sec. 8-4]. Although the general formula we derive forthe
bandwidth of an antenna involves the frequency derivativeof
resistance as well as the frequency derivative of reactance,it is
found that the half-power matched VSWR bandwidth of asimply tuned
lossy or lossless antenna is approximately equalto for all
frequencies if the bandwidth of the antenna is nottoo large. It is
proven in Appendix C that the of an antennaincreases extremely
rapidly as the maximum dimension of thesource region is decreased
while maintaining the frequency, effi-ciency, and far-field
patternmaking supergain above a few dBimpractical. It is also shown
in Appendix C that the quality fac-tors determined by previous
authors [1][3] are lower boundsfor our defined applied to
electrically small antennas withnondispersive and .
In Section V, we discuss how the internal energy, , and
band-width of an antenna would be affected by the presence of
mate-rial with negative values of or .
In Section VI, exact VSWR bandwidths are computed fromthe
magnitude of the reflection coefficient versus frequencycurves
obtained from the numerical solutions to tuned, thinstraight-wire
and wire-loop lossy and lossless antennas rangingin length from a
small fraction of a wavelength to many wave-lengths, as well as to
a tuned Yagi antenna and a straight-wireantenna embedded in a
frequency dependent dielectric material.The exact values of for
these antennas are computed fromthe general expression (80) derived
for the of tuned antennas.The exact values of VSWR bandwidth and
are compared tothe approximate values obtained from the derived
approximateformulas in (87) for VSWR bandwidth and . These
numericalcomparisons confirm that the approximate formulas in
(87)for VSWR bandwidth and of a tuned antenna give muchmore
accurate values in antiresonant frequency ranges than
theconventional formula (81) (or its absolute value) commonlyused
to determine bandwidth and quality factor.
Before leaving this Introduction, a few remarks about the
use-fulness of antenna may be appropriate. We can ask why
theconcept of antenna is introduced when it is the bandwidth ofan
antenna that has practical importance. One advantage ofis that the
inverse of the matched VSWR bandwidth of an an-tenna tuned at the
frequency is approximated by the value ofthe of the antenna at the
single frequency . The bandwidthof some antennas may be much more
difficult to directly com-
Fig. 1. Schematic of a general transmitting antenna, its feed
line, and itsshielded power supply.
pute, measure, or estimate than the , which is
fundamentallydefined in terms of the fields of the antenna, is
independent ofthe characteristic impedance of the antennas feed
line, and hasa number of lower-bound formulas derived in the
published lit-erature [1][3] (see Appendix C). The simple
approximate, yetaccurate formulas for exact bandwidth and that are
derived inthe present paper can be evaluated for an antenna and
comparedto the lower bounds for to decide if the antenna is nearly
op-timized with respect to and bandwidth. It is often possible
toincrease the bandwidth of electrically small antennas by
simplyrestructuring the antenna to reduce its interior fields and
there-fore its [11]. Moreover, because the of an antenna is
de-termined by the fields of the antenna, Maxwells equations canbe
used, as we do in Appendix C, to obtain fundamental limi-tations on
the bandwidth of antennas. Finally, regardless of theutility of the
concepts of and bandwidth, it seems quite re-markable that at any
frequency of most antennas, the , whichis defined in terms of the
fields of a simply tuned one-port linearpassive antenna, and the
bandwidth, which is defined in termsof the input reflection
coefficient of the same antenna, are ap-proximately inversely
proportional (provided the bandwidth isnarrow enough) and that this
approximate inverse relationshipis given by the simple formulas in
(87) below.
II. PRELIMINARY DEFINITIONS
Consider a general transmitting antenna (shown schemati-cally in
Fig. 1) composed of electromagnetically linear mate-rials and fed
by a waveguide or transmission line (hereinafterreferred to as the
feed line) that carries just one propagatingmode at the
time-harmonic frequency . (The feedline is assumed to be composed
of perfect conductors separatedby a linear, homogeneous, isotropic
medium.) The propagatingmode in the feed line can be characterized
at a reference plane
(which separates the antenna from its shielded power supply)by a
complex voltage , complex current , and complexinput impedance
defined as
(1)
where the real number is the input resistance and the realnumber
is the input reactance of the antenna. The voltageand current can
also be decomposed into complex coefficients
and of the propagating mode traveling toward (inci-dent) and
away (emergent) from the antenna, respectively, suchthat
(2)
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1300 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
4, APRIL 2005
Fig. 2. Schematic of a general transmitting antenna, its feed
line, its shieldedpower supply, and a series reactance X .
with equal to the feed-line characteristic impedance, whichcan
be chosen to be independent of frequency [12, pp.
255256].Alternatively, and can be defined in terms of and as
(3)
The reflection coefficient of the antenna is defined as
(4)
As indicated, the parameters , and , as well as ,and , are in
general functions of .
Assume the antenna is tuned at a frequency with a
seriesreactance (as shown in Fig. 2) comprised of either a
pos-itive series inductance or a positive series capacitance ,where
and are independent of frequency, to make the totalreactance
(5)
equal to zero at , that is
(6)
Then the derivative of with respect to can be writtenas
(7)
or simply as
(8)
at the frequency . The equations corresponding to (1)(4) forthe
tuned antenna can be written as
(9)
(10)
(11)
(12)
Because the tuning inductor or capacitor is assumed lossless
andin series with the antenna, . The frequency , atwhich , defines
a resonant frequency of the antennaif and an antiresonant frequency
of the antennaif .2 For the sake of brevity, we shall
sometimesrefer to the resonant or antiresonant frequency as simply
thetuned frequency. Note that we are defining a tuned antennaat the
frequency as an antenna that has a total input reactanceequal to
zero at . Therefore, a tuned antenna will not havea reflection
coefficient equal to zero unless the char-acteristic impedance of
its feed line is matched to the antenna
at the frequency . If an untuned antenna has, it is said to have
a natural resonant frequency at
if and a natural antiresonant frequency at if.
The tangential electric and magnetic fields onthe reference
plane of the feed line can be written in termsof real electric and
magnetic basis fields of thesingle propagating feed-line mode with
voltage and cur-rent ; specifically
(13)
There may be evanescent modes on the feed line, but the fieldsof
these evanescent modes are assumed to be negligible on thereference
plane . If the dimensional units of and arechosen as (meter) and
they are consistent with Maxwellsequations in the International
System of mksA units, thenhas units of Volts, has units of Amperes,
and the charac-teristic impedance of the feed line can be chosen as
a realpositive constant independent of frequency with units of
Ohms.It then follows that the normalization of the basis fields may
beexpressed as a nondimensional number equal to one, that is
(14)
where is the unit normal (pointing toward the antenna) onthe
plane . If the plane simply cuts two wire leads froma generator at
quasistatic frequencies, and refer to con-ventional circuit
voltages and currents that do not serve as gen-uine modal
coefficients. In that case, the equations in (11) be-come
definitions of and with equal to the internal re-sistance of the
generator whose internal reactance is tuned tozero. For the TEM
mode on a coaxial cable, the basis fields
, as well as the characteristic impedance , areindependent of
frequency. Also, one of the basis fields, either
or , in addition to , can always be made indepen-dent of
frequency for feed lines composed of perfect conductorsseparated by
linear, homogeneous, isotropic materials [12, pp.255256]. We shall
use this fact in deriving (64) below.
2These definitions of resonance and antiresonance come from the
behaviorof the reactance of series and parallel RLC circuits,
respectively, at their naturalfrequencies of oscillation. At the
resonant frequency of a series RLC circuitwith positiveL andC;X
> 0 and at the antiresonant frequency of a parallelRLC circuit
with positive L and C;X < 0.
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
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With the help of (14) we can determine various expressionsfor
the total power accepted by the antenna
(15)or
(16)
The superscript in (15) denotes the complex conjugate andin (16)
is the input conductance of the
antenna. The power accepted by the antenna equals the
powerdissipated by the antenna in the form of power radiatedby the
antenna plus the power loss in the material of theantenna. Defining
the radiation resistance of the antennaas and the loss resistance
of the antenna as
, we have
(17)
so that
(18)
The power radiated can also be expressed in terms of the
farfields of the antenna
(19)
where is a surface in free space surrounding the antenna andits
power supply, the solid angle integration element equals
with being the usualspherical coordinates of the position vector
, and the complexfar electric field pattern is defined by
(20)
with being the speed of light in free space. Theimpedance of
free space is denoted by in (19) and is theunit normal out of . The
radiation resistance is always equalto or greater than zero because
the power ra-diated by the antenna is always equal to or greater
than zero
. Also, the loss resistance is equal to or greaterthan zero if
the material of the antenna is passive
.
