Radiation Q bounds for small electric dipoles over a conducting ground plane

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Radiation Q bounds for small electric dipoles over a conducting ground plane

  1. 1. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 1 Radiation Q Bounds for Small Electric Dipoles Over a Conducting Ground Plane Hsieh-Chi Chang, Yong Heui Cho, Member, IEEE, and Do-Hoon Kwon, Senior Member, IEEE Abstract—The radiation quality factors of vertically and hor- izontally polarized single-mode dipole antennas over a ground plane are investigated and compared with their free-space coun- terparts. The theoretical Q results are validated using simulation and measurement results of small spherical helix dipole antennas. For vertically polarized antennas, the bandwidth can be enhanced approximately by a factor of two. Index Terms—Antenna bandwidth, dipole antennas, electri- cally small antennas, quality factor. I. INTRODUCTION ELECTRICALLY small antennas has been an important research topic for many years, with the bandwidth per- formance characterized by the radiation quality factor Q. For small dipoles, use of the fundamental spherical mode current can describe antenna characteristics accurately. Using the fundamental TM01 spherical mode, the Chu bound [1] QChu fs = 1 ka + 1 (ka)3 (1) is the wave physics-imposed fundamental limit, where k = 2π/λ is the free-space wavenumber in terms of the wavelength λ and a is the radius of the smallest circumscribing sphere. In (1), the subscript ‘fs’ signifies that the Chu bound applies to antennas in free space. Since the energy inside the sphere is ig- nored in the Chu limit, practical antennas cannot achieve QChu fs . For air-core spherical antennas with an electric source over the sphere surface, Thal derived a new bound [2] that includes the internal energy and thus may be closely approached in practice. An approximate expression was developed by Hansen and Collin [3] and it is equal to QThal fs = 0.71327 ka + 1.49589 (ka)3 . (2) Radiation Q of spherical antennas with material cores excited by electric or magnetic surface currents was reported in [4]. For small antennas of arbitrary shape, theoretical bandwidth lower bounds have been developed based on quasi-electrostatic and quasi-magnetostatic scattering properties of antenna vol- umes [5]–[7]. Antennas having bandwidths that closely approach the the- oretical bounds are being actively investigated. Best reported This work was supported by the SI Organization, Inc. H.-C. Chang and D.-H. Kwon are with Department of Electri- cal and Computer Engineering, University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA (e-mail: hchang@ecs.umass.edu; dhk- won@ecs.umass.edu). Y. H. Cho is with the School of Information and Communication Engi- neering, Mokwon University, Daejeon, 302-729, Korea (e-mail: yongheui- cho@gmail.com). spherical helix electric monopole/dipole [8], [9] and mag- netic dipole [10] antennas with air core, where bandwidths consistent with the Thal bounds for electric and magnetic antennas were obtained. A low-Q spherical helix antenna design with tunability was reported in [11]. In [12], [13], small magnetic dipole antennas with air or material cores have been studied and they were verified to have bandwidths close to the theoretical limits. A capped monopole antenna design using high permeability shells for reducing the internal stored energy and thereby closely approaching QChu fs was reported in [14]. Comprehensive reviews of fundamental limits and electrically small antenna designs are available in [15]–[17]. These theoretical Q bounds and the associated antenna designs are for antennas in free space. A conductor-backed scenario is another practical antenna operating environment. Antenna bandwidths can be significantly affected by a ground plane and thus the associated Q bounds are expected to be different from those for free space antennas. However, how a small antenna’s Q bound changes with the introduction of a ground plane has not been quantified thus far. In [18], Sten et al. presented approximate closed-form expressions for the Q of conductor-backed dipoles by considering a sphere in free space that contains both the original and image antennas. Since the energy internal to this large sphere was not included in the Q computation, their bounds tend to be overly conservative and thus may not be closely approached using practical antennas. In this paper, fundamental Q bounds for small spherical dipole antennas in vertical and horizontal polarizations over a ground plane are investigated. Two quality factors — QChu gnd and QThal gnd — are computed as a function of the antenna size and the ground separation, following the definitions by Chu and Thal. The theoretical Q results are validated using simulation and measurement of spherical helix dipole antennas [9] placed over a ground plane, where an approximate value of Q is obtained using the frequency-swept driving point impedance [19]. For horizontally polarized dipoles, Q increases significantly from the free-space values as the ground separation is reduced, as expected. For vertically polarized dipoles, it is found that Q can decrease by approximately a factor of two, i.e. doubling of the bandwidth, compared with the free-space antenna of the same size. This anticipated bandwidth enhancement is supported by antenna simulation and measurement. In the following development, an ejωt time convention is assumed and omitted throughout.
