Impedance Bandwidth and Q of Antennas

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

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)


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.


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)


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)


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


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].


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)


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


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).


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 ).


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


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


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


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


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


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


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


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)


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)


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)


and

(A.17)

where in (A.17) we have used th

Impedance Bandwidth and Q of Antennas

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