III. FORMULAS FOR THE BANDWIDTH OF ANTENNAS
The bandwidth of an antenna tuned to zero reactance is
oftendefined in one of two ways. The first way defines what is
com-monly called the conductance bandwidth and the second way
defines what is commonly called the matched VSWR bandwidth.We
shall show that the matched VSWR bandwidth, unlike theconductance
bandwidth, is well-defined for all frequenciesat which the antenna
is tuned to zero reactance.
A. Conductance BandwidthThe conductance bandwidth for an antenna
tuned at a fre-
quency is defined as the difference between the two frequen-cies
at which the power accepted by the antenna, excited by aconstant
value of voltage , is a given fraction of the poweraccepted at the
frequency . With the help of (9), the conduc-tance at a frequency
of an antenna tuned at the frequencycan be written as
(21)
We can immediately see from (21) that there is a problem
withusing conductance bandwidth, namely, that the derivative of
evaluated at equals
(22)and thus it is not zero at unless . This means thatin
general the conductance will not reach a maximum at the fre-quency
. Moreover, in antiresonant frequency ranges whereboth the
resistance and reactance of the antenna are changingrapidly with
frequency, the conductance may not possess a max-imum and
consequently the conductance bandwidth may notexist in these
antiresonant frequency ranges. (As we shall showin Section III-B,
the matched VSWR bandwidth does not sufferfrom these
limitations.)
Well away from the antiresonant frequency ranges of
mostantennas, we have is much smaller than
, the conductance will peak at a frequency much closerto than
the bandwidth, and a simple approximate expressionfor the
conductance bandwidth can be found as follows.
Having tuned the antenna at so that , wecan find the frequency
where by taking thefrequency derivative of the expression for in
(21) andsetting it equal to zero to get
(23)With , the functions andtheir derivatives can be expanded in
Taylor series about
(24a)(24b)(24c)(24d)
which can be substituted into (23) to obtain for small
(25)or
(26)
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1302 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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In resonant frequency ranges well away from antireso-nant
ranges, we can assume
so that (26) reduces to
(27)
That is, the frequency shift in the peak of the conductanceof an
antenna tuned at the frequency in a resonant fre-
quency range is given by the simple relationshipin (27)
involving only the input resistance and the first
frequencyderivatives of the input resistance and reactance of the
antennaat the tuned frequency . In other words, peaks at afrequency
given by
(28)
To determine the conductance bandwidth about the
shiftedfrequency at which peaks when the antenna is tunedat , we
find the two frequencies at whichthe accepted power is times its
value at is givenfrom (21) as
(29)
provided, as discussed above, we are well within the
resonantfrequency ranges where . The value of theconstant , which
lies in the range , is assumedchosen . We can re-express (29) as in
(30), shown at thebottom of the page, whose left-hand side is more
suitable to apower series expansion about than the left-hand side
of (29)because the function , which rapidly varies from its valueof
zero at , is not contained in the denominator of (30).
Since the conductance on the left-hand side of (30), and
itsfirst derivative, are zero at , a Taylor series expansionof the
left-hand side of (30) about recasts (30) in the formof (31), shown
at the bottom of the page, in which has beenreplaced by because
forwell within resonant frequency ranges. Evaluating the
secondderivative in (31), we find
(32)
where use has been made of . Again, in resonant fre-quency
ranges we can assume that
and, therefore, (32) yields
(33)
under the additional assumption that the termsare negligible, an
assumption that is generally satisfied if
.
The fractional conductance bandwidth is thereforegiven
approximately by
(34)
under the assumptions that we are well within resonant
fre-quency ranges where and
and do not change greatly over the bandwidth(an assumption that
holds if or, equivalently,
, which can always be satisfied if is chosensmall enough). The
expression (34) for the fractional conduc-tance bandwidth of tuned
antennas was derived previously byFante [3] for the half-power
bandwidth , assuming
.
Rhodes [13] postulates the half-power bandwidth of
anelectromagnetic system as
(35)
He then defines as the of the electromagnetic system andfinds
stored electric and magnetic energies that are consistentwith this
and (59) below. The shortcomings of thismethod are that (35) is
postulated as the half-power bandwidthof a general antenna and that
is defined as rather thanas a physical quantity determined
independently of from thefields of the antenna. Moreover, (35) as
well as (34) does notaccurately approximate the bandwidth of tuned
antennas in an-tiresonant frequency ranges (except at antiresonant
frequencieswith ).B. Matched VSWR Bandwidth
The matched voltage-standing-wave-ratio (VSWR) band-width for an
antenna tuned at a frequency is defined as the
(30)
(31)
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
1303
difference between the two frequencies on either side of atwhich
the VSWR equals a constant , or, equivalently, at whichthe
magnitude squared of the reflection coefficientequals (the constant
is assumedchosen ), provided the characteristic impedance ofthe
feed line equals . Then the magnitudesquared of the reflection
coefficient can be found from (12) as
(36)
Both and its derivative with respect to are zero at
.Consequently, has a minimum at for all values ofthe frequency at
which the antenna is tunedand matched to the feed line . This means
thatthe matched VSWR bandwidth, , determined by
(37)
unlike the conductance bandwidth, exists at all frequencies(for
small enough ), that is, throughout both the antiresonant
and resonant frequency ranges.Therefore, the matched VSWR
bandwidth is a more funda-mental, universally applicable definition
of bandwidth for ageneral antenna than conductance bandwidth.
Bringing the denominator from the left-hand side of (37) tothe
right-hand side and rearranging terms to remove the rapidlyvarying
function from the denominator on the left-handside of (37)
produces
(38)
Expanding the left-hand side of (38) in a Taylor series about
,we find
(39)
under the assumption that the terms are negligible.This
assumption is generally satisfied if . Thesolutions to (39) for
are
(40)
so that the fractional matched VSWR bandwidthtakes the simple
form
(41)
which holds for tuned antennas under the sufficient condi-tions
that and do not change greatly over thebandwidth (conditions that
hold if or, equiv-alently, , which can always be satisfied if
is chosen small enough). For half-power VSWR bandwidth,and .
A comparison of (41) with (34) reveals that under their
statedconditions of validity
(42)
wherever the conductance bandwidth exists,namely outside the
antiresonant frequency ranges. The matchedVSWR bandwidth has the
distinct advantage overthe conductance bandwidth of existing at
everytuned frequency (for small enough ), that is, within
bothresonant and antiresonant fre-quency ranges. Moreover, if and
do not changegreatly over the bandwidth (which can always be
satisfied if
is chosen small enough), is reasonably wellapproximated at all
tuned frequencies by the simple expression(41) even in frequency
bands where or are zero,close to zero, or negative (but not both
and tooclose to zero). As far as we know, (41) is a general result
forantennas that has not been established previously.
The approximate formula for bandwidth in (41) should be ap-plied
judiciously to antennas that are designed to have a combi-nation of
two or more natural resonances and antiresonances at
that are so close together thatthe curve of versus has closely
spaced peaksequal to unity at these frequen-cies. Then the
half-power bandwidth , for example, mayextend over all these
natural resonant and antiresonant frequen-cies even though there
will be a resonant peak inat each natural resonant or antiresonant
frequency that hasits own bandwidth for some such that (say
). The formula in (41) approximates the bandwidth ofeach of
these individual minor resonant and antiresonant peakswith some
that is less than .
IV. FORMULAS FOR IMPEDANCE AND AND ITS RELATIONSHIPTO
BANDWIDTH
The formula for matched VSWR bandwidth given in (41) re-quires,
in addition to , the derivative of impedancewith respect to
frequency evaluated at , that is,
. As we shall see, an explicit expression forin terms of the
electromagnetic fields is not needed in
the derivation of and its relationship to the bandwidth of
atuned antenna. On the other hand, the evaluation of the fre-quency
derivative of the reactance, , in terms of the elec-tromagnetic
fields of the antenna is crucial to the derivation of
and its relationship to bandwidth. As a lead-in to the
desiredexpression for , we begin by deriving general expres-sions
for the input impedance of an antenna tuned atin terms of the
fields of the antenna.
A. Field Expressions for Accepted Power and Input ImpedanceTo
obtain expressions for the input impedance of the
antenna shown in Fig. 2 tuned at the frequency , apply
thecomplex Poyntings theorem [10, (1-54)] to the infinite
volume
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1304 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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outside the volume of the shielded power supply. Thevolume
includes the volume of the antenna material thatlies to the right
of the feed-line reference plane . The closedsurfaces of the
volumes and have the feed-line referenceplane in common. Therefore,
the volume includes thevolume of the series tuning reactance .
