IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1

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1 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 From Maxwell Garnett to Debye Model for Electromagnetic Simulation of Composite Dielectrics Part II: Random Cylindrical Inclusions Muhammet Hilmi Nisanci, Student Member, IEEE, Francesco de Paulis, Student Member, IEEE, Marina Y. Koledintseva, Senior Member, IEEE, James L. Drewniak, Fellow, IEEE, and Antonio Orlandi, Fellow, IEEE Abstract A mixing rule in the theory of composites is intended to describe an inhomogeneous composite medium containing inclusions of one or several types in a host matrix as an equivalent homogeneous medium. The Maxwell Garnett mixing rule is widely used to describe effective electromagnetic properties (permittivity and permeability) of composites, in particular, biphasic materials, containing inclusions of canonical shapes (spherical, cylindrical, or ellipsoidal). This paper presents a procedure for deriving an equivalent Debye model that approximates the geometry-based Maxwell Garnett model for randomly distributed cylindrical inclusions. The derived Debye model makes the equivalent dielectric material suitable for any time-domain electromagnetic simulations. Index Terms Composite material, cylindrical inclusions, Debye model, frequency-dependent material. I. INTRODUCTION COMPOSITE materials are widely used in various electromagnetic applications from dc to optical frequencies. Engineering new composite materials with desirable properties and advanced characteristics for different RF and microwave applications, including electromagnetic compatibility (EMC)/ electromagnetic interference (EMI) problems, may need intense numerical simulations. Wideband time-domain electromagnetic simulations, e.g., based on the finite-difference time-domain (FDTD) numerical method, require representing material frequency responses as rational-fractional analytical functions. The simplest form of such an analytical function with a pole of the first order is the Debye dependence. The semianalytical Debye representation of the Maxwell Garnett (MG) mixing formula for biphasic mixtures containing Manuscript received August 23, 2010; revised January 28, 2011 and April 20, 2011; accepted July 17, M. H. Nisanci, F. de Paulis, and A. Orlandi are with the UAq Electromagnetic Compatibility Laboratory, Department of Electrical Engineering, University of L Aquila, L Aquila 67100, Italy ( antonio.orlandi@univaq.it; francesco.depaulis@univaq.it; muhammethilmi.nisanci@univaq.it). M. Y. Koledintseva and J. L. Drewniak are with the Electromagnetic Compatibility Laboratory, Missouri University of Science and Technology, Rolla, MO USA ( marinak@mst.edu; drewniak@mst.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TEMC spherical inclusions in a host matrix is proposed in Part I [1]. The limit for the MG model is the so-called quasi-static approximation. This means that the size of inhomogeneities must be much less than the wavelength in the medium. The MG formulation is applicable to dielectric dielectric mixtures, as well as dielectric-conducting inclusion mixtures at volume concentrations of conducting inclusions below the percolation threshold [2]. Inclusion shape affects the percolation threshold, and for elongated cylindrical inclusions the latter is inverse proportional to the aspect ratio a, which is defined as the ratio of the length to the diameter of inclusions. The percolation threshold for a mixture containing conducting sticks can be evaluated as approximately (1/a,...,5/a) [2]. As soon as the mixture is described by the MG formalism (within the limits of its validity), its frequency characteristics of the complex effective permittivity can be represented as a single-term or multiterm Debye dependence. However, consideration of composites containing elongated cylindrical inclusions, such as conducting fibers, is of a great practical importance, for example, for a design of absorbers with desired frequency characteristics of absorption and reflection of electromagnetic waves, or suppression of surface currents. Carbon-fiber filled polymer materials are widely used for these purposes, for example, to design light-weight shielding enclosures (see [3] and [4] and references therein). The objective of this paper is to derive an analytical procedure for defining parameters of a Debye model equivalent to the MG model for randomly oriented and spatially distributed conducting or dielectric cylindrical inclusions in a dielectric host matrix. The equivalent Debye model depends upon the original electromagnetic characteristics of the host and inclusions. Once the analytical expressions are developed, they can be easily implemented in the time-domain numerical electromagnetic solvers. There is no need in an intermediate stage of curve-fitting frequency characteristics of mixtures to Debye terms for implementation of composite materials in timedomain numerical simulations. There is a direct transition from material parameters of mixtures to their numerical modeling in electromagnetic structures. This will expedite the process of engineering electromagnetic materials with frequency dependences desirable for practical applications, including EMC purposes /$ IEEE

