In this paper, physical limitations on the partial gain G and relative bandwidth B are derived for antennas of arbitrary shape based on holomorphic properties of the forward scattering dyadic. The limitation on the performance of G and B is shown to be bounded from above by the long wavelength response of the antenna in terms of the electric and magnetic polarizability dyadics. The special case of ellipsoidal geometries are discussed

Gustafsson, Mats

Kristensson, Gerhard

Sohl, Christian

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Physical limitations on G and B for antennasSohl, Christian; Gustafsson, Mats; Kristensson, Gerhard2007Link to publicationCitation for published version (APA):Sohl, C., Gustafsson, M., & Kristensson, G. (2007). Physical limitations on G and B 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 G AND B FOR ANTENNASChristian Sohl, Mats Gustafsson, and Gerhard KristenssonDept. of Electroscience, Lund University, Box 118, S-211 00 Lund, Sweden{Christian.Sohl, Mats.Gustafsson, Gerhard.Kristensson}@es.lth.seAbstract: In this paper, physical limitations on the partial gain G and relative bandwidth B are derived forantennas of arbitrary shape based on holomorphic properties of the forward scattering dyadic. The limitationon the performance of G and B is shown to be bounded from above by the long wavelength response ofthe antenna in terms of the electric and magnetic polarizability dyadics. The special case of ellipsoidalgeometries are discussed.INTRODUCTIONThe physical limitations introduced in this paper generalize in many aspects the classical resultson the directivity and the Q-factor in Chu[1]. The advantages of the new limitations are at leastfourfold: 1) they are no longer restricted to the sphere, but holds for arbitrary antenna geometries;2) they are formulated in the antenna gain and bandwidth rather than the directivity and Q-factor;3) they permits studies of polarization effects such as diversity in MIMO applications; 4) theysuccessfully separate electric and magnetic antenna properties in terms of the intrinsic materialparameters. The present analysis is based on Gustafsson et al.[2] and is an application of theforward dispersion relation for broadband scattering introduced in Sohl et al.[3].SCATTERING AND ABSORBTION OF ANTENNASConsider an antenna of arbitrary shape surrounded by free space and subject to a plane-waveexcitation with Fourier amplitude E0 impinging in the k̂-direction, see Figure 1. The antennais modeled by the anisotropic and dispersive electric and magnetic susceptibility dyadics. Thebounding volume of the antenna is assumed to be of arbitrary shape but with the restriction thatthe complete power absorption of the incident wave is contained within this volume.Based on the physical principles of linearity, time-translational invariance, and causality, there isno fundamental difference between antennas and properly modeled scatterers. In fact, for a largeclass of antennas, the forward scattering dyadic S is holomorphic in the upper half plane Im k > 0and Cauchy’s integral theorem can be applied to the properly chosen function (k) = p̂∗e ·S(k, k̂) ·p̂e/k2, where p̂e = E0/|E0| denotes the electric polarization. Based on the optical theorem andthe long wavelength limit of , a dispersion relation for the extinction cross section σext can bederived in terms of the electric and magnetic polarizability dyadics, γe and γm, respectively, viz.,∫ ∞0σext(λ) dλ = π2(p̂∗e · γe · p̂e + p̂∗m · γm · p̂m), (1)where p̂m = k̂ × p̂e denotes the associated magnetic polarization. Note that the right hand sideof (1) only depends on the long wavelength response of the antenna. For details on the derivation,see Gustafsson et al.[2] and Sohl et al.