Assuming the in-tegral of the Poyntings vector is zero over the
shielded surfaceof the power supply and using (13)(14), we find
(43)
where, as in (17)
(44)
The power radiated is given in terms of the fields by (19)and
the power loss is given as
(45)
where in passive material. Then the power accepted,which equals
the total power dissipated by the antenna, can bewritten as
(46)
The efficiency of the antenna is defined as
(47)
which has a value equal to or less than unity. The usual
elec-tric and magnetic vectors are denoted by and ,respectively,
with , the vector being the cur-rent density. Since , we find from
(43) that
(48)and
(49)
Of course, the reactance of the antenna is equal to zero at
thetuned frequency , that is
(50)
Throughout the derivations in Sections II and III, it is
assumedthat the antennas are linear, that is, composed of materials
gov-erned by linear constitutive relations that relate and toand .
With the most general linear, spatially nondispersive con-stitutive
relations
(51)
where , and are the permeability dyadic, the permit-tivity
dyadic, and the magnetoelectric dyadics, respectively, thereactance
in (49) and (50) of the antenna tuned at the frequency
can be written as
(52)and
(53)
in which the subscript on a dyadic denotes its transpose.All the
field vectors as well as the real and imaginary partsof , and are,
in general, functions of both frequencyand the spatial position
vector . Outside the volume of theantenna material, , and in ,where
and are the permeability and permittivity of freespace and is the
unit dyadic.
With the constitutive relations in (51), the power loss andpower
accepted in (45) and (46) become
(54)
(55)
Since for all values of and in passive mate-rial, (54) implies
that a material is passive (lossy or lossless)if and only if its
associated Hermitian loss matrix is positivesemidefinite [14],
[15], [16, Sec. 5.2], a property that can be ex-pressed
symbolically as
(56)
In lossless material and the loss matrix is zero, thatis
(57)
If the material is reciprocal, , and [16,Sec. 5.1].
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
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For the simple isotropic constitutive relations
(58a)
with complex permeability and permittivity given by [10,
p.451]
(58b)
these equations for , and become
(59)
(60)
(61)
and
(62)
wherein it can be noted that and must both be positive (orzero)
in passive material to ensure that a result thatalso follows from
(57) and (58).
B. Field Expressions for the Frequency Derivative ofImpedance
and for Internal Energies
The formulas for conductance bandwidth and matchedVSWR bandwidth
given in (34) and (41), respectively, requirethe derivative of
impedance with respect to frequency evaluatedat , that is, .
The derivative of the resistance, , can be written from(48)
as
(63)
in which , that is, the fre-quency derivative of holding the
feed-line current con-stant with frequency and evaluated at the
tuned frequency .The expression (55) or (62) can be inserted for in
(63) toget in terms of the electromagnetic fields of the
tunedantenna. However, as we shall see below, the expression
for
given in (63) is not needed in the derivation of thequality
factor and its relationship to the bandwidth of thetuned antenna
and thus such an exercise proves unnecessary. Onthe other hand, the
evaluation of the frequency derivative of thereactance, , in terms
of the electromagnetic fields of theantenna is crucial to the
derivation of and its relationship tothe bandwidth.
Taking the frequency derivative of (52) and settingobviously
produces an expression for in terms of theantenna fields.
Unfortunately, however, this expression cannotbe used directly in
the derivation of . A more useful expres-
sion for is derived in Appendix A (see (A.15)) by com-bining
Maxwells equations with the frequency derivative ofMaxwells
equations to get
(64)
The primes indicate derivatives with respect to evaluated atthe
tuned frequency , and the subscripts indicate thatthe input current
at the reference plane in the feed line ofthe antenna is held
constant with frequency during the indicateddifferentiations. The
volume is capped by a sphere ofradius surrounding the antenna
system. Each of the two inte-gral terms inside the square brackets
of (64) approaches a posi-tive infinite value as , but together
they approach a finitevalue because all the other terms in (64) are
finite. As ,the second integral term inside the square brackets,
the one in-volving , subtracts the infinite energy in the radiation
fieldsfrom the infinite energy in the total fields to leave a
finite averagereactive energy involving static and induction
fields. Theexpression in (64) was derived by Rhodes [13], [17] in
the lessgeneral form given with the isotropic permeability and
permit-tivity in the constitutive relations (58). The expression
corre-sponding to (64) for perfectly conducting lossless antennas
wasderived by Levis [18] and Fante [3].
For antennas with asymmetric far-field magnitude patterns,we
shall show in Section IV-E that the square-bracketed
energy(reactive energy) and the last integral in (64) are each
depen-dent on the position of the origin of the coordinate system
inwhich the integrals are evaluated. The values ofand the integral
over in (64) are both independent of the po-sition of the origin of
the coordinates, and thus the sum of thesquare-bracketed energy
plus the last integral in (64) is indepen-dent of the position of
the origin of the coordinates.
We could have combined the entire last integral in (64) withthe
square brackets of (64) to get an alternative reactive en-ergy that
is independent of the chosen origin for all antennas.We do not want
to do this, however, for two reasons. First, thereis little
physical justification for including this integral as partof the
average reactive energy. Second, in all of our numericalwork (see
Section VI) we have found that the inverse relation-ship between
the exact defined with this alternative reactiveenergy and the
exact bandwidth does not hold accurately in theantiresonant bands
of tuned antennas. The ideal choice of theorigin of the coordinate
system is discussed in Section IV-E.
Using the general constitutive relations in (51), (64) can
berewritten as
(65)
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with
(66)and
(67a)
(67b)
(67c)
(67d)(67e)
Note that the finite magnetic, electric, and magnetoelectric
ener-gies [ , and ] at any frequency can bedefined by the formulas
(67a)(67c) evaluated at any frequency
instead of . However, the formula in (52) for the reactanceat
any frequency can generally only be rewritten in terms ofthese
finite energies as
(68)
if and the contributions to and fromand are negligible (so that
and ).
For lossless antenna material, (57) shows that . More-over,
energy relations are used in Appendix B to prove that if theantenna
material is lossless in a frequency window about , then
(69)which is equivalent to the associated Hermitian
susceptibilityenergy matrix being positive semidefinite [14], a
property ex-pressible symbolically as
(70a)
The energy relations in Appendix B also reveal that the real
partsof the elements of lossless (in a frequency window about
)constitutive parameters obey the inequalities
(70b)The inequalities in (70) and thus (69) can also be proven
fromthe Kramers-Kronig dispersion relations in a manner
analo-gously to the proofs in [15] and [19, Sec. 84]. The last part
ofAppendix B proves that in a lossless medium, the left-handside of
(69) equals the average reversible kinetic plus potentialenergy of
the charge carriers in a final time-harmonic field thatis built up
gradually from a zero magnitude at .
Using the terminology of Brillouin [20, p. 88] and Landauet al.
[19, p. 275] for our purposes, we shall refer to
, and as the average internalmagnetic, electric, and
magnetoelectric energies of the tunedantenna, respectively. They
are finite and have dimensions ofenergy, and have the far-field
radiation energies sub-tracted from them, and if they were the
energies in quasistaticfields in free space or nondispersive media,
they would equalthe amounts of energy one could quasistatically
extract fromthese magnetic and electric fields. In reality,
however, they arenot just quasistatic energies and, in addition,
the antenna maycontain dispersive materials, that is, constitutive
parametersthat are strongly frequency dependent. Nonetheless,
treating
, and as internal energies of the antenna in orderto define a
quality factor for the antenna, we shall find thesatisfying result
that provided thebandwidth of the antenna is narrow enough.
If in addition to the antenna being lossless in a
frequencywindow about , it is also nonradiating, (69) reduces
(65)(67)to
(71)
This equation implies that the frequency derivative of the
re-actance (actually ) of a lossless and nonradiatingantenna is
equal to the internal energy, which is greater than orequal to and
thus always greaterthan or equal to zero (Foster reactance theorem
for lossless non-radiating antennas or purely reactive one-port
passive termina-tions [21, Sec. 4.3]. Equation (71) remains valid
if the tunedfrequency is replaced by any frequency at which the
an-tenna is lossless and nonradiating but untuned.
In lossy media, it is possible to have negative values of the
firstintegral in (71) and, thus, it is conceivable that could
benegative for certain lossy antennas.
The energies in (65) and (67d)(67e) denoted by andare dispersive
quantities (in that they depend on the frequencyderivative of the
fields) associated with the power dissipated by
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
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the antenna as power loss and power radiated , re-spectively.
Unlike the power loss and power radiated (each ofwhich cannot be
negative), however, the sum of these disper-sion energies can be
negative as well as positive or zero and
in (65) can be negative as well as positive or zero. There-fore,
the Foster reactance theorem, which says that fora one-port linear,
lossless, passive network is always positive,does not hold for
antennas even if unless the antennais not only lossless but does
not radiate, in which case (71) holds.(Because both and are missing
from the expression for
in [5, eq. (43)], it is mistakenly concluded in [5] that
theFoster reactance theorem holds at all frequencies for
antennaswith .)
