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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 4, APRIL 2005
Impedance, Bandwidth, and Q of Antennas
Arthur D. Yaghjian, Fellow, IEEE, and Steven R. Best, Senior Member, IEEE
Abstract—To address the need for fundamental universally valid definitions of exact bandwidth and quality factor ( ) of tuned antennas, as well as the need for efficient accurate approximate formulas for computing this bandwidth and , exact and approximate expressions are found for the bandwidth and of a general single-feed (one-port) lossy or lossless linear antenna tuned to resonance or antiresonance. The approximate expression derived for the exact bandwidth of a tuned antenna differs from previous approximate expressions in that it is inverly proportional to the magnitude 0 ( 0 ) of the frequency derivative of the input impedance and, for not too large a bandwidth, it is nearly equal to the exact bandwidth of the tuned antenna at every frequency 0 , that is, throughout antiresonant as well as resonant frequency of bands. It is also shown that an appropriately defined exact a tuned lossy or lossless antenna is approximately proportional to 0 ( 0 ) and thus this is approximately inverly proportional to the bandwidth (for not too large a bandwidth) of a simply tuned antenna at all frequencie
s. The exact of a tuned antenna is defined in terms of average internal energies that emerge naturally from Maxwell’s equations applied to the tuned antenna. The internal energies, which are similar but not identical to previously defined quality-factor energies, and the associated are proven to increa without bound as the size of an antenna is decread. Numerical solutions to thin straight-wire and wire-loop lossy and lossless antennas, as well as to a Yagi antenna and a straight-wire antenna embedded in a lossy dispersive dielectric, confirm the accuracy of the approximate expressions and the inver relationship between the defined bandwidth and the defined over frequency ranges that cover veral resonant and antiresonant frequency bands. Index Terms—Antennas, antiresonance, bandwidth, impedance, quality factor, resonance.
I. INTRODUCTION
T
HE primary purpo of this paper is twofold: first, to define a fundamental, universally applicable measure of bandwidth of a tuned antenna and to derive a uful approximate expression for this bandwidth in terms of the antenna’s input impedance that holds at every frequency, that is, throughout the entire antiresonant as well as resonant frequency ranges of the antenna; and cond,
to define an exact antenna quality factor independently of bandwidth, to derive an approximate expression for this exact , and to show that this is approximately inverly proportional to the defined bandwidth. The average “internal” electric, magnetic, and magnetoelectric energies that we u to define the exact of a linear antenna are similar though not identical to tho of previous authors [1]–[8]. The approximate expression for the bandwidth
Manuscript received October 2, 2003; revid September 14, 2004. This work was supported by the U.S. Air Force Office of Scientific Rearch (AFOSR). The authors are with the Air Force Rearch Laboratory, Hanscom AFB, MA 01731 USA (e-mail: arthur.yaghjian@hanscom.af.mil). Digital Object Identifier 10.1109/TAP.2005.844443
and its relationship to are both more generally applicable and more accurate than previous formulas. As part of the derivation of the relationship between bandwidth and , exact expressions for the input impedance of the antenna and its derivative with respect to frequency are found in terms of the fields of the antenna. The exact of a general lossy or lossless antenna is also re-expresd in terms of two dispersion energies and the frequency derivative of the input reactance of the antenna. The value of the total internal energy, as well as one of the dispersion energies, for an antenna with an asymmetric far-field magnitude pattern, and thus the value of for such an antenna, is shown to depe
nd on the chon position of the origin of the coordinate system to which the fields of the antenna are referenced. A practical method is found to emerge naturally from the derivations that removes this ambiguity from the definition of for a general antenna.1 The validity and accuracy of the expressions are confirmed by the numerical solutions to straight-wire and wire-loop, lossy and lossless tuned antennas, as well as to a Yagi antenna and a straight-wire antenna embedded in a frequency dependent dielectric material, over a wide enough range of frequencies to cover veral resonant and antiresonant frequency bands. The remainder of the paper, many of the results of which were first prented in [9], is organized as follows. Preliminary definitions required for the derivations of the expressions for impedance, bandwidth, and of an antenna are given in Section II. In Section III, the fractional conductance bandwidth and the fractional matched voltage-standing-wave-ratio (VSWR) bandwidth are defined and determined approximately for a general tuned antenna in terms of the input resistance and magnitude of the frequency derivative of the input impedance of the antenna. It is shown that the matched VSWR bandwidth is the more fundamental measure of bandwidth becau, unlike the conductance bandwidth, it exists in general for all frequencies at which an antenna is tuned. (Throughout this paper, we are considering only the bandwidth relative to a change in the accepted power and not to any additional loss of bandwidth caud, 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 in terms of the antenna fields, and the relationship between bandwidth and is determined. In particular, the frequency derivative of the input reactance is expresd in terms of integrals of the electric and magnetic fields of the tuned antenna. The integrals of the fields are then re-expresd in terms of internal
1This ambiguity in the values of internal energy and engendered by subtracting the radiation-field energy of an antenna with an asymmetric far-field magnitude pattern is not mentioned or addresd in [1]–[8], probably becau the references concentrate on defining the of individual spherical multipoles which have far-field magnitude patterns that are symmetric about the origin.
