In this paper, physical limitations on the directivity D and Q-factor Q are derived for antennas of arbitrary shape. The quotient D/Q is shown to be bounded from above by the antenna volume and certain shape coefficients in terms of the eigenvalues of the high-contrast polarizability dyadic. The theory is exemplified by numerical results for the half-wave antenna with astonishing agreement

Gustafsson, Mats

Kristensson, Gerhard

Sohl, Christian

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Physical limitations on D/Q for antennasGustafsson, Mats; Sohl, Christian; Kristensson, Gerhard2007Link to publicationCitation for published version (APA):Gustafsson, M., Sohl, C., & Kristensson, G. (2007). Physical limitations on D/Q for antennas. 1-3. Paperpresented at International Symposium on Electromagnetic Theory, 2007, Ottawa, Canada.Total number of authors:3General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.PHYSICAL LIMITATIONS ON D/Q FOR ANTENNASMats Gustafsson, Christian Sohl, and Gerhard KristenssonDept. of Electroscience, Lund University, Box 118, S-211 00 Lund, Sweden{Mats.Gustafsson, Christian.Sohl, Gerhard.Kristensson}@es.lth.seAbstract: In this paper, physical limitations on the directivity D and Q-factor Q are derived for antennasof arbitrary shape. The quotient D/Q is shown to be bounded from above by the antenna volume andcertain shape coefficients in terms of the eigenvalues of the high-contrast polarizability dyadic. The theoryis exemplified by numerical results for the half-wave antenna with astonishing agreement.INTRODUCTIONThe limitations presented in this paper generalize the classical results by Chu[1]. The most impor-tant advantage of the new limitations is that they are not restricted to the smallest circumscribingsphere, but instead hold for arbitrary geometries. In fact, the sphere is far from optimal with respectto the volume for many antennas, cf., the half-wave antenna. The new formulation also permitsstudies of polarization effects, with important applications to diversity in MIMO systems.In contrast to the numerous papers that have been published on the subject, the present analysis issolely based on the fundamental principles of linearity, time-translational invariance and causalityrather than the spherical vector waves. Details on the analysis is given in Gustafsson et al.[2] andSohl et al.[3].LIMITATIONS ON THE DIRECTIVITY AND THE Q-FACTORConsider an arbitrary antenna V subject to an incident plane-wave Ein impinging in the k̂-direction.Let E0 denote the Fourier amplitude of Ein and introduce the electric and magnetic polarizationsp̂e = E0/|E0| and p̂m = k̂ × p̂e, respectively. The scattering properties of V in the forwarddirection are then described by the far-field amplitude via a linear and causal time-translationalinvariant convolution of Ein and a certain temporal dyadic St. Let S denote the Fourier transformof St and introduce (k) = p̂∗e · S(k, k̂) · p̂e/k2. The function  is then holomorphic in the upperhalf-plane Im k > 0 for a large class of dyadics St. Particular useful in the forthcoming analysis isthe static limit (0) = (p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) /4π, where γe and γm denote the electric andmagnetic polarizability dyadics, respectively.A resonance model for  in the vicinity of the resonance at k = k0 is(k) = (0)iQk0/(2k)1 − iQ (k/k0 − k0/k) /2 , (1)where k is real-valued and Q coincides with the Q-factor of the antenna when the relative band-width B is based on the half-power threshold, see Figure 1. For the generalization of (1) to multipleresonances, see Gustafsson et al.