Engineering of Composite Media for Shields at Microwave Frequencies

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1 Engineering of Coposite Media for Shields at Microwave Frequencies Marina Koledintseva, Poorna Chander avva, ichard DuBroff, Jaes Drewniak EMC Laboratory, ECE Departent, University of Missouri-olla, olla, MO, USA e-ail: Konstantin ozanov Institute for Theoretical and Applied Electroagnetics, ussian Acadey of Sciences, Moscow, ussia e-ail: Bruce Archabeault IBM Co. esearch Triangle Park, NC, USA e-ail: Abstract Analytical and nuerical odeling of coposites with an isotropic dielectric base and ultiphase conducting inclusions for the developent of wideband icrowave shields is considered. The odel uses Maxwell Garnett foralis for ultiphase ixtures. Such coposites are required in any engineering applications, including electroagnetic copatibility. Keywords- Maxwell Garnett foralis; shielding; conducting inclusions; dielectric base aterial I. INTODUCTION Engineering of coposite aterials with the desired frequency characteristics is an iportant proble for various applications in radio frequency and icrowave electronic devices, including design of filters and shields for electroagnetic copatibility purposes. Analytical and nuerical odeling of frequency response for a desired coposite aterial prior to anufacturing and testing of ultiple aterials will save resources and tie for the developent cycle. To characterize electroagnetic properties of coposite edia, it is iportant to know the electroagnetic paraeters of a host (atrix, base) aterial and inclusions. There are any ective edia theories (EMT) allowing hoogenization of coposite edia [-4]. However, hoogenization of a ixture is always an approxiation. Macroscopic paraeters, such as an ective perittivity, can be used only if the sources and fields are varying slowly, so that quasi-static approxiation in the frequency-doain is eployed. Factors affecting frequency characteristics of coposites are as follows: frequency dependence of constitutive paraeters of a host aterial; frequency dependences of constitutive paraeters of inclusions (perittivity, pereability, conductivity, etc.); shape of inclusions; volue fractions of a host aterial and inclusions (related to size of inclusions and packing density); orientation and alignent of electric and/or agnetic dipole oents of inclusions (due to their shape and crystallographic polarizability/agnetization); statistical distribution of the paraeters of inclusions; orphology of the coposite (contact between the inclusions, presence of clusters, etc. [5]). The Maxwell Garnett (MG) odel [6] is the siplest and the ost widely used for description of coposite edia at coparatively low concentrations of inclusions. Another iportant feature of the MG forulation is its linearity with respect to frequency for a ultiphase ixture. It allows frequency characteristics of edia to be represented in the for of rational-fractional functions convenient for recursive convolution procedures in nuerical tie-doain electroagnetic codes, such as FDTD [7]. Herein, we apply the extended MG foralis for frequency-dependent dielectric properties of inclusions and host aterial and electrodynaically anisotropic ixtures, in particular, with a rando in-plane distribution of particles in a thin layer. II. MAXWELL GANETT FOMALISM FO MULTIPHASE MIXTUES Assue that there is a ultiphase ixture of etallic or dielectric particles in a hoogeneous dielectric base that satisfies the following conditions: this ixture is electrodynaically isotropic; the ixture is linear, that is, none of its constitutive paraeters depends on the intensity of electroagnetic field; the ixture is non-paraetric, that is, its paraeters do not change in tie according to soe law as a result of soe external forces electrical, echanical, etc.; inclusions are at the distances greater than their characteristic size; the characteristic size of inclusions is sall copared to the wavelength in the ective ediu; inclusions are arbitrary randoly oriented ellipsoids; if there are conducting inclusions, their concentration should be lower than the percolation threshold [8]; /5/$. (C) 5 IEEE