With the simple isotropic constitutive relations in (58),
theenergy expressions in (67a)(67d) become
(72a)
(72b)(72c)(72d)
The inequalities in (70), which hold in material that is
losslessin a frequency window about , reduce to
(73)regardless, incidentally, of whether the values of
arepositive or negative.
The far-field dispersion energy given by (67e) can beevaluated
from the antennas complex far electric field pattern
defined in (20). The material-loss dispersion energygiven by
(67d) or (72d) requires a knowledge of the electricand magnetic
fields in the material of the antenna. For thin-wirelossy antennas
with , where is the con-ductivity of the wire material and , the
dispersion energy
in (72d) reduces to
(74)
If the cross section of the wire is circular and the skin
depthof the current density is much smaller than the diameter ofthe
wire, (74) further reduces to
(75)
under the approximation ,where is the current density at the
surface of the wire and
is the radial distance from the center of the wire. In (75),is
the resistance per unit length of wire and
is the total current flowing in the wire at the positionalong
the wire. As usual, the primes indicate differentiationwith respect
to frequency and the subscript means thatthe frequency derivative
is taken with the input current thatfeeds the antenna held constant
(independent of frequency).If the diameter or resistivity of the
wire varies along the wire,
will be a function of . If the current were uniform acrossthe
wire as in a lumped circuit resistor carrying a current
, (75) is replaced by
(76)
The formula in (75) is used in Section VI to numerically
evaluatefor lossy wire antennas, and (76) is applied in Appendix
D
to lumped resistors in series and parallel RLC circuits.
Withinresonant frequency ranges , the tuned antennacan usually be
approximated by a series RLC circuit. For thatapproximation, in
(76) and since , it followsthat within resonant frequency
ranges.
Because, and , the energies ,
and include the derivative terms with ,and . In [9] we
defined
and the associated in (78) below without thesederivative terms.
However, in order to define a in (78) thatis proportional to the
inverse of the matched VSWR fractionalbandwidth given approximately
in (41), the energy mustequal when , and arenegligible. It then
follows from (64) that
must be defined as shown in (67a)(67c)with the derivative terms
included. For example, at a naturalresonant frequency of an antenna
that can be modeled byan RLC series circuit with negligible but
and
nonnegligible (because the inductor and capacitor isfilled with
a material that has a nonnegligible and ,respectively)
(77a)
Similarly, at a natural antiresonant frequency of an antennathat
can be modeled by an RLC parallel circuit withnegligible but and
nonnegligible
(77b)
For the parallel RLC circuit, and. If were defined without
the
and terms included in , then for these RLCantennas would not
include the derivatives of and ,and the would not closely
approximate the inverse of
. The need to include the derivative terms in theused to define
is confirmed in the Numerical Re-
sults Section VI-D for a straight-wire antenna embedded in
afrequency dependent dielectric material (see Figs. 18 and 19).
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C. Definition and Exact Expressions of QThe quality factor for
an antenna tuned to have zero
reactance at the frequency can now be definedas
(78)
Absolute value signs are placed about in the definitionof in
(78) to allow for hypothetical antennas (mentionedin the previous
subsection) with ; see in Fig. 19.Formulas for in terms of fields
are given by means of(66)(67) and formulas for the power accepted
by the antennaare given by means of (15)(19) and (55). In
particular,
and can be written from (65) as
(79)
so that can be expressed as
(80)
The expressions on the right-hand sides of (78) and (80) arevery
different in form, yet they are exact and thus produce thesame
value of . In Section VI, the formula (80) ratherthan (78) is used
to compute the exact values of for variousantennas because it is
easier to numerically computefor these antennas than to numerically
evaluate the integrals in(67a)(67c) that define used in (78).
Especially note that the in (80) differs from both
theconventional formula for the quality factor [1]
(81)
and from Rhodess formula in (35) above, namely,, because of the
term
. (Rhodes [13] assumes (mistakenly)that the right-hand side of
(80) is not a valid expression forbecause it does not, in general,
equal . Fante [3]assumes that (80) is a valid expression for if
,and as well as are negligible.) The formula in(81) is commonly
used to determine the quality factor and thebandwidth ( for
half-power conductance bandwidth, asin (34), and for half-power
matched VSWR bandwidth)of tuned antennas. In general, neither in
(81) nor
accurately approximates the exact in (78) and(80) of tuned
antennas in antiresonant frequency ranges.
It is proven in Appendix C that the of an antenna in-creases
extremely rapidly as the maximum dimension of theeffective source
region is decreased while maintaining the fre-quency, efficiency,
and far-field pattern. This implies that su-pergain above a few dB
is impractical. It is also shown in Ap-pendix C that the quality
factors determined by previous authors[1][3] are lower bounds for
our defined applied to electri-cally small antennas with
nondispersive and .
D. Approximate Expression for and Its Relationship
toBandwidth
We can estimate the total dispersion energy,, in (80) to get an
approximate expression for
that can be immediately related to the bandwidth of the
tunedantenna. Away from antiresonant frequency ranges of
tunedantennas, and usually . Fur-thermore, for the sake of
evaluating , we assume the powerloss and power radiated can both be
approximated by ohmicloss in a resistor of a series RLC circuit,
where R can be a func-tion of . Evaluating in Appendix D forsuch a
series RLC circuit reveals that its value is small enoughto make
the second term on the right-hand side of (80) negli-gible compared
to the first. Therefore, away from antiresonantfrequency ranges,
that is, within resonant frequency ranges
(82)
or, since away from antiresonant fre-quency ranges
(83)
At an antiresonant frequency , we assume that tuned an-tennas
can be approximated by a tuning inductor or capac-itor in series
with a parallel RLC circuit. An evaluation of
in Appendix D for such a tuned parallelRLC circuit reveals
that
(84)
so that
(85)
Inserting (85) into (80) yields
(86)
which, combined with (83), holds for all .Comparing the
approximate formula for the quality factor
in (86) with the approximate formula for the matchedVSWR
fractional bandwidth in (41), one finds
(87)
provided and do not change greatly over the band-width of the
antenna (assumptions that hold if the bandwidthis narrow enough.)
As noted in (35), Rhodes [13] definesby the expression in (87) with
replaced by(and ). Such an expression does not produce an
accurateapproximation to and bandwidth in antiresonant
frequencybands (except at antiresonant frequencies with ).
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
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In concluding this section, it is emphasized that not everytuned
antenna has to obey the inverse relationship betweenbandwidth and
given in (87). The derivation of (41) and (86)assumes that the
antenna is linear, passive, and tuned by a linearpassive circuit.
If the antenna contains nonlinear or active ma-terials and tuning
elements, the bandwidth could conceivablybe appreciably widened
without decreasing commensuratelythe internal energy and (as
defined in (78)) of the antenna.Then, of course, (41), (86), and
the inverse relationship betweenbandwidth and would not necessarily
hold.
Also, the derivation of (86) approximates byin resonant
frequency ranges. Consequently, the ap-
proximation (86) could be inaccurate in a resonant
frequencyrange if the resistivity of the antenna changed rapidly
enoughwith frequency to make . In general, thederivation of (86)
breaks down if or in theRLC series and parallel circuit antenna
models of Appendix Dbecome too large and one would not expect the
exact to be ahighly accurate approximation to the inverse of the
exact band-width. In all our numerical simulations to date with
practicalantenna models, however, the approximations in (86) and
(87)have exhibited high accuracy throughout both resonant
andantiresonant frequency ranges.
Nonetheless, the derivation of (86) in Appendix D that
usesseries and parallel RLC circuits to model antennas in
theirresonant and antiresonant frequency ranges, respectively, has
aserious limitation. In this derivation, the radiation resistance
ofthe antenna is lumped into the antennas resistive loss so thatthe
term is replaced by a contribution to . As discussedin Section
IV-B, the value of for antennas with asymmetricfar-field magnitude
patterns depends on the position of theorigin of the coordinate
system. Therefore, in replacingwith a contribution to , which is
independent of the origin ofthe coordinates, it is implicitly
assumed that the antennasis either independent of the origin or
that the origin is chosen tomake of the antenna approximately equal
to the of theRLC circuit that is used in Appendix D to model the
antenna.In the following Section IV-E, we shall give a practical
methodfor determining approximately such an ideal location for
theorigin of the coordinates at each tuned frequency . Moreover,a
simpler alternative method is given in the last paragraph ofSection
IV-E for obtaining an approximate value of theassociated with the
ideal origin at each tuned frequency .