  2. 2. 2 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION x y z O h (a) x y z O h h− (b) J aO ax az (c) aO J ax az (d) 2a Infinite PEC ground ss Fig. 1. Problem configuration of a spherical antenna over an infinite PEC ground plane. (a) The problem geometry. (b) The free-space configuration after applying image theory. (c) Electric surface current over the antenna surface for vertical polarization given by Js = −ˆθaJ0 sin θa. (d) Surface current for horizontal polarization given by Js = − ˆψaJ0 sin ψa. II. METHOD OF ANALYSIS A. Conductor-Backed Small Antennas In Fig. 1(a), a spherical antenna over a perfect electric conductor (PEC) ground plane is illustrated. The image theory is applied and an image spherical antenna is placed below the plane of the original ground, as illustrating in Fig. 1(b). The radius of the spherical antenna and the sphere center- to-ground separation are equal to a and h, characterized in terms of wavelength via ka and kh, respectively. Referenced to the antenna center Oa, a fundamental-mode electric surface current flows over the original antenna surface. For a vertically polarized antenna, Fig. 1(c) shows the electric surface current given by Js = −ˆθaJ0 sin θa, where θa is the angle measured from the +za axis and J0 is a constant. For a horizontally polarized antenna, the current is equal to Js = − ˆψaJ0 sin ψa in term of the angle ψ measured from the +xa axis, as illustrated in Fig. 1(d). In the presence of a ground plane of finite or infinite extent, part or all of the ground plane is sometimes considered to be part of the overall antenna. However, in this study, an antenna specifically refers to the spherical radiating structure only, without any part of the ground plane included. The energy stored inside the antenna refers to the non-propagating energy inside the original spherical antenna volume. It will be the differentiating quantity in computing QChu gnd and QThal gnd . In order to obtain a tight bound on Q, all non-propagating energy stored in all space should be accurately evaluated and accounted for. To this end, inclusion of the stored energy inside the sphere that tightly encloses both the original and image antennas is the key. For spherical antennas that excite only the dominant spherical mode, M and N spherical vector wave functions [20], [21] are the best suited for this purpose. Whenever possible, volume energy densities and radiation power densities as well as their volume and surface integrals will be represented in terms of vector wave functions. B. Field Expressions for a Single Spherical Antenna Outside the source volume, the electric and magnetic fields radiated by a current distribution are expanded in terms of the M and N vector functions by E = ∞ n=1 n m=−n AmnM(4) mn + BmnN(4) mn (3) H = j η ∞ n=1 n m=−n AmnN(4) mn + BmnM(4) mn (4) where η is the free-space intrinsic impedance. The expansion coefficients Amn and Bmn can be obtained using an inner product between the source and mode functions. For the electric surface current under consideration, they are given by Amn Bmn = − k2 η 2λmn M (1)∗ mn N (1)∗ mn · Jsds′ (5) where λmn = 2πn(n + 1) 2n + 1 (n + m)! (n − m)! . (6) In (3)–(5), superscripts ‘(1)’ and ‘(4)’ refer to the radial dependence of the first and fourth kinds [20], represented by the spherical Bessel function jn(kr) and the spherical Hankel function of the second kind h (2) n (kr), respectively. The current over the original antenna surface is designed to radiate the fundamental spherical mode, which is a reasonable approximation for small dipole antennas. The fields outside the sphere can be matched with those of a Hertzian dipole located at the sphere center having a proper dipole moment. The fields internal to the source distribution are obtained by the same set of equations (3)–(5) after the superscripts ‘(1)’ and ‘(4)’ are interchanged. The spherical mode of a ˆz-directed Hertzian dipole that generates the same fields as the spherical antenna outside the sphere in Fig. 1(c) is the TM01 mode. The modes of an ˆx- directed Hertzian dipole that generate the same fields as the current in Fig. 1(d) are a combination of TM11 and TM−11 modes. C. Vector Addition Theorem When a problem configuration involves multiple sources at different locations, it is not easy to analyze them in mixed coordinates. To compute the total fields from multiple sources, we can represent the fields from each source with respect to a common origin. To move the origin away from the sphere center, a vector addition theorem [21] can be employed. It is applied to the fields of both antennas in Fig. 1(b) to express the total fields referenced at O, which is the mid-point between the two sphere centers.