2 inclusions are static, that is, they do not ove; the ixture is cheically stable, that is, inclusions do not dissolve in the host aterial, and do not interact cheically. The generalized Maxwell Garnett ixing forula for ultiphase ixtures with randoly oriented ellipsoidal inclusions is [9], n b fi ( i b ) + i j b Nij ( i b) b + () n Nij f ( ) i i b + i j b Nij ( i b where b is the relative perittivity of a base dielectric; i is the relative perittivity of the i-th sort of inclusions; f i is the volue fraction occupied by the inclusions of the i-th sort; N are the depolarization factors of the i -th sort of ij inclusions, and the index j,, corresponds to x,y, and z Cartesian coordinates. Forulas for calculating depolarization factors of ellipsoids and the table of depolarization factors of canonical spheroids - spheres, disks, and cylinders, can be found in [7]. Equation () allows that within the sae phase (aterial of inclusions), particles can have different depolarization factors. However, in reality it is alost ipossible to have perfect ellipsoidal or spheroidal particles, so, for any arbitrary shape a reasonable approxiation is needed. If the inclusions are thin cylinders, their two depolarization factors are close to /, and the d third can be calculated as in [], N ln l l is the length of fibers, and d is their diaeter. ) l d, where Equation () adits that the perittivities of the base and inclusions can be coplex functions of frequency in the for of b b + χ b ; χ, () i i + where b, i are the high-frequency perittivity values for the base aterial and for inclusions of the i-th type, respectively; and χ b, i are the corresponding dielectric susceptibility functions. Since the MG forula is linear, the resultant ective perittivity of the ixture can be also represented through ective high-frequency perittivity and susceptibility function, i χ. () + Thus, the MG foralis can serve as a basis for engineering coposite icrowave aterials. Their frequency responses can be synthesized taking the paraeters to describe base and inclusion aterials fro experiental, anufacturer s, or published reference data. This eans that atheatical odeling based on ()-() will provide an icient tool for the analysis of the frequency behavior of a coposite depending on the physical properties and geoetry of its constituents. Analytical and nuerical odeling of ixtures with the desired frequency characteristics (as well as odeling coatings, filters, or other passive devices using these ixtures) prior to anufacturing and testing of ultiple real aterials will save resources and tie for the developent cycle. As shown in [7], the ective susceptibility function of a ixture, in the siplest case can be approxiated by a Debye or Lorentzian dependence; the arbitrary coplex-shaped frequency dependences can then be approxiated by a series of Debye-like ters (with the first-order poles), N Ak χ. (4) jωτ k + If inclusions are conducting (etallic) particles, their frequency characteristic in ters of relative perittivity is σ i j j (5) ω As entioned above, the Maxwell Garnett ixing rule is applicable when the concentration of the conducting particles in the ixture is below the percolation threshold, p 4.5/ a<<, where a is an aspect (axis) ratio for the inclusions in the for of highly prolate spheroids [8]. Otherwise, the different approxiations fro the general ective ediu theories should be used, for exaple, McLachlan [] or Ghosh-Fuchs approxiations []. The base aterial ight be quite transparent over the frequency range where high shielding ectiveness is desired. However, if there are conducting inclusions, the shielding ectiveness will be provided by absorption of electroagnetic energy due to conductivity loss and to the diensional resonance in the particles. Presence of conductive particles will also increase reflection fro coposite layer. If there are agnetic inclusions, they ight absorb electroagnetic energy due to the phenoenon of natural ferroagnetic resonance, like in hexagonal ferrite powders []. As an exaple, the real and iaginary parts of the relative ective perittivity of a ixture of carbon fibers and PMMA base are shown in Figures and. PMMA is assued to be a Debye dielectric [9] with the paraeters S. 7,., and τ 8 s. For carbon inclusions, the relative k

3 pereability is µ, and the d.c. conductivity is r 5 σ 6.78 S/. Since << σ /( ω ), the real part in (5) can be neglected. The volue fraction is chosen as f i.5%. The length of fibers is l, while the diaeter is about d.5 µ (aspect ratio a l / d 4 ). This concentration is still below the percolation threshold [8]. It should be entioned that with the frequency increase, due to skin ect not all the bulk of an inclusion affects the ective perittivity, but only a thin layer on its surface. However, for carbon inclusions at the frequencies of consideration (below 5 GHz) the skin ect can be neglected. Taking into account reflections between the boundaries, the overall reflection and transission coicient values can be calculated as and where γ γ l + e, (6) γ l + e γ l TT e T, (7) γ l + e is the coplex propagation constant in the layer of thickness l, and the reflection and transission coicients at the boundaries z and zl are calculated in an ordinary way through the characteristic ipedances of the corresponding edia [4], and ; + T ; + (8) + T + (9) AI SLAB AI Figure. eal part of relative perittivity of carbon-fiber and PMMA ixture. P inc P abs P trans P ref l o o z z l Figure. Air-coposite layer-air geoetry for calculating reflection and transission coicients. In these forulas, index corresponds to the first subscripts in the corresponding ipedances, and to the second ones. Figure. Iaginary part of relative perittivity of carbon-fiber and PMMA ixture. III. EFLECTION AND TANSMISSION COEFFIENTS FO A LAYE OF AN ISOTOPIC COMPOSITE MATEIAL A. Noral incidence of a plane wave upon a coposite isotropic layer For a plane electroagnetic wave norally incident on the air-coposite layer-air structure shown in Fig., the reflection and transission coicients, as well as shielding ectiveness are calculated using a transission-line approach. The coposite aterial is assued to be nonagnetic (its relative pereability is µ r ). The ective perittivity is calculated through the ixing forula. The characteristic ipedance of the air is µ /, and characteristic ipedance of the coposite ediu is µ r /. Both coicients and T are the coplex values, since is coplex, jδ j e. () The propagation constant in the coposite aterial is calculated as

