Gustafsson, Mats

Sohl, Christian

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00A priori bounds on the onset frequency of wideband antennasSohl, Christian; Gustafsson, Mats2008Link to publicationCitation for published version (APA):Sohl, C., & Gustafsson, M. (2008). A priori bounds on the onset frequency of wideband antennas. 104-105.Paper presented at Nordic Radio Science and Communication Conference, 2008, Växjö, Sweden.Total number of authors:2General 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.A PRIORI BOUNDS ON THE ONSET FREQUENCY OF WIDEBANDANTENNASChristian Sohl and Mats GustafssonDepartment of Electrical and Information Technology, Lund University, SwedenEmail: {Christian.Sohl, Mats.Gustafsson}@eit.lth.seABSTRACTThis paper presents new bounds on the onset fre-quency and partial realized gain of wideband anten-nas. The result is a sum rule quantifying the antennaperformance in terms of its low-frequency propertiesvia certain static boundary-value problems. The the-oretical findings are compared with numerical simula-tions using the method of moments.1. INTRODUCTIONThis conference paper is based on a recent approachon physical bounds on antennas set forth in Refs. 1and 2. For this purpose, consider an antenna of arbi-trary shape modeled by linear and time-translationalinvariant constitutive relations in terms of the elec-tric and magnetic susceptibilities χe = χe(x) andχm = χm(x), respectively.1 Based on the holomorphicproperties of the forward scattering dyadic, a sum rulefor the partial realized gain g (with respect to the spa-tial k̂-direction and electric ê-polarization) is derivedin Refs. 1 and 2, viz.,∫ ∞0g(k; k̂, ê)k4dk =η2(ê∗ · γ(χe) · ê + (k̂ × ê∗) · γ(χm) · (k̂ × ê)), (1)where η = η(−k̂, ê∗) is a real-valued number in theunit interval [0, 1). Here, the static polarizabilitydyadic γ is defined by (` takes any of the values e andm depending on whether the problem is of electric ormagnetic nature)γ(χ`) =3∑i,j=1(âi · γij)âiâj ,where â1, â2 and â3 form an arbitrary set of linearlyindependent unit vectors, andγij =∫R3χ`(x)(âj −∇ψj(x)) dVx.1The results in this paper are formulated for isotropic suscep-tibilities, but they can easily be extended to include anisotropicor bi-anisotropic material models.The scalar potential ψj is the unique solution of thestatic boundary-value problem{∇ · ((χ`(x) + 1)∇ψj(x)) = âj · ∇χ`(x)ψj(x) = O(x−2) as x →∞x ∈ R3,where x = |x|. It is surprising to see that the integralon the left-hand side of (1) is related to the static orlow-frequency behavior of the antenna.As an example of how (1) can be used in mod-ern antenna design, consider a planar antenna Λ en-closed by a circular disk Λ+ = {x ∈ R3 : x ≤ a} ofradius a.2 Let n̂ denote the outward-directed unitnormal vector of the disk, and choose k̂ = n̂ andê = ρ̂, corresponding to a direction of observationand an electric polarization which are perpendicularand parallel to the disk, respectively. Introduce thefrequency band f ∈ [3.1, 10.6] GHz, or equivalentlyk ∈ [0.65, 2.22] cm−1, as the appropriate frequencyband for ultra-wideband (UWB) communication inNorth America. Assume that Λ is specified to havea partial realized gaing(k; n̂, ρ̂) ≥gp(n̂, ρ̂)k4/k41 k ∈ [0, k1]gp(n̂, ρ̂) k ∈ [k1, k2]0 otherwise, (2)where k1 = 0.65 cm−1 and k2 = 2.22 cm−1. Then, fora given threshold gp(n̂, ρ̂), it is desirable to determinethe smallest radius a such that it is feasible for Λ tohave a partial realized gain which satisfies (2).Based on (2), a straightforward calculation of (1)yields∫ ∞0g(k; n̂, ρ̂)k4dk ≥ gp(n̂, ρ̂)(1k31+∫ k2k1dkk4)=gp(n̂, ρ̂)34k32 − k31k31k32.From the analysis in Ref. 1, it follows that the polar-izability dyadics for the perfectly electric conductingcircular disk are γ(χe) = 16a3I⊥/3 and γ(χm) = 0,2Here, the support Λ is defined by Λ = Λe ∪ Λm, whereΛ` = {x ∈ R3 : χ`(x) 6= 0} and ` takes any of the values e andm.RVK08/MMWP08, June 9-13, 2008, Växjö University104respectively, where I⊥ = I − n̂n̂ denotes the projec-tion dyadic in R3. Hence, by inserting (??) into (1),one obtainsgp(n̂, ρ̂)a3≤ 0.55η(−n̂, ρ̂),where a now measures the radius of the disk in unitsof centimeters. For example, by invoking the upperbound η(−k̂, ê∗) < 1, it is concluded that the min-imum radius of the disk is 1.8 cm for gp(n̂, ρ̂) = 3,and 1.9 cm for gp(n̂, ρ̂) = 4. For many antennas, ηis close to 1/2 and a more realistic bound is therefore2.2 cm and 2.4 cm for gp(n̂, ρ̂) = 3 and gp(n̂, ρ̂) = 4,respectively.The conference presentation will focus on the use ofthis sum in antenna design, and how static considera-tions can offer fundamental insights into the behaviorof wideband antennas, e.g., by establishing estimateson the onset antenna frequency. The theoretical find-ings will be compared with several numerical simula-tions using the method of moments.2. REFERENCES[1] M. Gustafsson, C. Sohl, and G. Kristensson. Phys-ical limitations on antennas of arbitrary shape.Proc. R. Soc. A, 463:2589–2607, 2007.[2] C. Sohl and M. Gustafsson. A priori estimates onthe partial realized gain of ultra-wideband (UWB)antennas. Quart. J. Mech. Appl. Math., 2008.RVK08/MMWP08, June 9-13, 2008, Växjö University105

A priori bounds on the onset frequency of wideband antennas

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A priori bounds on the onset frequency of wideband antennas

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