  3. 3. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 3 x Original antenna Image antenna aO iO O z 1 2 3 4 5 5 a Fig. 2. Division of the entire space into five different regions having volumes Vi (i = 1, 2, . . . , 5). D. Division of Space and Computation of Q Since the equivalent two-sphere configuration in Fig. 1(b) does not possess a spherical symmetry, it is not easy to preform integrations involving the surface and volume of a sphere having the center away from O. To circumvent this difficulty, the entire space is split into five regions, as indicated in Fig. 2.1 Since only non-propagating energy should be taken into account when computing the radiation Q, different regions of space need to be identified depending on the presence or absence of energy associated with radiation. At any point outside a spherical antenna surface, the surface current distribution can be replaced by an equivalent Hertzian dipole located at the sphere center. Hence, at any field point outside the two antenna spheres in Fig. 2, the total fields can be obtained by a vector sum of fields generated by two equivalent Hertzian dipoles located at (x, y, z) = (0, 0, h) and (0, 0, −h). This makes the spherical surface indicated by a black dashed contour the boundary between two volumes having standing- wave (inside) and propagating-wave (outside) characteristics. The red dashed circles indicate region boundaries for using different methods used for evaluating volume integrals. Vol- ume integrations for energy computation can be obtained in a closed form as an infinite series inside the small red dashed circle (region 1), outside the large red dashed circle (region 4), and inside the two spheres (region 5). In the two remaining regions (2 and 3), closed-form expressions cannot be obtained for the stored energies and thus volume integrals need to be evaluated numerically. The non-propagating stored electric and 1Stored energies in each region Wi m and Wi e (i = 1, 2, . . . , 5) as well as the radiated power P in the following development will account only for the z > 0 range. However, due to the relative simplicity of power and energy evaluations over the entire angular range of 0 ≤ θ ≤ π, volume Vi will include both the z > 0 portion and its image in the z < 0 range. A multiplication of a resulting integral by 1/2 will give appropriate values for the energies and power. magnetic energies are We = 1 2 V wedv′ (7) Wm = 1 2 V wmdv′ (8) where we and wm represent the electric and magnetic volume energy densities. They are equal to we = ǫ 4 |E| 2 − wr e (9) wm = µ 4 |H| 2 − wr m (10) where wr e and wr m are the electric and magnetic volume energy densities associated with radiation, respectively. In (7)–(8), the integration volume V is all space and the factor of 1/2 is due to the validity of the free-space configuration being limited to z > 0. For the stored energy in region 5, we choose the center of the original antenna sphere as the common origin and use the vector addition theorem to express the total fields due to both antennas. A volume integration over the original volume can be performed to find the energy stored inside the spherical antennas, which results in a covering infinite series. Finally, the radiated power of the original antenna above the ground plane is P = 1 2 Re r=h+a 1 2 E × H∗ · ds′ (11) where again the extra factor 1/2 is due to the limited validity of the two-sphere system. Any spherical surface of radius r > a + h can be used for closed-form evaluation of (11). At this point, all needed quantities for Q evaluation are computed. We can substitute all quantities into the definition of the quality factor Q = 2ω max {We, Wm} P . (12) In (12), omission or inclusion of the stored energies in region 5 leads to QChu gnd or QThal gnd as appropriate. E. Non-Propagating Stored Energies and Radiated Power 1) Region 1: The radial function is the spherical Bessel function. The electric and magnetic fields in this region represent standing waves, so there is no energy associated with radiation to the far zone. Therefore, both wr e and wr m are equal to zero. Expressions for non-propagating, stored electric and magnetic energies are given in Appendix A for both vertically and horizontally polarized spherical antennas. 2) Region 2: Here, the integration volume does not have boundaries that conform to constant-coordinate surfaces, re- quiring numerical integration. The fields are expressed by a superposition of contributions made by the two equivalent Hertzian dipoles, referenced at Oa and Oi as illustrated in Fig. 3. The coordinate transformations between (ra, θa, φa) referenced at Oa and (r, θ, φ) referenced at O are given by ra = r2 − 2rh cos θ + h2 (13) θa = cos−1 r cos θ − h ra . (14)