4 γ jω µ j For a layer of coposite edia placed in air, the shielding ectiveness in db is S ().E. log ( T ). () T T + ; + () ; T (4) + + ; + ; T (5) (6) Transission and reflection coicients at different angles of incidence for a coposite layer with carbon fibers 5 ( σ 6.78 S/) and the sae geoetry as in Section II, are shown in Fig The results differ for perpendicular and parallel polarizations. Figure 4. Transission coicient versus frequency for the 5- thick coposite layer at various conductivities of inclusions. Noral incidence. Figure 6. Transission coicient versus frequency for the 5- thick coposite layer at various angles of oblique incidence. Perpendicular polarization. Figure 5. eflection coicient versus frequency for the 5- thick coposite layer at various conductivities of inclusions. Noral incidence. Exaples of transission and reflection coicients for ixtures siilar to that described in Section II, but with varying values of conductivity, are shown in Fig. 4 and 5. B. Oblique incidence of a plane wave upon a coposite isotropic layer The sae approach is used for oblique incidence of plane waves upon a coposite layer. Equations (6) and (7) can be used for the overall reflection and transission coicients in the case of an air-coposite layer-air structure. However,, and T, will be different for parallel and perpendicular polarization [4], Figure 7. eflection coicient versus frequency for the 5- thick coposite layer at various angles of oblique incidence. Perpendicular polarization.

5 an equal aount of dipole oents oriented along three axes (x,y, and z), and. Consider a plane wave norally incident on a thin layer described by the tensor (7). Assuing that the wave propagates along the z-axis, with coponents E z H z ; and Ex, E y, H x, H y, H z. Based on Maxwell s equations, it can be shown that the dispersion equation for the propagation constant is siilar to that obtained in [5] for anisotropic ferrite ediu, 4 γ ω µ ( + γ + ω µ. (8) 4 ) The equation (8) yields a siple solution Figure 8. Transission coicient versus frequency for the 5- thick coposite layer at various angles of the oblique incidence. Parallel polarization. µ o γ ± ω ( + ) ± ( + ) + 4( ) (9) The first two ± signs correspond to the positive and negative directions of the wave propagation along z, and the other two ± signs correspond to the ordinary and extraordinary waves in anisotropic ediu. However, since, there is a single physically reasonable solution, like for the isotropic case. Its for is siilar to (), but instead of the value. 5 should be used. Then, for calculating the reflection and transission coicients in a single coposite D layer, (6)-(9) can be applied, as in the isotropic case, reebering that the characteristic ipedances and propagation constants are related through Figure 9. eflection coicient versus frequency for the 5- thick coposite layer at various angles of oblique incidence. Parallel polarization. ωµ ; γ ωµ () γ C. Noral incidence of a plane wave upon a thin coposite anisotropic layer An iportant case is a thin layer of a coposite aterial. In any practical applications, the inclusions, like conducting fibers, are long copared to the thickness of a layer. A practical case is when all fibers are parallel to the plane of the layer. The layer of a coposite aterial is then anisotropic. If the aterial is non-texturized, i.e., dielectric dipole oents of inclusions are randoly oriented in-plane, this is a special two-diensional ( D ) case, and the ective perittivity is then described by a tensor in the for of, (7) where. 5 and b, is calculated according to (), and b is the perittivity of the base aterial. The coicient.5 coes fro the re-distribution of dipole oents copared to the isotropic case, when there is Figure. Shielding ectiveness for an isotropic and anisotropic coposite layers; each is 5 thick. S.E. levels for isotropic and D anisotropic layers are shown in Fig.. The isotropic layer paraeters are as in Section II. In the D layer all the paraeters are the sae as in isotropic case, but the length of the inclusions is 8, and