Kuester [22] has pointed out that an RLC circuit can be
con-structed with an arbitrary value of by separating the
resistorfrom the inductor and capacitor by a length of transmission
linewhose characteristic impedance is equal to the resistance of
theresistor that terminates this line. The input impedance and
band-width of such an RLC circuit is independent of the length of
thistransmission line, whereas the internal energy and as definedby
(78) or (80) will increase with the length of this transmis-sion
line. Increases in internal energy and without a changein the input
impedance can also occur using surplus capaci-tors and inductors
[23, p. 176]. These spurious contributions tothe exact that create
discrepancies between the exact valueof in (78) or (80) and the
approximate value in (86), as wellas the ambiguity in with respect
to the chosen origin for the
far-field pattern of the antenna, can be removed by the
simpleprocedure given in the last paragraph of Section IV-E.
E. Determination of the Ideal Location for the Origin of
theCoordinates and the Associated
The values of , and may depend on the choiceof the origin of the
coordinates to which the far-field pattern isreferenced. To prove
this, let the origin of the coordinate systembe displaced by an
amount with respect to the antenna. Thenthe far-field pattern (at
frequency ) with respect to this newcoordinate system is given
by
(88)and, thus
(89)Inserting and into the last integral in (64), that is,
into
, shows that the change in this integral caused by a
displace-ment of the origin of the coordinates is given by
(90)
which has a magnitude that is less than or equal to. If the
magnitude of the far-field pat-
tern is symmetric about the origin, then the change given in(90)
is zero, that is, the value of the last integral in (64), andthus
the square-bracketed energy in (64), is independent of theorigin of
the coordinate system.
If (for example, if is symmetric aboutthe origin), the choice of
the origin of the coordinate system isirrelevant. If , then the
radiation-field energy(second integral in the square brackets of
(64)) that subtractsfrom the total-field energy (first integral in
the square bracketsof (64)) may either overcompensate or
undercompensate for theradiation energy if the origin is too far
from the center of thesource region of the antenna. Thus, it is
reasonable, though notnecessarily ideal, to choose the origin of
the coordinates at thecenter of the imaginary spherical surface
that circumscribes thesource region of the antenna. Nonetheless, we
ultimately have tolive with the fact that our defined reactive and
internal energiesof an antenna (like that of previous authors
[1][8]) and thus its
defined in (78) depends to some degree on the choice of
theorigin of the coordinate system relative to the antenna
(unless
). This nonuniqueness in reactive energy andof an antenna arises
because of the need to subtract the infi-
nite energy in the radiation fields from the infinite energy in
thetotal fields of the antenna to obtain finite values of reactive
andinternal energies, which turn out to depend on the point to
whichthe far field is referenced if . The above deriva-tion shows
that the amount that changes with a shiftin the origin is less than
, where is the efficiencyof the antenna [see (47)] and .
The quality factor is most often determined for antennaswhose
maximum linear dimensions are on the order of a wave-length or less
because it is these relatively small antennas thatusually determine
the bandwidth of a one-port antenna system.For example, the
bandwidth of a reflector antenna or an array
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1310 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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fed by one element is usually determined mainly by the
band-width of the feed element. Choosing the origin near the
centerof the dominant radiating sources of an antenna that is not
muchlarger than a wavelength across involves an ambiguity of nomore
than about a wavelength and, thus, an ambiguity in ofno more than
about . Nonetheless, itwould be desirable to determine an ideal
criterion for choosingthe origin of the coordinate system.
Fortunately, the results ofSections IV-C and IV-D reveal such a
criterion that we can beused to specify a practical way to choose a
reasonable positionof the origin for each tuned frequency .
As discussed in the previous subsection, it is assumed in
thederivation of (86) and thus in the derivation of the first
equationin (87), namely
(91)
that either , so that is independent of thelocation of the
origin, or if then the locationof the origin is chosen to produce a
that maintains the rela-tionship (86) and thus (91). If the
location of the origin is chosensuch that (86) remains valid when ,
then (86)and (80) imply
(92)
At a natural resonant frequency of an untuned antennawhere and ,
wehave shown (see Sections IV-B and IV-D) that
. Therefore, in order for (86) and (91) to hold at a nat-ural
resonant frequency when , (92) impliesthat should be . From (67e)
it is seen that thismeans that at a natural resonant frequency one
should choosethe position of the origin of the coordinate system to
make
(93)
If we know the far field pattern of the antenna, it is
straight-forward to evaluate the integral in (93) for different
positionsof the origin to find an origin that makes
at a natural resonant frequency . (Note from(90) that any vector
perpendicular to can be usedto shift without changing the value of
. Also, the valueof at a natural resonant frequency is usually
neg-ligible.)
At a natural antiresonant frequency of the untuned an-tenna
where and , wealso have (so that )and (92) along with (67e)
imply
(94)
Thus, one can find the position of the origin that makesin (94)
equal to in (94). To find
numerically from (94), the value of must bedetermined. This can
be done either by directly computing thederivative of the input
reactance of the antenna or by indirectlycomputing the derivative
of from the fields of the antennain expressions (52) or (59).
Once the origins and are found at the natural reso-nant and
antiresonant frequencies and of the untunedantenna, one can
linearly extrapolate between the positions ofthese origins to
obtain approximate values of the ideal posi-tion of the origin at
every frequency. [For tuned frequenciesbetween 0 and the lowest
natural resonant or antiresonant fre-quency , that is, for where is
the smallestfrequency (either a resonant or antiresonant frequency)
that sat-isfies , one can use (with equal to if isa natural
resonant frequency or if is a natural antireso-nant frequency).] In
Section VI-C, (93) and (94) are used to nu-merically evaluate the
ideal origin positions and fora Yagi antenna at two natural
resonant and two natural antires-onant frequencies. The numerical
results show that with theseorigins, the approximation in (86) and
(91) hold with consid-erable accuracy throughout the resonant and
antiresonant fre-quency bands.
We emphasize that this procedure for finding the ideal loca-tion
of the origin for determining an unambiguous exact of an-tennas
with is given for the sake of academiccompleteness and for
comparing the approximate formulas in(87) with an exact . The
formulas in (87) are the ones that areconvenient and useful in
numerical practice provided it is pos-sible to directly compute .
Even if an unambiguous exact
is desired, it can be found from (78) or (80) using any
posi-tion of the origin if . If , areasonable exact can be found
from (78) or (80) by choosingthe origin as the center of the sphere
that circumscribes the dom-inant sources of the antenna.
Once this origin of the circumscribing sphere is chosen, aneven
better exact Q can be obtained by adjusting the values of
to equal at the natural resonantand antiresonant frequencies
(that is, at or ).At other frequencies, the values of can be
adjusted by anamount equal to the linear extrapolation of the
adjustments atthe adjacent natural resonant and antiresonant
frequencies. For
, the linear extrapolation can be formed betweenan adjustment of
zero at and the adjustment at .This simple procedure can be used
independently of the valueof to define an exact that will
reasonably com-pensate for both a nonideal origin and spurious
contributionsto mentioned in the last paragraph of Section IV-D. In
Sec-tion VI-C, this simple procedure is applied to the Yagi
antennamentioned in the previous paragraph to obtain an
alternativeexact curve that agrees reasonably well with the exact Q
curveobtained by shifting the origin of the coordinates.
V. NEGATIVE VALUES OF AND
Our definitions of the internal energy and quality factorof
antennas are quite general and, in particular, in (72) it is
not
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
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assumed that the values of the real parts of the permeability
andpermittivity of the antenna material are greaterthan or equal to
zero. For low-loss materials (73) shows that
and even if andare negative [25][28]. Thus, it seems reasonable
to assumethat the defined by (78) with the given in (66)and
(72a)(72c) remains valid for low-loss materials withnegative and .
The approximate formulas for and
in (87) and the inverse relationship between themmay become less
accurate the faster and changeover the bandwidth of the antenna,
regardless of whether thevalues of and are positive or negative
(assuming theamount of this material is large enough to
significantlyaffect the bandwidth of the antenna).
For lossy materials, and can be less thanzero near
antiresonances of the material where the loss is verylarge, and it
is conceivable that or could benegative enough in the antenna
material to produce a negativevalue of (but not a negative value of
, which isdefined in terms of ). However, the antiresonances
inlossy media that produce negative values of orwould likely have
such narrow bandwidths or high loss that theywould make the antenna
impractical if they contributed signifi-cantly to and ; see Section
VI-D.
Finally, consider tuning an electrically small capacitive
orinductive antenna, that is, an antenna with , witha low-loss
series inductor or capacitor (having reactance )filled with either
a or material that can be positive or neg-ative. Since (73) implies
from (71) that , the tuned an-tenna has a reactance derivative
. It follows from (41), therefore, that the bandwidth of
anelectrically small capacitive or inductive antenna cannot be
dra-matically increased by tuning with a negative capacitance or
in-ductance instead of a positive inductance or capacitance,
respec-tively, as long as the capacitors and inductors are linear,
passive,low-loss circuit elements. Tretyakov et al. [28] conclude
that thebandwidths of radiating electric or magnetic line currents
cannotbe increased by covering them with electrically thin lossless
dis-persive materials having negative permeability or negative
per-mittivity, respectively.