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energies ud to define the of the antenna and two dispersion energies: the first dispersion energy determined by an integral involving the far field and the frequency derivative of the far field of the antenna; and the cond determined by an integral involving the fields and the frequency derivative of the fields within the antenna material. The dependence in the value of the far-field dispersion energy on the origin of the coordinate system, and thus the ambiguity in (mentioned above), is removed by the procedure derived in Section IV-E. An apparently new energy theorem proven in Appendix B is ud to derive a number of inequalities that the constitutive parameters must satisfy in lossless antenna material. We find that the Foster reactance theorem, which states that the frequency derivative of the reactance of a one-port linear, lossless, passive network is always positive, does not hold for antennas (whether or not the antenna is lossless becau the radiation from the antenna acts as a loss) [10, Sec. 8-4]. Although the general formula we derive for the bandwidth of an antenna involves the frequency derivative of resistance as well as the frequency derivative of reactance, it is found that the half-power matched VSWR bandwidth of a simply tuned lossy or lossless antenna is approximately equal to for all frequencies if the bandwidth of the antenna is not too large. It is proven in Appendix C that the of an antenna increas extremely rapidl
y as the maximum dimension of the source region is decread while maintaining the frequency, efficiency, and far-field pattern—making supergain above a few dB impractical. It is also shown in Appendix C that the quality factors determined by previous authors [1]–[3] are lower bounds applied to electrically small antennas with for our defined and . nondispersive In Section V, we discuss how the internal energy, , and bandwidth of an antenna would be affected by the prence of mateor . rial with negative values of In Section VI, exact VSWR bandwidths are computed from the magnitude of the reflection coefficient versus frequency curves obtained from the numerical solutions to tuned, thin straight-wire and wire-loop lossy and lossless antennas ranging in length from a small fraction of a wavelength to many wavelengths, as well as to a tuned Yagi antenna and a straight-wire antenna embedded in a frequency dependent dielectric material. for the antennas are computed from The exact values of the general expression (80) derived for the of tuned antennas. The exact values of VSWR bandwidth and are compared to the approximate values obtained from the derived approximate formulas in (87) for VSWR bandwidth and . The numerical comparisons confirm that the approximate formulas in (87) of a tuned antenna give much for VSWR bandwidth and more accurate values in antiresonant frequency ranges than the conventional formula (81) (or its absolute value) commonly ud to determine bandwidth and quality factor. Before leaving this Introduction, a few remarks about the ufulness of antenna may be appropriate. We can ask why the concept of a
ntenna is introduced when it is the bandwidth of an antenna that has practical importance. One advantage of is that the inver of the matched VSWR bandwidth of an anis approximated by the value of tenna tuned at the frequency the of the antenna at the single frequency . The bandwidth of some antennas may be much more difficult to directly com-
Fig. 1. Schematic of a general transmitting antenna, its feed line, and its shielded power supply.