6 the concentration is reduced 8 ties to keep the sae volue fraction of carbon. For the oblique incidence upon a thin coposite D layer, the solution is straightforward as soon as the corresponding dispersion equation for propagation constant is obtained and its roots are found. In this case the coponents E z and H z. It is then necessary to consider corresponding polarizations; and the corresponding dispersion equation contains all three coponents of the tensor (7). For these reasons, the solution of the oblique incidence proble in anisotropic case is cubersoe. IV. FDTD MODELING OF A SHIELDING ENCLOSUE The frequency dependences of the coposite aterial PMMA-carbon fibers considered in Section II can be approxiated by the closest Debye dependence, using the genetic algorith (GA) [7]. The paraeters of this Debye dependence are ;. 9; σ. 4 S/; S and τ s. These Debye paraeters can be used to odel a shielding enclosure ade of this coposite using the full-wave E-FDTD software, as is proposed in [7]. The size of the enclosure assued in coputations is c x 4.5 c x c. A pseudo wire source placed at the center of the enclosure along the y-direction is used to excite the enclosure. An electric field probe, placed in the far zone fro the source in the x-direction is used to easure the electric field. Figure shows, that a coposite enclosure reduces the far field intensity. The increase of the enclosure thickness leads to the lower far field intensity. Figure. Spectru of the electric field easured in far field V. CONCLUSION The Maxwell Garnett (MG) ective ediu foralis allows engineering coposite aterials, that is, synthesizing their frequency responses, using paraeters for the base and inclusion aterials fro experiental, anufacturer s, or published reference data. The MG forulation is linear with respect to frequency for a ultiphase ixture; it allows representing frequency characteristics of edia in a for convenient for incorporating the into nuerical tie-doain electroagnetic codes, such as FDTD. It is shown that the MG foralis can be applied for frequency-dependent dielectric properties of inclusions and host aterials and electrodynaically anisotropic ixtures, in particular, with a rando in-plane distribution of particles in a thin layer. eflection and transission coicients of plane waves for both noral and oblique incidence on a layer are calculated on the basis of the MG foralis for coposite isotropic aterials. eflection and transission coicients at noral incidence upon a thin layer of aterial with in-plane rando distribution of dipole oents are also considered. The oblique incidence case is straightforward, but ore cubersoe. A siilar approach can be also applied for ultilayer structures, using the cascading approach based on transission atrix characterized by ABCD network paraeters. EFEENCES [] P.S.Neelakanta, Handbook of Electroagnetic Materials, Boca aton, FL: CC Press, 995. [] E.F.Kuester, C.L.Holloway, Coparison of approxiations for ective paraeters of artificial dielectrics, IEEE Trans. Microw. Theory Techn., vol., pp , 99. [] P.Sheng, Theory of dielectric function of granular coposite edia, Phys. ev. Letters, vol.45, no., pp. 6-6, 98. [4] W.T.Doyle and I.S.Jacobs, The influence of particle shape on dielectric enhanceent in etal-insulator coposites, J.Appl.Phys., vol. 7, no. 8, pp , 99. [5].E.Diaz, W.M.Merrill, and N.G.Alexopoulos, Analytical fraework for the odeling of ective edia, J. Appl. Phys., vol. 84, no., pp , 998. [6] J.C.Maxwell Garnett, Colours in etal glasses and etal fils, Philos. Trans.. Soc. London, Sect. A, vol., pp.85-4, 94. [7] M. Y. Koledintseva, J. Wu, J. hang, J. L. Drewniak, and K. N. ozanov, epresentation of perittivity for ulti-phase dielectric ixtures in FDTD odeling, Proc. IEEE Syp. Electroag. Copat., 9- Aug. 4, Santa Clara, CA, vol., pp.9-4, 4. [8] A.N.Lagarkov and A.K.Sarychev, Electroagnetic properties of coposites containing elongated conducting inclusions, Physical eview B, vol. 5, no., March, 996, pp [9] A.Sihvola, Effective perittivity of dielectric ixtures, IEEE Trans. Geosc. eote Sens., vol. 6, no. 4, pp. 4-49, 988. [] S.M.Matitsine, K.M.Hock, L.Liu, et.al., Shift of resonance frequency of long conducting fibers ebedded in a coposite, J. Appl. Phys., vol. 94, no., pp ,. [] D.S. McLachlan, A.Priou, I.Chernie, E. Isaac, and E.Henry, Modeling the perittivity of coposite aterials with general ective ediu equation, Journal of Electroagn. Waves and Applications, vol. 6, no. 6, pp.99-, 99. [] K. Ghosh and. Fuchs, Spectral theory for two-coponent porous edia, Phys. eview B, v. 8, pp. 5 56, 988. [] L.K.Mikhailovsky, A.A.Kitaytsev, and M.Y.Koledintseva, Advances of Gyroagnetic Electronics for EMC probles, Proc. IEEE Syp. Electroag. Copat, Washington, DC, August -5,, V., pp ,. [4] D.M.Pozar, Microwave Engineering, nd ed., New York: Wiley, 998. [5] N.N.Fedorov, Fundaentals of Electrodynaics, Moscow: Vysshaya Shkola, 98, pp (in ussian).

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