VI. NUMERICAL RESULTS
In this section, the expressions for the exact bandwidth
andquality factor as well as for the approximate bandwidth
andquality factor derived in Sections IIIV are evaluated
numeri-cally for representative lossless and lossy tuned antennas.
Thesenumerical solutions are determined over a wide enough rangeof
frequencies to allow the antenna to vary in size from a
smallfraction of a wavelength to several wavelengths across. The
an-tennas considered here are the thin straight-wire antenna,
thecircular wire-loop antenna, a three-element directive Yagi
an-tenna, and a straight-wire antenna embedded in a frequency
de-pendent dielectric material. The numerical analysis of all but
thelast of these antennas is performed using the Numerical
Elec-tromagnetics Code, Version 4 (NEC) [29], which is capable
ofdetermining the current, input impedance, and far-fields of
theseantennas over a wide range of operating frequencies. For
each
of these tuned antennas, close agreement is found between
thenumerically computed exact and approximate formulas for
thebandwidth and quality factor over the full range of
frequencies.Moreover, the inverse relationship (87) between
bandwidth andquality factor is confirmed for each of the tuned
antennas atevery frequency.
Using the computed impedance data from NEC, the exactmatched
VSWR bandwidth is obtained by tuning the antennaat the desired
operating frequency with a lossless series in-ductor or capacitor.
A lossless inductor is used to tune theantennas initial reactance
to zero
if is less than zero, and a lossless capac-itor is used to tune
the antennas initial reactance to zero
if is greater thanzero. Once the antenna is tuned at the desired
operating fre-quency, the VSWR of the tuned antenna is determined
for all fre-quencies under the condition that the characteristic
feed-lineimpedance is equal to the antennas tuned impedance at
, that is, . The exact matchedVSWR bandwidth about the tuned
frequency is computed fora specific value of the VSWR by finding
the frequency range
in which the VSWR is less than or equal to . As de-fined in
Section III-B, the fractional matched VSWR bandwidthis thenwith .
To compare the inverse of the exactmatched VSWR bandwidth with the
antennas exact and ap-proximate quality factor, the exact matched
VSWR bandwidthis converted to an equivalent quality factor defined
by[see (87)]
(95)
In our numerical examples, the bandwidth VSWR is given by.
In addition to determining the inverse of the exact matchedVSWR
bandwidth , the exact of the antenna isfound using (80). The first
term on the right-hand side of(80), , is evaluated directly from
theantennas feed-point impedance with evaluatedfrom (8). The second
term on the right-hand side of (80),
, is evaluated numer-ically from the antennas current,
conductivity, and complexfar-field pattern.
The far-field dispersion energy is evaluated directlyfrom (67e).
The frequency derivative and integral in (67e) areevaluated
numerically for each observation angle and frequencyas necessary to
accurately compute the frequency derivativewith a finite
difference. These quantities are calculated withthe input (feed)
current held at a constant value independentof frequency. This is
accomplished in NEC by feeding theantenna with a voltage source
having a voltage equal to theantennas input impedance at each
frequency, thereby settingthe current to 1 A at all frequencies.
For the lossy antennas, thematerial-loss dispersion energy was
evaluated from(75).
The approximate conventional quality factor is givenin (81) and
we shall designate the newly derived approximate
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1312 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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Fig. 3. Input impedance of the center-fed, untuned, lossless
straight-wireantenna having a total length of 1 m and a wire
diameter of 1 mm.
quality factor in (86) by . We can rewrite from (86)and (8)
as
(96)where and are the resistance and reac-tance of the untuned
antenna. These approximate expressionsfor the quality factors given
by the conventional formula (81)(and/or its absolute value) and by
our newly derived formula(86), which is re-expressed in (96), are
evaluated using the an-tennas impedance and compared to the exact
values of andthe inverse of the exact VSWR bandwidth .
A. Bandwidth and Quality Factor of the Lossless
Straight-WireAntenna
The first antenna we consider is the lossless,
center-fed,straight-wire antenna that has an overall length of 1 m
and awire diameter of 1 mm. The NEC-calculated impedance ofthis
untuned antenna is given in Fig. 3 for a frequency rangecovering
the first natural resonance and antiresonance. Usingthe calculated
feed-point impedance of the corresponding tunedantenna, the exact
matched VSWR bandwidth was calculatedfor a bandwidth VSWR of .
A comparison of the exact , and for the tunedlossless
straight-wire antenna is shown in Fig. 4, which demon-strates
excellent agreement between the exact , the equivalent
obtained from the exact bandwidth, and the approximatequality
factor obtained from the frequency derivative of theantennas input
impedance. These latter three quality factors aredetermined in
significantly different manners, yet they remainin excellent
agreement throughout the entire frequency range.Fig. 4 also reveals
that the conventional approximation tothe quality factor,
determined from the frequency derivative ofthe antennas reactance,
does not provide a reasonable estimateof the exact or the inverse
of the antennas exact matchedVSWR bandwidth for frequencies about
the natural an-tiresonance.
Fig. 4. Comparison of the exactQ; Q ;Q , andQ (1.5:1 matched
VSWRbandwidth) for the center-fed, tuned, lossless straight-wire
antenna.
Fig. 5. Input impedance to higher frequencies of the center-fed,
untuned,lossless straight-wire antenna having a total length of 1 m
and a wire diameterof 1 mm.
Beyond the antennas first natural resonant and
antiresonantfrequency ranges, the antennas input impedance will
undergosuccessive alternating regions of natural resonances and
antires-onances, as seen in Fig. 5 for a frequency range of 450
MHzthrough 2000 MHz. At frequencies near the natural resonances,the
antennas input resistance is relatively low in value, while
atfrequencies near the natural antiresonances, it is relatively
highin value. A comparison of the exact , and for thetuned antenna
over this frequency range is given in Fig. 6, whereit can be seen
that the values of exact , and remainin excellent agreement over
the full frequency range. Fig. 6 re-veals again, however, that the
conventional approximate qualityfactor does not provide an accurate
estimate of the exactor inverse bandwidth in antiresonant frequency
ranges.
Considering the form of the exact expression in (80),
thereasonable agreement that exists between the exact and
theconventional approximation at low frequencies and in res-onant
frequency ranges, and the disagreement between the exact
and in antiresonant frequency ranges, we can concludethe
following. At very low frequencies, where the antenna is
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Fig. 6. Comparison of the Q; Q ;Q , and Q (1.5:1 matched
VSWRbandwidth) for the center-fed, tuned, lossless straight-wire
antenna over awider range of frequencies.
Fig. 7. Comparison of the Q; jQ j; Q , and Q (1.5:1 matched
VSWRbandwidth) for the center-fed, tuned, lossless straight-wire
antenna.
electrically small, and in resonant frequency ranges, the
dom-inant factor in determining the quality factor of the tuned
an-tenna is the frequency derivative of its input reactance .This
implies from (80) that the values of the dispersion ener-gies and
are close to zero in these resonant frequencyregions. Also, in
these resonant frequency regions, the value of
is relatively small. In antiresonant frequency regions,the
frequency derivative of the antennas input reactance is
lessdominant, the magnitude of can be significant, and thevalues of
the dispersion energies and become impor-tant in determining the
quality factor of the antenna.
As mentioned in Section IV-C, Rhodes [13] has
appropriatelysuggested that should be used instead of in
theconventional approximation (81) for the quality factor in
orderto increase its accuracy in antiresonant frequency ranges.
Theapproximate quality factor over the full range of frequencieswas
calculated using (81) with replaced by .This approximate quality
factor, which equals , is com-pared with the exact , and in Fig. 7.
Other than rightat the natural antiresonant frequencies of the
antenna, doesnot provide an accurate estimate of the exact or
inverse band-width in antiresonant frequency ranges. Using in
Fig. 8. Input impedance of the untuned, lossless circular-loop
antenna with aradius of .348 m and a wire diameter of 1 mm.
Fig. 9. Comparison of the Q;Q ;Q , and Q (1.5:1 matched
VSWRbandwidth) for the tuned, lossless circular-loop antenna.
(81) provides an accurate approximation to the exact and
in-verse bandwidth right at the natural antiresonant frequenciesof
the antenna because at these frequencies, and
.