[2].The extinction cross section σext is directly related to the imaginary part of  via the optical theoremσext = 4πk Im . For the resonance at k = k0, (1) yields that the partial realized gain (1 − |ρ|2)GEMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, CanadaRe kIm k££k0% holomorphic/Qk0{k0kk0(0)Q/2%Im %/Q2k0k0(1{Q ){1 k0(1+Q ){1(0)Q/4%Figure 1: The symmetrically distributed poles × of  in the complex k-plane (left figure) and the corre-sponding single resonance model of Im  when Q  1 (right figure).satisfies(1 − |ρ|2)G = k2σaπ≤ k2σextπ= (0)2k2Qk01 + Q2(k/k0 − k0/k)2/4 , (2)where ρ and σa are the reflection coefficient and absorption cross section of the antenna, respec-tively. The upper bound in (2) is in general not isoperimetric but can be sharpened with a prioriknowledge of the absorption efficiency η = σa/σext. Recall that σext, σa, and G depend on theincident direction k̂ as well as the electric polarization p̂e.For an antenna which is perfectly matched (ρ = 0) at k = k0, the partial realized gain (1 − |ρ|2)Gcoincides with the partial directivity D. Under this assumption, (1) and (2) yields D/Q ≤ 2k30(0),orDQ≤ k302π(p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) . (3)Inequality (3) is fundamental and holds for any linear, time-translational invariant and causal an-tenna. Recall that γe and γm scales with the volume of the antenna, i.e., the right hand side of (3)is proportional to (k0a)3, where a is the radius of, say, the volume-equivalent sphere.From the analysis in Gustafsson et al.[2] it is clear that both γe and γm are bounded from aboveby the high-contrast polarizability dyadic γ∞. This fact can be used to estimate the quotient D/Qindependently of the electric and magnetic polarizations, viz.,supp̂e·p̂m=0DQ≤ k302π(γ1 + γ2), (4)where γ1 and γ2 denote the largest and second largest eigenvalues of γ∞, respectively. A simpleexample of (4) is given by the sphere of radius a. In this case the right hand side of (4) is equal to4(k0a)3, which is sharper than the corresponding result 6(k0a)3 for the generalization of Chu[1] inthe case of both TE- and TM-polarizations and k0a  1.The upper bounds in (3) and (4) differ only by a factor of π compared to the corresponding resultsin Gustafsson et al.[2] for the partial realized gain, viz., GΛB ≤ k30(γ1 + γ2)/2, where GΛ =mink∈K(1−|ρ|2)G is the minimum realizable gain on the interval K = [k0(1−B/2), k0(1+B/2)].EMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.82a2b4a=¸0.5 1 1.5 2 2.5 300.20.40.60.811.21.41.61.8¾=2¼a2 (1{j½j4a=¸2)G¾a¾extFigure 2: The extinction and absorption cross sections (left figure) and the partial realized gain (right fig-ure) for the half-wave antenna. The different curves correspond to Hallén’s integral equation (solid), theresonance model for  (dashed) and the physical limitation on GΛB (shaded).A NUMERICAL EXAMPLE: THE HALF-WAVE ANTENNAThe method of moments (MoM) solution of Hallén’s integral equation together with a gap feedmodel is used to determine the cross sections and partial realized gain for a cylindrical half-waveantenna with axial ratio b/a = 10−3, see Figure 2. The antenna is resonant for 2b ≈ 0.47λ0 withthe associated directivity D = 1.64. The relative bandwidth is about 8% at SWR = 2 and thecorresponding Q-factor based on a differentiation of the reflection coefficient is Q = 8.4. Notethat the absorption efficiency at the resonance is η ≈ 0.5, corresponding to a minimum scatteringantenna.The finite element method (FEM) is used to determine the eigenvalues γ1 och γ2 for the half-waveantenna. Under the assumptions of non-magnetic material parameters, the result is γ1 = 0.71a3with γ2 negligible. These eigenvalues inserted into (4) together with D = 1.64 and η = 0.5 yieldsQ ≥ 8.4 which coincides with the Q-factor above. In Figure 2, it is observed that the singleresonance model with Q = 8.4 is a good approximation of the cross sections and the realized gain.The corresponding bound on the product GΛB is illustrated by the rectangular region (GΛ = 1.64and B = 38%) for an arbitrary minimum scattering antenna confined within the cylindrical region.REFERENCES[1] L. J. Chu. “Physical limitations of omni-directional antennas”. Appl. Phys., vol. 19, pp. 1163–1175, 1948.[2] M. Gustafsson, C. Sohl, and G. Kristensson. “Physical limitations on antennas of arbitraryshape”. Submitted to IEEE Transactions on Antennas and Propagation.[3] C. Sohl, M. Gustafsson, and G. Kristensson. “Physical limitations on broadband scattering byheterogeneous obstacles”. Submitted to Journal of Mathematical Physics.EMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada

Physical limitations on D/Q for antennas

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Physical limitations on D/Q for antennas

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