  4. 4. 4 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION , ,a iz z z aO O iO ax ix x aθ iθ θ ar ir r P Fig. 3. Coordinate transformation between three systems having origins at Oa, Oi, and O. Similarly, the transformations between the two coordinate systems centered at Oi and O are ri = r2 + 2rh cos θ + h2 (15) θi = cos−1 r cos θ + h ri . (16) The electric and magnetic fields can be represented by E(r, θ, φ) = Ea(ra, θa, φa) + Ei(ri, θi, φi) (17) H(r, θ, φ) = Ha(ra, θa, φa) + Hi(ri, θi, φi). (18) The fields in this region are standing waves. Hence, both wr e and wr m are equal to zero. The non-propagating, stored energies can be computed numerically using integrations (7)– (8) over the region volume. 3) Region 3: The fields in this region represent propagating waves, necessitating subtraction of the radiation energies from the total energies to obtain the non-propagating, stored electric and magnetic energies. Following the approach by Collin and Rothschild [22], the difference between stored magnetic and electric energies is found from the complex Poynting theorem [23] as W3 m − W3 e = 1 2 Im 1 2ω ∂V3 1 2 E × H∗ · ds′ (19) where the surface integral is over the bounding surface ∂V3 of the volume V3. The total energy density associated with radiation is equal to the real part of the radial component of the complex Poynting vector divided by the speed of energy flow, which is the speed of the light. Hence, the sum of electric and magnetic volume energy densities is wr e + wr m = ˆr · √ µǫRe 1 2 E × H∗ . (20) The sum of the stored energies can be obtained by first subtracting (20) from the total energy density and performing a volume integration over V3, i.e. W3 m + W3 e = 1 2 V3 ǫ 4 |E| 2 + µ 4 |H| 2 − (wr m + wr e) dv′ . (21) From (19) and (21), non-propagating electric and magnetic energies in region 3 are found. aO iO Original antenna Image antenna Fig. 4. Transformation of the coordinate system from Oi to Oa. 4) Region 4: This region extends out to infinity and the fields represent propagating waves. Hence, the energies asso- ciated with radiation should be subtracted from total energies. The same approach as for region 3 can be applied, as was also done by Fante [24]. Closed-form expressions for W4 e + W4 m and W4 m − W4 e in an infinite series format lead to individual stored energies. 5) Region 5: As illustrated in Fig. 4, the fields generated by the image antenna can be expanded around the center Oa of the original antenna. The total fields are then represented by a superposition of spherical vector waves referenced at Oa. The fields in this region are standing waves, so wr e and wr m are both equal to zero. Expressions for the fields and energies for vertically and horizontally polarized antennas summarized in Appendix B. 6) Radiated Power: The total fields are available in region 4, expanded in spherical wave functions referenced at O. Owing to orthogonality between different spherical modes, the total radiated power is the sum of radiated powers of individual modes. The radiated power of a vertically polarized conductor- backed spherical antenna for the radial mode index v is written as Pv = 1 2 4π 2v + 1 |αout E | 2 2k2η v(v + 1) A01 0v 2 (22) where αout E is defined in (29). The radiated power of a horizontally polarized conductor-backed spherical antenna for mode v is Pv = 1 2 4π 2v + 1 |αout E | 2 2k2η 1 2 A11 1v 2 + 1 2 B11 1v 2 [v(v + 1)] 2 + A−11 −1v 2 + B−11 −1v 2 . (23) Then, the total radiated power is P = ∞ v=1 Pv. (24) III. THEORETICAL RESULTS Stored energies, radiated power, and radiation Q are ana- lyzed for small antenna dimensions (ka ≤ 0.5) over a ground
  5. 5. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 5 0.1 0.2 0.3 0.4 0.5 10 0 10 1 10 2 10 3 10 4 ka (kh=0.5252) Q Qfs Thal Qfs Chu Qgnd Thal Qgnd Chu [18] Ant. sim. 0.1 0.2 0.3 0.4 0.5 10 0 10 1 10 2 10 3 10 4 ka (kh=2ka) Q Qfs Thal Qfs Chu Qgnd Thal Qgnd Chu [18] Ant.sim. (a) (b) Fig. 5. Radiation Q for vertically polarized antennas over a ground plane with respect to ka, together with those of the same-size antennas in free space. (a) For a fixed ground separation kh = 0.5252. (b) For a fixed ratio of ground separation to antenna size kh = 2ka. In both cases, the quality factor from [18] is also shown for comparison. separation up to one wavelength (kh ≤ 2π). For both antenna polarizations, it was found that the electric energy dominates for all combinations of ka and kh considered. A. Vertically Polarized Spherical Antenna Fig. 5 compares five different Q values over 0.1 ≤ ka ≤ 0.5 for a vertically polarized spherical antenna. In Fig. 5(a), the ground separation is fixed (kh = 0.5252), which makes the horizontal axis correspond to antenna size at a fixed height over the ground at a given frequency. In Fig. 5(b), the ratio between the antenna size and ground separation is fixed (kh = 2ka). In this case, the horizontal axis is proportional to frequency for fixed physical antenna size and ground separation. In each case, a Q curve based on the result from [18, Eq. (25)]2 is displayed as a reference, which is overly conservative (too low). Too low a minimum Q from [18] results from the omission of stored energies inside the imaginary sphere containing both the antenna and its image (the outer boundary of region 3 in Fig. 2 in our configuration), where high energy densities are expected. Hence, the minimum Q in [18] cannot be approached using practical antennas. In 2Note that the length a in [18] is the radius of the sphere that encloses both the original and image antennas. It corresponds to h + a in this work. 0 1 2 3 4 5 6 0.4 0.6 0.8 1 1.2 1.4 kh Q gnd Chu /Q fs Chu ka=0.5 0.4 0.3 0.2626 0.2 0.1 0 1 2 3 4 5 6 0.4 0.6 0.8 1 1.2 1.4 kh Q gnd Thal /Q fs Thal Ant. sim. (a) (b) Fig. 6. The Q lower bounds for vertical dipole antennas over a ground plane normalized to the bounds in free space. (a) QChu gnd /QChu fs for the Chu bound. (b) QThal gnd /QThal fs for the Thal bound. contrast, by including all (for QThal gnd ) or part (for QChu gnd ) of the energies inside the same sphere, the Q values in this study are tighter bounds