pute, measure, or estimate than the , which is fundamentally defined in terms of the fields of the antenna, is independent of the characteristic impedance of the antenna’s feed line, and has a number of lower-bound formulas derived in the published literature [1]–[3] (e Appendix C). The simple approximate, yet accurate formulas for exact bandwidth and that are derived in the prent paper can be evaluated for an antenna and compared to the lower bounds for to decide if the antenna is nearly optimized with respect to and bandwidth. It is often possible to increa the bandwidth of electrically small antennas by simply restructuring the antenna to reduce its interior fields and therefore its [11]. Moreover, becau the of an antenna is determined by the fields of the antenna, Maxwell’s equations can be ud, as we do in Appendix C, to obtain fundamental limitations on the bandwidth of antennas. Finally, regardless of the utility of the concepts of and bandwidth, it ems quite remarkable that at any frequency of most antennas, the , which is defined i
n terms of the fields of a simply tuned one-port linear passive antenna, and the bandwidth, which is defined in terms of the input reflection coefficient of the same antenna, are approximately inverly proportional (provided the bandwidth is narrow enough) and that this approximate inver relationship is given by the simple formulas in (87) below. II. PRELIMINARY DEFINITIONS Consider a general transmitting antenna (shown schematically in Fig. 1) compod of electromagnetically linear materials and fed by a waveguide or transmission line (hereinafter referred to as the “feed line”) that carries just one propagating frequency . (The feed mode at the time-harmonic line is assumed to be compod of perfect conductors parated by a linear, homogeneous, isotropic medium.) The propagating mode in the feed line can be characterized at a reference plane (which parates the antenna from its shielded power supply) , complex current , and complex by a complex voltage defined as input impedance (1) is the input resistance and the real where the real number is the input reactance of the antenna. The voltage number and current can also be decompod into complex coefficients and of the propagating mode traveling toward (incident) and away (emergent) from the antenna, respectively, such that (2)
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(12) Becau the tuning inductor or capacitor is assumed lossless and in ries with the antenna, . The frequency , at , defines a resonant frequency of the antenna which and an antiresonant frequency of the antenna if if .2 For the sake of brevity, we shall sometimes as simply the refer to the resonant or antiresonant frequency “tuned frequency”. Note that we are defining a “tuned antenna” at the frequency as an antenna that has a total input reactance equal to zero at . Therefore, a tuned antenna will not have equal to zero unless the chara reflection coefficient 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 on of the feed line can be written in terms the reference plane of the of real electric and magnetic basis fields and cursingle propagating feed-line mode with voltage rent ; specifically (13) There may be evanescent modes on the feed line, but the fields of the evanescent modes are assumed to be negligible on the reference plane . If the dimensional units of and are chon as (meter) and they are consistent with Maxwell’s equations in the International System of mksA units, then has units of Volts, has units of Amperes, and the characof the feed line can be chon as a real teristic impedance positive constant independent of frequency with units of Ohms. It then follows that the normalization of the basis fields may be expresd as a nondimensional number equal to one, that is (14)
Fig. 2. Schematic of a general transmitting antenna, its feed line, its shielded power supply, and a ries reactance X .
with equal to the feed-line characteristic impedance, which can be chon to be independent of frequency [12, pp. 255–256]. 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 . with a ries Assume the antenna is tuned at a frequency (as shown in Fig. 2) comprid of either a posreactance or a positive ries capacitance , itive ries inductance where and are independent of frequency, to make the total reactance
(5) equal to zero at , that is (6) Then the derivative of as with respect to can be written
(7) or simply as (8) at the frequency . The equations corresponding to (1)–(4) for the tuned antenna can be written as
(9) (10) (11)
is the unit normal (pointing toward the antenna) on where simply cuts two wire leads from the plane . If the plane and refer to cona generator at quasistatic frequencies, ventional circuit voltages and curr
ents that do not rve as genuine modal coefficients. In that ca, the equations in (11) beequal to the internal recome definitions of and with sistance of the generator who internal reactance is tuned to zero. For the TEM mode on a coaxial cable, the basis fields , as well as the characteristic impedance , are independent of frequency. Also, one of the basis fields, either or , in addition to , can always be made independent of frequency for feed lines compod of perfect conductors parated by linear, homogeneous, isotropic materials [12, pp. 255–256]. We shall u this fact in deriving (64) below.
2The definitions of resonance and antiresonance come from the behavior of the reactance of ries and parallel RLC circuits, respectively, at their natural frequencies of oscillation. At the “resonant frequency” of a ries RLC circuit with positive L and C; X > 0 and at the “antiresonant frequency” of a parallel RLC circuit with positive L and C; X < 0.