B. Bandwidth and Quality Factor of Lossless and
LossyCircular-Loop Antennas
The lossless and lossy circular wire-loop antennas consid-ered
here have a total wire length of approximately 2.18 mand a wire
diameter of 1 mm such that the first natural reso-nant frequency of
the circular-loop, that is, the first where
and , equals the first natural resonantfrequency of the
straight-wire antenna discussed above, namely,
MHz. The input impedance of the loss-less circular-loop
calculated with NEC is plotted in Fig. 8. Oneof the important
differences to note in comparing the impedanceof the circular-loop
antenna to that of the straight-wire antennais that the
circular-loop antenna undergoes a natural antireso-nance (at
approximately 66 MHz) prior to the frequency whereit undergoes its
first natural resonance.
Fig. 9 compares the exact and for the loss-less circular-loop
antenna. Again the exact , the inverse of theexact matched VSWR
bandwidth , and the approximate
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1314 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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Fig. 10. Comparison of different methods for computing the
quality factorin the first antiresonant region of the tuned, lossy
circular-loop antenna with aradius of .348 m and a wire diameter of
.5 mm.
quality factor computed from (96) are in excellent agree-ment
over the whole frequency range. The conventional qualityfactor
computed from (81) gives accurate results at low fre-quencies and
in resonant frequency ranges. However, it does notproduce an
accurate approximation to and inverse of band-width in antiresonant
frequency ranges.
To demonstrate the significant contribution from the
mate-rial-loss dispersion-energy term to the exact of a lossy
an-tenna, the quality factor and bandwidth of the lossy
circular-loopantenna were computed. Loss was included in the NEC
modelof the circular wire-loop by specifying a finite copper-wire
con-ductivity ( (Ohm-m) ). The wire diameter wasreduced from 1 mm
to 0.5 mm to increase the resistance of thewire. The exact , and
the approximate quality factorsand computed for the lossy
circular-loop antenna are pre-sented in Fig. 10 in the frequency
range around the first antires-onance. In Fig. 10, the terms in
(80) comprising the expressionfor exact are shown separately to
illustrate the significance ofthe and terms in calculating the
exact . The conven-tional quality factor equals the first term of
the exact in(80). The curve labeled by in Fig. 10 is a
calculationof exact using only and the far-field
dispersion-energyterm . Note that the summa-tion of these two terms
does not give an accurate calculationof the exact . Once the
material-loss dispersion-energy term
, as well as the far-field dis-persion-energy term , is included
in the calculation of exact
, close agreement is obtained with the inverse of the
matchedVSWR bandwidth and with the approximate quality
factordetermined from (96).
In Figs. 11 and 12, the quality factors (as approximated by) for
the lossless and lossy straight-wire and circular wire-
loop antennas, both having wire diameters of 0.5 mm, are
com-pared to the Collin-Rothschild lower bounds on quality
factorfor the tuned electric or magnetic dipole antenna. This
lower-bound quality factor is given by [2], [4]
(97)
Fig. 11. Quality factors (as approximated by Q ) for the tuned,
lossless andlossy straight-wire (wire diameter equal to .5 mm)
antennas compared to theCollin-Rothschild lower bound for an
electric-dipole antenna.
Fig. 12. Quality factors (as approximated by Q ) for the tuned,
lossless andlossy circular-loop antennas (wire diameter equal to .5
mm) compared to theCollin-Rothschild lower bound for a
magnetic-dipole antenna.
where is the free-space wave number and is theradius of an
imaginary sphere circumscribing the electricallysmall dipole
antenna. For a lossy antenna, the lower bound in(97) is multiplied
by the NEC-computed radiation efficiencyof each antenna [30]. It is
obvious from Figs. 11 and 12 that thelower bounds on are
dramatically lower than the actual forlossless and lossy
straight-wire and circular wire-loop antennasat these frequencies
below the first resonance or antiresonance.This discrepancy between
the lower bound and actual qualityfactors implies that the
contribution to the from the electricand magnetic fields inside the
circumscribing sphere of radius
is the dominant contribution to the total even as, and
espe-cially as, the electrical size of the antennas becomes
small.
C. Bandwidth and Quality Factor of Lossless Yagi AntennaIn
Section IV-E, two techniques were described that allow
us to reasonably remove the ambiguity in determining the
exactthat may arise from the value of depending upon the
chosen position of the origin of the coordinates with respect
tothe antenna. (This ambiguity does not exist for antennas
havingfar-field magnitude patterns satisfying .) To
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Fig. 13. Schematic of a 3-element, lossless Yagi antenna.
Fig. 14. Input impedance of the untuned, lossless, 3-element
Yagi antenna.
illustrate the effectiveness of these techniques, the exact ofa
lossless Yagi antenna is determined and compared with theequivalent
obtained from the inverse of the exact matchedVSWR bandwidth. We
emphasize, however, that one can ap-proximate the exact bandwidth
of a tuned antenna through allfrequency ranges using (41), that is,
using the inverse of ,if the impedance of the antenna is known,
whether or not onecomputes an exact .
The Yagi antenna considered here consists of three
perfectlyconducting elements as shown in Fig. 13. Its untuned
impedanceis plotted in Fig. 14 over a frequency range that covers
two nat-ural resonant and two natural antiresonant frequencies. At
fre-quencies near its first natural resonance, where the Yagi
antennais designed to operate, it has a directive radiation pattern
and,as a result, is generally not equal to zero at
thesefrequencies. For this reason, the exact determined from
(80)might not accurately predict the inverse of the exact
bandwidthof the antenna.
To illustrate these points the inverse of the exact bandwidthand
the exact and approximate quality factors are plotted inFig. 15.
The center of the feed element is chosen as the coordi-nate origin
for the calculation of the exact from (80). Fig. 15shows that the
approximate quality factor determined from(96) and the determined
in (95) from the inverse of the exact
Fig. 15. Comparison of the Q; Q ;Q , and Q (1.5:1 matched
VSWRbandwidth) for the tuned, lossless, 3-element Yagi antenna with
the coordinateorigin placed at the center of the driven
element.
Fig. 16. Comparison of the Q;Q , and Q (1.5:1 matched
VSWRbandwidth) for the tuned, lossless, 3-element Yagi antenna with
the coordinateorigin shifted for each frequency by an amount
determined from the shifts atthe natural resonant and antiresonant
frequencies.
matched VSWR bandwidth are in excellent agreement at all
fre-quencies, whereas is inaccurate in the antiresonant
regions.Comparing the exact with reveals that near the Yagisfirst
natural resonance, where it is designed to operate with a
di-rective radiation pattern, the agreement is relatively poor.
As explained in Section IV-E, one can improve the agree-ment
between the exact and the inverse of bandwidthby shifting the
origin of the coordinate system with respect tothe antenna to make
at the natural resonant and an-tiresonant frequencies. Once the
positions of the shifted originsare determined at the natural
resonant and antiresonant frequen-cies, a linear interpolation
between these shifted origins is per-formed to compute the
appropriate shifted origins for frequen-cies between each natural
resonance and antiresonance. Theshifted origin for each natural
resonant frequency is found bycomputing through trial and error the
location of the coordinateorigin that results in a calculated . The
shifted originfor each natural antiresonant frequency is found by
computingthe location of the coordinate origin that results in a
calculated
. (Since the Yagi is lossless, .) Fig. 16
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1316 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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Fig. 17. Comparison of the Q; Q , and Q (1.5:1 matched
VSWRbandwidth) for the tuned, lossless, 3-element Yagi antenna with
the coordinateorigin placed at the center of the driven element,
but with the exact Q ateach frequency determined by interpolating
between its values at the naturalresonant and antiresonant
frequencies.
compares the exact computed with these shifted origins toand .
As one might expect, the major improvement in accu-racy of the
shifted-origin exact in Fig. 16 over the feed-ele-ment-origin exact
in Fig. 15 occurs near the first natural res-onance of the Yagi
antenna.
The drawback of this shifted-origin technique is the
largecomputer time required to calculate the frequency derivative
ofthe far field of the antenna at each natural resonant and
antireso-nant frequency as the origin of the coordinates is shifted
by trialand error to obtain the proper value of . As an
alternativeto this shifted-origin technique, we can first compute
an initialexact using an origin near the center of the imaginary
spherethat circumscribes the antenna. This exact will be
calculatedknowing that an ambiguity exists associated with the
specifiedlocation of the coordinate origin. However, the ambiguity
canbe corrected at the natural resonant and antiresonant
frequen-cies knowing at each of these natural frequencies. If
thedifferences between the exact calculated with the origin nearthe
center of the circumscribing sphere and are taken ascorrections at
the natural frequencies, we can interpolate thesecorrections
between the natural resonant and antiresonant fre-quencies to
arrive at a full set of corrections for all frequen-cies. This
allows us to compute a corrected exact withouthaving to determine
the far fields and their frequency derivativesat each frequency for
different coordinate origins. This interpo-lation technique was
applied to the values of the exact initiallycalculated with the
center of the feed element as the referencecoordinate origin. Fig.