that can be closely approached in practice. Curves for QChu fs and QThal fs are identical in Figs. 5(a)–5(b). For a vertically polarized antenna, it is observed that QChu gnd and QThal gnd are both significantly lower than their free-space counterparts. In other words, placing a spherical dipole of vertical polarization over a PEC ground plane can decrease the quality factor. This implies that the bandwidth will become broader than that of a same-sized antenna in free space. It is interesting to note that QThal gnd < QChu fs ; the effect of a ground plane is significant enough to make the realizable Q of a conductor-backed dipole lower than the fundamental limit for a free-space antenna of the same size. Fig. 5 and several following figures also have circled data points marked “antenna simulation.” They are Q values for real antenna designs associated with a given set of ka and kh values, which will be discussed later in Section IV. For several values of ka as a parameter, Fig. 6 presents the Q bounds for vertically polarized dipoles over a ground plane as a function of kh. The Chu bound QChu gnd and the Thal bound QThal gnd are normalized to their free-space bounds (1) and (2), respectively. Only the fundamental spherical spherical mode current is assumed over the antenna surface in deriving these two bounds. Since excitation of any high-order mode contributes only to increasing Q, QChu gnd in Fig. 6(a) represents
  6. 6. 6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 0 1 2 3 4 5 6 0.8 1 1.2 1.4 1.6 1.8 2 kh ka=0.5 0.4 0.3 0.2626 0.2 0.1 gnd Thal/Wfs Thal P W Thal/Pfs Thal gnd Fig. 7. The radiated power and stored energy for vertical polarization with respect to kh normalized to their free-space values. the fundamental bound that no Q of any passive physical antenna can fall below. The Thal bound QThal gnd in Fig. 6(b) is the fundamental bound for spherical electrical dipoles with air core excited by electric sources. It is expected that the two ratios in Fig. 6 approach unity for kh ≫ ka as the antenna bandwidth will not be significantly affected by a ground plane at a large electrical distance. This is observed for the two small antenna sizes ka = 0.1 and 0.2626 in the range of kh considered. There is a wide range of kh on the lower end where QChu gnd /QChu fs < 1, QThal gnd /QThal fs < 1, and they monotonically decrease with respect to decreasing kh. They reach values 0.5 or lower for the two small antenna sizes, which correspond to bandwidth enhancement by a factor of two or larger. In order to understand the reduction of Q better, Fig. 7 shows ratios for the stored energy WThal gnd /WThal fs and the radiated power PThal gnd /PThal fs that combine to describe the Q behavior. Here, we focus on the quantities for the Thal bound because it is QThal gnd that will be tested and validated in Section IV. It can be observed that WThal gnd is not significantly different from WThal fs . The radiated power is a stronger function of kh and PThal gnd /PThal fs → 2 as kh → 0 because the fields from the original and image antennas add constructively in all directions in the upper hemisphere in the far zone. This increase in the radiate power is the primary reason for the decrease in Q for small ground separations relative to the free-space case. B. Horizontally Polarized Spherical Antenna Fig. 8 compares different quality factors for a conductor- backed horizontally polarized spherical dipole antenna. The ground separation is fixed at kh = 0.5252 in Fig. 8(a) and the ground separation-to-antenna size ratio is fixed at kh = 2ka in Fig. 8(b). Both QChu gnd and QThal gnd are significantly higher than QChu fs and QThal fs . The Q results from [18] show significantly low bounds in this horizontal polarization case as in the vertical polarization case. For a fixed ground separation in Fig. 8(a), Q increases by approximately a constant factor with the intro- duction of a ground plane for all ka considered. For a fixed ratio kh/ka in Fig. 8(b), it is noted that QChu gnd , QThal gnd approach QChu fs , QThal fs as the ground separation is increased. Hence, it 0.1 0.2 0.3 0.4 0.5 10 0 10 1 10 2 10 3 10 4 ka (kh=0.5252) Q Q gnd Thal Q gnd Chu Q fs Thal Q fs Chu [18] Ant. sim. 0.1 0.2 0.3 0.4 0.5 10 0 10 1 10 2 10 3 10 4 ka (kh=2ka) Q Q gnd Thal Q gnd Chu Q fs Thal Q fs Chu [18] Ant. sim. (a) (b) Fig. 8. Radiation Q factors for horizontally polarized antennas over a ground plane with respect to ka, together with those of the same-sized antennas in free space. (a) For fixed ground separation kh = 0.5252. (b) For a fixed ratio of ground separation to antenna size kh = 2ka. In both cases, the quality factor from [18] is also shown for comparison. is anticipated that Q of a conductor-backed horizontal dipole may become lower than that of a free-space dipole for some kh. Fig. 9 presents the new bounds QChu gnd and QThal gnd for horizon- tally polarized dipoles over the ground plane as a function of kh for several electrically small values of ka. Similarly to the vertical polarization case, they represent fundamental bounds for an arbitrary dipole antenna in horizontal polarization that is contained in a sphere of radius a and for a spherical antenna of radius a with air core excited by an electric source. Q increases dramatically as kh → 0, but there exists a broad range of ground separation around kh = 2 in which Q is lower than the free-space value for both Chu and Thal bounds. A horizontal dipole that is quarter-wavelength above the ground plane belongs in this range. For small ka and large kh, both Q’s are not noticeably affected by the presence of the ground plane as expected. From Fig. 10, it is seen that WThal gnd /WThal fs is a slowly varying function of kh, especially for small ka. In contrast, PThal gnd /PThal fs is more sensitive to the ground separation and it approaches zero as kh → 0. This is the reason for rapidly increasing Q as kh is reduced. Since the image antenna points in the opposite direction from the original antenna, the dipole mode is suppressed and the quadrupole mode becomes the