[3].EMTS 2007 International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canadaarbitraryelementmatchingnetworkk̂x̂½antennareference plane¸(1{j½j2)G¸0 ¸2¸1B¸0G¤Figure 1: Illustration of a hypothetic antenna subject to an incident plane-wave (left figure) and the genericphysical limitation addressed in this paper (right figure).LIMITATIONS ON GAIN AND BANDWIDTHFor an unmatched antenna, the absorption cross section σa is given by σa = (1 − |ρ|2)σa0, whereσa0 is the absorption cross section for the corresponding perfectly matched antenna. Thus, for anywavelength λ ∈ [0,∞),σext ≥ σa = (1 − |ρ|2)σa0 = 14π(1 − |ρ|2)λ2G, (2)where the partial gain G = 4πσa0/λ2 have been introduced. The upper bound in (2) is in generalnot isoperimetric but can be sharpened with a priori knowledge of the absorption efficiency η =σa/σext. Recall that G, σa, and, σext depend on the incident direction k̂ as well as the electricpolarization p̂e.Introduce the wavelength interval Λ = [λ1, λ2] with center wavelength λ0 and associated relativebandwidth B. Then, (2) implies∫ ∞0σext(λ) dλ ≥ 14π∫Λ(1 − |ρ|2)λ2G(λ) dλ ≥ λ30GΛB(1 +112B2)(3)where GΛ = minλ∈Λ(1 − |ρ|2)G denotes the minimum partial realized gain over Λ. The factor1+B2/12 can be estimated from below by unity since B > 0. Recall that B  1 in many antennaapplications. Based on this fact, (1) inserted into (3) yields the following limitation on the productGΛB valid for any linear, time-translational and causal antenna:GΛB ≤ 4π3λ30(p̂∗e · γe · p̂e + p̂∗m · γm · p̂m). (4)Since both γe and γm are proportional to the volume of the antenna, it follows that the upper boundin (4) scales as (k0a)3, where a denotes the radius of, say, the volume-equivalent sphere.In many antenna applications it is desirable to bound GΛB independently of the material properties.For this purpose, introduce the high-contrast polarizability dyadic γ∞ as the limit of either γe orγm when the elements of the electric and magnetic susceptibility dyadics simultaneously becomeinfinitely large. From the variational properties of γe and γm discussed in Sohl et al.[3], it thenfollows thatsupp̂e·p̂m=0GΛB ≤ 4π3λ30(γ1 + γ2), (5)EMTS 2007 International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, CanadaProlate:Oblate:Prolate:Oblate:Prolate MSA:Oblate MSA:°1+°2°1+°2°1°1°1°10.2 0.4 0.6 0.8 100.511.522.53»° =VsjProlate: °2=°3Oblate: °1=°2Prolate: °1Oblate: °3(TE+TM)Chu (TE)D/Q/(k0a3)0.2 0.4 0.6 0.8 100.511.522.533.54»Figure 2: The eigenvalues γ1, γ2 and γ3 (left figure) and the the ratio D/Q (right figure) for the prolate andoblate spheroids as function of the semi-axis ratio ξ. Note the normalization with the volume Vs = 4πa3/3of the smallest circumscribing sphere.where γ1 and γ2 denote the largest and second largest eigenvalue of γ∞, respectively. For non-magnetic material parameters, γ2 = 0, and the right hand side of (5) can be improved by at most afactor of two. Recall that γ1 and γ2 are easily calculated for arbitrary geometries using either thefinite element method (FEM) or the method of moments (MoM).THE ELLIPSOIDAL GEOMETRIESClosed-form expressions of γ1, γ2 and γ3 exist for the ellipsoidal geometries, see Gustafsson etal.[2]. These eigenvalues are depicted in Figure 2 for the prolate and oblate spheroids togetherwith the bounds on D/Q in Gustafsson et al.[2] corresponding to GΛB in (5). The eigenvaluesγ1, γ2 and γ3 are seen to degenerate for the sphere (ξ = 1). In the case of a minimum scatteringantenna (MSA) with non-magnetic material parameters, (5) is, for any semi-axis ratio ξ ∈ [0, 1],sharper than the limitation in Chu[1] for the TE-polarization. For the generalization of Chu[1] toboth TE- and TM-polarizations, the present theory provides even sharper physical limitations, seeFigure 2.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

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Physical limitations on G and B for antennas

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