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With the help of (14) we can determine various expressions accepted by the antenna for the total power
defines what is commonly called the matched VSWR bandwidth. We shall show that the matched VSWR bandwidth, unlike the conductance bandwidth, is well-defined for all frequencies at which the antenna is tuned to zero reactance. A. Conductance Bandwidth
(15) or
(16) in (15) denotes the complex conjugate and in (16) is the input conductance of the antenna. The power accepted by the antenna equals the power dissipated by the antenna in the form of power radiated by the antenna plus the power loss in the material of the of the antenna antenna. Defining the radiation resistance and the loss resistance of the antenna as as , we have The superscript
The conductance bandwidth for an antenna tuned at a freis defined as the difference between the two frequenquency cies at which the power accepted by the antenna, excited by a constant value of voltage , is a given fraction of the power accepted at the frequency . With the help of (9), the conductance at a frequency of an antenna tuned at the frequency can be written as (21) We can immediately e from (21) that there is a problem with using conductance bandwidth, namely, that th
e derivative of evaluated at equals (22) unless . This means that and thus it is not zero at in general the conductance will not reach a maximum at the frequency . Moreover, in antiresonant frequency ranges where both the resistance and reactance of the antenna are changing rapidly with frequency, the conductance may not posss a maximum and conquently the conductance bandwidth may not exist in the antiresonant frequency ranges. (As we shall show in Section III-B, the matched VSWR bandwidth does not suffer from the limitations.) Well away from the antiresonant frequency ranges of most is much smaller than antennas, we have , the conductance will peak at a frequency much clor to than the bandwidth, and a simple approximate expression for the conductance bandwidth can be found as follows. so that , we Having tuned the antenna at where by taking the can find the frequency frequency derivative of the expression for in (21) and tting it equal to zero to get
(17) so that (18) The power radiated can also be expresd in terms of the far fields of the antenna
(19) where is a surface in free space surrounding the antenna and its power supply, the solid angle integration element equals with being the usual spherical coordinates of the position vector , and the complex is defined by far electric field pattern (20) with being the speed of light in free space. The impedance of free space is denoted by in (19) and is the unit normal out of . The radiation resistance
is always equal becau the power rato or greater than zero diated by the antenna is always equal to or greater than zero . Also, the loss resistance is equal to or greater if the material of the antenna is passive than zero . III. FORMULAS FOR THE BANDWIDTH OF ANTENNAS The bandwidth of an antenna tuned to zero reactance is often defined in one of two ways. The first way defines what is commonly called the conductance bandwidth and the cond way
(23) , the functions With their derivatives can be expanded in Taylor ries about and (24a) (24b) (24c) (24d) which can be substituted into (23) to obtain for small
(25) or (26)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 4, APRIL 2005
In resonant frequency ranges well away from antiresonant ranges, we can assume so that (26) reduces to (27) in the peak of the conductance That is, the frequency shift of an antenna tuned at the frequency in a resonant freis given by the simple relationship quency range in (27) involving only the input resistance and the first frequency derivatives of the input resistance and reactance of the a
ntenna peaks at a at the tuned frequency . In other words, given by frequency (28) To determine the conductance bandwidth about the shifted at which peaks when the antenna is tuned frequency at which at , we find the two frequencies the accepted power is times its value at is given from (21) as (29) provided, as discusd above, we are well within the resonant . The value of the frequency ranges where constant , which lies in the range , is assumed . We can re-express (29) as in (30), shown at the chon bottom of the page, who left-hand side is more suitable to a than the left-hand side of (29) power ries expansion about becau the function , which rapidly varies from its value of zero at , is not contained in the denominator of (30). Since the conductance on the left-hand side of (30), and its , a Taylor ries expansion first derivative, are zero at of the left-hand side of (30) about recasts (30) in the form has been of (31), shown at the bottom of the page, in which becau for replaced by well within resonant frequency ranges. Evaluating the cond derivative in (31), we find
. Again, in resonant frewhere u has been made of quency ranges we can assume that and, therefore, (32) yields (33) under the additional assumption that the terms are negligible, an assumption that is generally satisfied if . is therefore The fractional conductance bandwidth given approximately by
(34) under the assumptions that we are well within resonant frewhere and quency ranges and do not change greatly over the bandwidth or, equivalently, (an assumption that holds if , which can always be satisfied if is chon small enough). The expression (34) for the fractional conductance bandwidth of tuned antennas was derived previously by , assuming Fante [3] for the half-power bandwidth . Rhodes [13] postulates the “half-power bandwidth” of an “electromagnetic system” as (35) as the of the electromagnetic system and He then defines finds “stored electric and magnetic energies” that are consistent and (59) below. The shortcomings of this with this method are that (35) is postulated as the half-power bandwidth rather than of a general antenna and that is defined as as a physical quantity determined independently of from the fields of the antenna. Moreover, (35) as well as (34) does not accurately approximate the bandwidth of tuned antennas in antiresonant frequency ranges (except at antiresonant frequencies ). with B. Matched VSWR Bandwidth (32) The matched voltage-standing-wave-ratio (VSWR) bandwidth for an antenna tuned at a frequency is defined as the
(30)
(31)