  7. 7. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 7 0 1 2 3 4 5 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 kh Q gnd Chu /Q fs Chu ka=0.5 0.4 0.3 0.2626 0.2 0.1 0 1 2 3 4 5 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 kh Q gnd Thal /Q fs Thal Ant. sim. (a) (b) Fig. 9. The Q lower bounds for horizontal dipole antennas over a ground plane normalized to the bounds in free space. (a) QChu gnd /QChu fs for the Chu bound. (b) QThal gnd /QThal fs for the Thal bound. 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 kh ka=0.5 0.4 0.3 0.2626 0.2 0.1 gnd Thal/Wfs Thal P W Thal/Pfs Thal gnd Fig. 10. The radiated power and stored energy for horizontal polarization with respect to kh normalized to their free-space values. dominant radiation mechanism with an associated higher mode Q. A surge of PThal gnd /PThal fs above unity around kh = 2 and the associated range of kh for PThal gnd /PThal fs > WThal gnd /WThal fs define the range of ground separation that allows a wider bandwidth for a conductor-backed antenna. IV. ANTENNA SIMULATION AND MEASUREMENT To validate the Q results for conductor-backed dipoles in Section III, several folded spherical helix wire dipoles based TABLE I RESONANT VERTICALLY POLARIZED ANTENNA PROPERTIES OVER A PEC GROUND PLANE FOR DIFFERENT GROUND SEPARATIONS No. of Armsa No. of Turnsa kh Q imp gnd QThal gnd 3 1.59 0.5252 47.76 46.21 4 1.64 π/2 68.07 65.77 4 1.635 π 95.99 92.49 4 1.633 4.5 86.66 82.58 4 1.63 2π 88.77 88.56 a The exact geometrical parameters of the spherical helix antennas in this study are given by [8, Eqs. (1)–(5)] using these parameters together with the radius of the sphere. TABLE II RESONANT HORIZONTALLY POLARIZED ANTENNA PROPERTIES OVER A PEC GROUND PLANE FOR DIFFERENT GROUND SEPARATIONS No. of Arms No. of turns kh Q imp gnd QThal gnd 8 1.634 0.5252 399.77 371.64 4 1.631 π/2 77.11 72.05 4 1.675 π 92.94 88.37 4 1.637 2π 90.55 89.57 on [8], [9] were designed for matching to a 50-Ω transmis- sion line. Full-wave simulation results were obtained for the frequency-swept input impedance using a method-of-moments based tool FEKO version 6.1 from EMSS. An antenna Q was computed from the input impedance [19, Eq. (96)] at the design frequency and it is denoted by Qimp gnd. An electrically small antenna dimension ka = 0.2626 was chosen at the design frequency of 300 MHz (antenna diameter 2a = 84 mm) and several ground separations ranging from kh = 0.5252 to 2π were selected. For a given polarization and ground separation, the number of helical arms and the number of turns were optimized to obtain an input resistance at resonance close to 50 Ω. The reference antenna for comparison is the same-sized spherical dipole having four helical arms of 1.588 turns each from [9], which is designed for free space. The associated reference Q values are found to be Qimp fs = 90.17 from the simulated input impedance and QThal fs = 85.32 from (2). Tables I and II list the design parameters and Q values for each case. For all antenna designs, a good agreement was obtained between QThal gnd and Qimp gnd and this validates the new Q values for conductor-backed small electric dipole antennas. For kh = 2ka = 0.5252, the values of Qimp gnd are indicated by circles in Figs. 5 and 8, which shows an excellent agreement with QThal gnd for both polarizations. Data points from Tables I and II indicated by circles in Figs. 6(b) and 9(b) closely follow the curves associated with ka = 0.2626 for all values of kh considered. In both figures, the data points from antenna simulations represent normalized values to QThal fs . Among the designs reported above, one vertically polarized and one horizontally polarized spherical antennas were fabri- cated and their input reflection coefficients were measured. Copper wire of 2.6-mm diameter was bent to form each helical arm. For vertical polarization, the kh = 0.5252 variant (1st entry in Table I) was chosen. An aluminum plate of 0.9 m × 0.9 m was used as the ground plane after confirming by simulation that the ground size does not significantly affect the bandwidth and resonance frequency compared with the