17 shows that the resulting interpolatedexact compares favorably
with and as well as withthe shifted-origin exact shown in Fig.
16.
D. Bandwidth and Quality Factor of a Straight-WireEmbedded in a
Lossy Dispersive Dielectric
Our definitions of internal energies in (67a)(67c) or (72),
andthus include terms involving the frequency derivatives of
theconstitutive parameters. To confirm that these derivative
termsshould indeed be included as part of the energy used to
define
, we embed the lossless, center-fed, straight-wire antenna
de-
Fig. 18. Input impedance of the center-fed, untuned, lossless
straight-wireantenna having a total length of 1 m and a wire
diameter of 1 mm, and embeddedin a lossy dispersive dielectric.
scribed in Section VI-A in a lossy dispersive dielectric
materialwith Lorentz permittivity given by
(98)
for through the first resonant frequency of the antenna,
wherethe electric susceptibility constant , the offset
relativepermittivity constant , the loss constant , andthe Lorentz
antiresonant frequency Hz.For frequencies MHz, we can model this
embeddedantenna by a constant inductance ( henrys) inseries with a
frequency dependent radiation resistance (
Ohms) and a lossy capacitance (farads, Ohms). The
impedance and of the untuned embedded antennais shown in Fig.
18, which agrees closely with the impedance(not shown) computed
with the NEC code. The efficiency
of this antenna is less than 5% for frequencies less than80 MHz
and thus it is not a practical antenna throughout aboutthe first
half of the frequency range shown in Fig. 18.
Fig. 19 demonstrates the close agreement between the inverseof
the exact bandwidth and the approximation for theinverse of the
bandwidth of the tuned antenna, as well as thefailure in the
antiresonant region of the conventional expression
for the quality factor of the tuned antenna. In the
frequencyrange from about 30 to 70 MHz, the exact does not
agreewell with the inverse of the exact bandwidth (or with
theapproximation ) because the antenna material is both highlylossy
and dispersiveso dispersive, in fact, that the value of
can become negative to make equal to zero atfrequencies near 40
MHz and 60 MHz. Thus, as pointed out inSection IV-D, one would not
expect the exact to be a highlyaccurate approximation (in this
frequency range) to the inverseof the exact bandwidth. Most
noteworthy in Fig. 19 is the qualityfactor where
(99)
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Fig. 19. Comparison of the exact Q; Q ;Q ;Q (1.5:1 matched
VSWRbandwidth), and Q Q for the center-fed, tuned, lossless
straight-wireantenna embedded in a lossy dispersive dielectric.
which is the amount that the is reduced by the omission of
thederivative term in the first integral of (72b). Fig. 19 shows
that
the omission of this would produce a much less accuratevalue of
the exact quality factor over the frequency range wherethe term
contributes significantly to .
VII. CONCLUSIONThe input reactance of a general one-port linear
antenna can
vary over a large range of negative and positive values as
thefrequency of the antenna sweeps through successive natural
res-onances and antiresonances of the antenna. If, however, the
an-tennas input reactance is tuned to a value of zero at any
fre-quency by means of a series inductor or capacitor, a
matchedVSWR fractional bandwidth of the antenna can be defined
thatexists at every frequency . Moreover, this fractional
band-width, , is given approximately by the simple for-mula in (41)
for any frequency at which the antenna is tuned.
The internal energy of the same general one-portlinear antenna
tuned at a frequency is defined in (66)(67) interms of an
integration of the electric and magnetic fields of theantenna. This
internal energy excludes the radiation fields butincludes terms
involving the frequency derivative of the con-stitutive parameters.
In defining asin (78), the inclusion of these frequency-derivative
terms al-lows to remain inversely proportional to the
fractionalbandwidth , as expressed in (87), for antennasthat
contain materials with frequency dependent constitutiveparameters;
see Figs. 18 and 19. Although it is not impossiblefor to have a
negative value, and the values of theconstitutive parameters may
even be negative, it is proventhat the internal energy density in
lossless antenna material isalways greater than or equal to zero;
see (69).
For antennas with asymmetric far-field magnitude patterns,the
value of the internal energy and thus willdepend upon the position
of the origin of the coordinate systemwith respect to the antenna
(because of the radiation fields thatare subtracted to yield a
finite value for the internal energy). Areasonable choice for the
origin of the coordinate system is thecenter of the sphere that
circumscribes the dominant sources of
the antenna. Nonetheless, a simple procedure is given at the
endof Section IV-E for eliminating this ambiguity and determininga
well-defined exact value of .
Although the value of and thus can be de-termined, in principle,
by integrating the electric and magneticfields of the antenna
throughout all space, it may be prohibitivein numerical practice to
evaluate the exact by means ofthis direct volume integration.
Therfore, an alternative expres-sion for the exact is given in (80)
in terms of the fre-quency derivative of the input reactance of the
antenna and twodispersion energies. One of the dispersion
energiesrequires an integration of the far-field and the
otherrequires an integration of the fields over the portions of the
an-tenna material that exhibit loss. Each of these integrations
ismuch less demanding than integrating the fields of the
antennaover all space to compute the exact .
In Section VI-D, this alternative expression in (80) for
theexact is evaluated numerically for straight-wire and wire-loop
lossy and lossless antennas, as well as for a Yagi antennaand a
straight-wire antenna embedded in a lossy dispersive di-electric,
over frequency ranges that cover several resonant andantiresonant
frequency bands. In all cases, except for the last an-tenna in a
frequency range where the efficiency was less than 5%(rendering the
antenna impractical in this frequency range), theexact agreed
closely with the inverse of the exact computedbandwidth and with
the approximate formula for the qualityfactor and inverse bandwidth
given in (87). In fact, all our com-parisons to date for practical
antennas indicate that the simpleapproximation, , in (87) for
andthe inverse of the matched VSWR bandwidth is so accurate thatit
makes the evaluation of the exact and exact bandwidth prac-tically
unnecessary unless the frequency derivative of the inputimpedance
of the antenna, specifically , is not readilycomputable.
APPENDIX ADERIVATION OF EXPRESSION IN (64) FOR
To derive the expression (64) for , begin by taking thefrequency
derivative of Maxwells equations
(A.1)to get
(A.2)Scalar multiply the first equation in (A.2) by and the
com-plex conjugate of the second equation in (A.1) by , then
sub-tract the two resulting equations to get
(A.3)Similarly, scalar multiply the complex conjugate of the
secondequation in (A.2) by and the first equation in (A.1) by ,then
subtract the two resulting equations to get
(A.4)Subtracting (A.3) from (A.4) yields
(A.5)
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1318 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO.
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Integrate (A.5) over the volume between the shieldedpower supply
and a sphere of radius that surrounds the an-tenna system, apply
the divergence theorem, and take the limitas to get
(A.6)
The left-hand side of (A.6) results from (13)(14) and the
factmentioned in Section II that one of the basis fields in the
feedline of the antenna can always be made independent of
fre-quency. This implies from the frequency derivative of (14)
that
(A.7)If the frequency derivatives in (A.6) are taken while
holdingconstant with frequency, on the left-hand side of (A.6).Then
substituting and takingthe imaginary part of (A.6) produces
(A.8)
Taking the frequency derivative of, which is real, shows
that
(A.9)
and thus (A.8) can be rewritten as
(A.10)
where we have used the fact that the real part of the last
integrandin (A.10) is zero for in the part of that lies outside
theantenna material (that is, outside ).
Lastly, we evaluate the integral in (A.10) in terms of thefar
electric field pattern defined in (20) by expandingand in a Wilcox
series [31]
(A.11)
(A.12)
and by taking the derivative with respect to frequency of
in(A.11) to get
(A.13)
where is the speed of light in free space. (Recall that.)
Crossing from (A.13) into obtained from (A.12)
and using a number of vector identities gives
(A.14)
which when substituted into (A.10) yields
(A.15)
an expression equal to (64) if .The term in (A.14) does not
appear in (A.15) be-
cause it integrates to zero. This can be proven by expanding
thefields of the antenna (outside an enclosing sphere) in a
completeset of vector spherical wave functions [32, Secs.
7.117.14],[33, ch. 9] (see Appendix C) and noting that each of the
spher-ical modes has a that is 90 degrees out of phase with
. Therefore, for each vector spherical mode,. Orthogonality of
the vector spherical modes then
demands that this result also holds for the complete sum
ofvector spherical modes, that is, for the total fields of the
antenna.Specifically
(A.16)
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YAGHJIAN AND BEST: IMPEDANCE, BANDWIDTH, AND OF ANTENNAS
1319
and
(A.17)
where in (A.17) we have used th