  8. 8. 8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 250 260 270 280 290 300 310 320 330 340 350 −20 −15 −10 −5 0 Frequency (MHz) |S 11 |(dB) Measurement Simulation Sim: ref. ant. 250 260 270 280 290 300 310 320 330 340 350 −20 −15 −10 −5 0 Frequency (MHz) |S 11 |(dB) Measurement Simulation Sim: ref. ant. (b) (a) Mirror Ground spacer Foam Ground Fig. 11. Reflection coefficients of spherical antennas over a ground plane. (a) Vertically polarized dipole corresponding to the 1st entry in Table I. (b) Monopole version of the horizontally polarized dipole corresponding to the 2nd entry in Table II. For S11 associated with the monopole measurement, a reference impedance of 25 Ω was used. infinite ground case. In order to avoid the adverse effect of cable leakage current, an indirect impedance measure- ment technique without direct antenna excitation [25] was employed. Fig. 11(a) compares the simulated and measured input reflection coefficient with that of the free-space reference dipole. The indirect measurement approach requires high-SNR signals for accurate results and reliable result is obtained around the resonance frequency. Simulated and measured responses show an excellent agreement around the design frequency and the bandwidth is distinctly broader than that of the reference antenna, validating the prediction by the Q theory [Fig. 6(b)]. For horizontal polarization, the kh = π/2 version (2nd entry in Table II) was selected. In this case, symmetry in geometry, excitation, and polarization allows a monopole configuration using a conducting electric mirror in the symmetry plane of the antenna. Hence, a monopole version was fabricated. It was mounted on a 0.9 m × 0.9 m mirror (positioned horizontally) and then placed on a 1.2 m × 1.2 m ground plane (positioned vertically), both of aluminum plates. The measured reflection coefficient is compared in Fig. 11(b) with the simulation result together with the reference free- space case. The measured and simulated responses show an excellent agreement and the bandwidth is similar to that of the free-space antenna of the same size, as predicted by QThal gnd in Fig. 9(b). V. CONCLUSION Fundamental Chu radiation Q lower bounds for conductor- backed small electric dipole antennas have been established for vertical and horizontal polarizations. Assuming a funda- mental spherical mode radiation from a dipole over a ground plane, the fundamental Q bound was computed from the radiated power and the stored energies outside the antenna- circumscribing spheres in a free-space environment after the application of image theory. Due to portions of volume and surface integration bounds that do not conform to constant- coordinate surfaces and contours in spherical coordinates, the final results for Q are numerical in nature with the antenna size and ground separation as parameters. In applicable volumes, spherical vector wave functions and vector addition theorems were used to find the stored energies in a closed-form series format. Appropriate for air-core spherical antennas excited by an electric source over the sphere surface, Thal bounds on radiation Q were found for the same conductor-backed dipoles by including the stored energies within the spherical antenna volume. As were confirmed using antenna simulation and measurement, they may be closely approached using simple, practical antennas unlike the Chu bounds. For vertically polarized small dipoles over a ground plane, it was found that the Q may decrease by a factor around 0.5 for small ground separations, signifying the possibility of doubling the bandwidth compared with an antenna of the same size in free space. For horizontally polarized dipoles, the radiation Q dramatically increases as the ground separation is reduced, which is a qualitatively well-known phenomenon. As the ground separation is increased, there is a broad range of antenna height over the ground plane where a wider bandwidth is expected than in free space. Using simulation and measurement of of folded spherical helix dipole antennas of 84-mm diameter for 300 MHz over a ground plane, the newly established Thal bounds for air- core spherical dipole antennas were validated. The Q values computed from frequency-swept input impedance showed an excellent agreement with the predicted Thal bounds. APPENDIX A NON-PROPAGATING, STORED ENERGIES IN REGION 1 The expansion coefficients Amn uv and Bmn uv are Amn uv = Amn uv,a + Amn uv,i (25) Bmn uv = Bmn uv,a + Bmn uv,i (26) where Amn uv,a and Bmn uv,a are the expansion coefficients due to the original antenna; Amn uv,i and Bmn uv,i are the expansion coefficients due to the image antenna. All coefficients are referenced to O. When referenced individually to Oa and Oi for the original and image antennas, only one (vertical polarization) or two (horizontal polarization) spherical modes are excited. Translation of the origin for the associated M and N functions from Oa and Oi to O results in an infinite number of modes.
  9. 9. CHANG et al.: RADIATION Q BOUNDS FOR SMALL ELECTRIC DIPOLES OVER A CONDUCTING GROUND PLANE 9 Specifically, Amn uv,a and Bmn uv,a are given by Cruzan [21, Fig. 1 and Eq. (26)] using a ← h and θ0 ← π. Similarly, Amn uv,i and Bmn uv,i are obtained using a ← h and θ0 ← 0 in the same equation. A. Vertically Polarized Spherical Antenna The field expressions in region 1 are E = αout E ∞ v=1 A01 0vN (1) 0v (r, θ, φ) (27) H = αout H ∞ v=1 A01 0vM (1) 0v (r, θ, φ) (28) where αout E = jηJ0 ka 3 [2j0(ka) − j2(ka)] (29) αout H = j η αout E . (30) The non-propagating, electric energy is W1 e = 1 2 ∞ v=1 ǫ 4 αout E A01 0v 2 V1 N (1) 0v 2 dv′ . (31) The non-propagating, magnetic energy is W1 m = 1 2 ∞ v=1 µ 4 αout H A01 0v 2 V1 M (1) 0v 2 dv′ . (32) B. Horizontally Polarized Spherical Antenna The field expressions in region 1 are E = αout E ∞ v=1 1 2 A11 1vN (1) 1v (r, θ, φ) + B11 1vM (1) 1v (r, θ, φ) − A−11 −1vN (1) −1v(r, θ, φ) − B−11 −1vM (1) −1v(r, θ, φ) (33) H = αout H ∞ v=1 1 2 A11 1vM (1) 1v (r, θ, φ) + B11 1vN (1) 1v (r, θ, φ) − A−11 −1vM (1) −1v(r, θ, φ) − B−11 −1vN (1) −1v (r, θ, φ) . (34) The non-propagating, electric and magnetic energies are given by (35) and (36), respectively, shown on the following page. APPENDIX B NON-PROPAGATING, STORED ENERGIES IN REGION 5 Here, the expansion coefficients are referenced to Oa. The coefficients Amn uv,i and Bmn uv,i are given by Cruzan [21, Fig. 1 and Eq. (26)] using a ← 2h and θ0 ← 0. A. Vertically Polarized Spherical Antenna The field expressions in region 5 are E = ∞ v=1 αout E A01 0v,i + δ1vαin E N (1) 0v (ra, θa, φa) (37) H = ∞ v=1 αout H A01 0v,i + δ1vαin H M (1) 0v (ra, θa, φa) (38) where αin E and αin H are the same coefficients with αout E and αout H in (29)–(30) with the radial function changed to Hankel func- tion of the second kind. The symbol δ1v refers to Kronecker delta. The non-propagating, stored electric energy is W5 e = 1 2 ∞ v=1 ǫ 4 αout E A01 0v,i + δ1vαin E 2 V5 N (1) 0v 2 dv′ . (39) The non-propagating, stored magnetic energy is W5 m = 1 2 ∞ v=1 ǫ 4 αout H A01 0v,i + δ1vαin H 2 V5 M (1) 0v 2 dv′ . (40) B. Horizontally Polarized Spherical Antenna The field expressions in region 5 are E = ∞ v=1 1 2 αout E A11 1v,i − αin Eδ1v N (1) 1v + B11 1v,iM (1) 1v − αout E A−11 −1v,i − αin Eδ1v N (1) −1v + B−11 −1v,iM (1) −1v (41) H = ∞ v=1 1 2 αout H A11 1v,i − αin Hδ1v M (1) 1v + B11 1v,iN (1) 1v − αout H A−11 −1v,i − αin Hδ1v M (1) −1v + B−11 −1v,iN (1) −1v . (42) The non-propagating, stored, electric energy is W5 e = 1 2 V5 ǫ 4 |E| 2 dv′ (43) and the non-propagating, stored, magnetic energy is W5 m = 1 2 V5 µ 4 |H| 2 dv′ . (44) ACKNOWLEDGMENT The authors would like to thank Yutong Yang for fabrication and measurement of the antennas. REFERENCES [1] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [2] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2757–2763, Oct. 2006. [3] R. C. Hansen and R. E. Collin, “Physical limitation of omni-directional antennas,” IEEE Antennas Propag. Mag., vol. 54, no. 5, pp. 38–41, Oct. 2009. [4] T. V. Hansen, O. S. Kim, and O. Breinbjerg, “Stored energy and quality factor of spherical wave functions–in relation to spherical antennas with material cores,” IEEE Trans. Antennas Propag., vol. 60, no. 3, pp. 1281– 1290, Mar. 2012. [5] M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. A, vol. 463, pp. 2589–2607, 2007. [6] ——, “Illustrations of new physical bounds on linearly polarized an- tennas,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1319–1327, May 2009. [7] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3114–3121, Oct. 2010. [8] S. R. Best, “The radiation properties of electrically small folded spher- ical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 953–960, Apr. 2004. [9] ——, “Low Q electrically small linear and elliptical polarized spherical dipole antenna,” IEEE Trans. Antennas Propag., vol. 3, no. 3, pp. 1047– 1053, Mar. 2005.
  10. 10. 10 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION W1 e = 1 2 ∞ v=1 ǫ 4 αout E 2 V1 1 2 A11 1v 2 N (1) 1v 2 + A−11 −1v 2 N (1) 1v 2 + 1 2 B11 1v 2 M (1) 1v 2 + B−11 −1v 2 M (1) −1v 2 dv′ (35) W1 m = 1 2 ∞ v=1 µ 4 αout H 2 V1 1 2 A11 1v 2 M (1) 1v 2 + A−11 −1v 2 M (1) 1v 2 + 1 2 B11 1v 2 N (1) 1v 2 + B−11 −1v 2 N (1) −1v 2 dv′ . (36) [10] ——, “A low Q electrically small magnetic (TE mode) dipole,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 572–575, 2009. [11] J. J. Adams and J. T. Bernhard, “Tuning method for a new electrically small antenna with low Q,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 303–306, 2009. [12] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1898–1906, Jun. 2010. [13] O. S. Kim, “Low-Q electrically small spherical magnetic dipole anten- nas,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2210–2217, Jul. 2010. [14] H. R. Stuart and A. D. Yaghjian, “Approaching the lower bounds on Q for electrically small electric-dipole antennas using high permeability shells,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3865–3872, Dec. 2010. [15] R. C. Hansen, Electrically Small, Superdirective, and Superconducting Antennas. Hoboken, NJ: Wiley, 2006. [16] J. L. Volakis, C.-C. Chen, and K. Fujimoto, Small Antennas: Miniatur- ization Techniques & Applications. New York: McGraw-Hill, 2010. [17] D. F. Sievenpiper, D. C. Dawson, M. M. Jacob, T. Kanar, S. Kim, J. Long, and R. G. Quarfoth, “Experimental validation of performance limits and design guidelines for small antennas,” IEEE Trans. Antennas Propag., vol. 60, no. 1, pp. 8–19, Jan. 2012. [18] J. C.-E. Sten, A. Hujanen, and P. K. Koivisto, “Quality factor of an electrically small antenna radiating close to a conducting ground plane,” IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 829–837, May 2001. [19] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298– 1324, Apr. 2005. [20] J. A. Stratton, Eletromagnetic Theory. New York: McGraw-Hill, 1941. [21] O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math., vol. 20, no. 1, pp. 33–40, 1962. [22] R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. 12, no. 2, pp. 23–27, 1964. [23] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Piscataway, NJ: Wiley, 2001. [24] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, no. 2, pp. 151–155, Mar. 1969. [25] Y. Yang and D.-H. Kwon, “Impedance measurement approach for small antennas without direct cable attachment,” in Proc. 2013 IEEE Antennas Propag. Soc. Int. Symp., Orlando, FL, Jul. 2013, pp. 782–783.

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