2 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY II. LIMIT-BASED ANALYTICAL APPROACH Herein, dielectric dielectric and conductor dielectric mixtures containing elongated cylindrical inclusions are considered. The MG rule in form (1) as given in [5] [8] can be applied to mixtures with inclusions of nonspherical shape, since it contains depolarization, or form, factors of inclusions N ik ε eff-mg = ε e + (1/3)f(ε i ε e ) 3 k=1 (ε e/ε e + N ik (ε i ε e )) 1 (1/3)f(ε i ε e ) 3 k=1 (N ik/ε e + N ik (ε i ε e )). In (1), ε e and ε i are the host and inclusion relative permittivities, respectively, and f is the volume fraction of the inclusions. The MG rule is known to be applicable to conducting inclusions at concentrations below percolation [1]. Depolarization factors N ik of elongated cylindrical inclusions, in the assumption that their diameter d is much less than its length l, are calculated [6] through their aspect ratio a = l/d as N i3 = 1 ln ( a + a 2 1/a a 2 1 ) a 2 a 2 1 ( 2 a2 1 ) 3 (1) (2a) N i1 = N i2 = 1 N i3 (2b) 2 where indices k = 1, 2, and 3 correspond to the Cartesian coordinates associated with each individual inclusion. The Debye model for a dispersive dielectric material is ε D = ε + (ε s ε ) (3a) 1+jωτ and a conductive inclusion material can be described as ε D = ε + σ (3b) jωε 0 where ε s and ε are the static and optic limit permittivity values, τ is the relaxation time, ω is the angular frequency, σ is the material conductivity, and ε 0 is the permittivity in free space. The objective of this study is to obtain analytical expressions for the Debye parameters from the MG formulation (1) and to reproduce the frequency-dependent behavior of the effective permittivity (1). A well-known approach for extracting this equivalent Debye model relies on various optimization procedures. Two examples of the optimization procedures are a genetic algorithm as described in [9] [12], and the curve-fitting procedure based on regression analysis and Legendre polynomials [13]. Any curve-fitting procedure needs programming of optimization procedures and setting initial search parameters and criteria specific for each particular case. This might require substantial computer resources, and the user must have an advanced expertise in the particular optimization procedure, since determining ranges of initial parameter pool and optimization settings is a kind of an art that might affect not only computational time and memory consumption, but also accuracy of the fit. TABLE I CASES OVERVIEW At the same time, derivation and implementation of analytical expressions for the direct evaluation of the equivalent Debye model is fast and does not require particular programming resources. Herein, six different cases are considered taking into account several combinations of host/inclusion permittivity types; this is done because the derivation of the Debye parameters from the original MG model may differ depending on the host and inclusion types. These cases are the same as those used in Part I [1] for the spherical inclusions, but they are related to elongated cylindrical inclusions: 1) Case 1c: ε e = constant, ε i = constant; 2) Case 2c: ε e = constant, ε i = Debye; 3) Case 3c: ε e = Debye, ε i = constant; 4) Case 4c: ε e = Debye, ε i = Debye; 5) Case 5c: ε e = constant, ε i = lossy; 6) Case 6c: ε e = Debye, ε i = lossy. The c stands for cylindrical inclusions, to the contrast of s for spherical inclusions in Part I [1]. The cases are summarized in Table I. The values in the table are given just as particular cases, for which computations have been run. For example, ε s = 2.2 corresponds to Teflon; inclusions in Cases 2c-A and 4c-A are Barium Titanate (they provide high dielectric contrast with the Teflon host material); ε s = 2.5 corresponds to chloroprene rubber inclusions, whose dielectric contrast with Teflon is comparatively low. As for conducting inclusions, the conductivity values are chosen in the range for carbon. Case 1c is included in the list even though it provides a constant effective permittivity ε eff-mg that is already suitable for time-domain numerical simulations. The other cases are dealing with frequency-dependent parameters of the host matrix and/or inclusions, resulting in the frequency-dependent ε eff-mg.fig.1 shows the real and imaginary parts of ε eff-mg as functions of frequency for the subcases A of the cases 2c 5c in Table I. In this figure, the curves are calculated for the inclusion volume fraction of f = 20.1%, and the aspect ratio of cylinders a = 5. The five considered examples present a Debye-like behavior; a similar trend can be found in the same cases associated to spherical inclusions, as in [1]. A one-term equivalent Debye model is associated with Cases 2c, 3c, and 5c. The two cases, 4c and 6c, may be described by a two-term equivalent Debye model. The goal herein is to find analytical expressions for the Debye parameters (ε sd, ε D, τ D ) as the functions of the original parameters (ε e, ε i, f, N ik ) used in the MG mixing rule.

3 NISANCI et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION OF COMPOSITE DIELECTRICS 3 position of the relaxation peak τ D = lim Re (τ D (ω)) ω 0 ( ) (εsd ε D /ε eff-mg ε D ) 1 = lim Re. (7) ω 0 jω This approach is employed for each case listed in Table I to get analytical expressions of the equivalent Debye model. Fig. 1. ε eff -MG for Cases 2c-A 6c-A with f = (a) Real part. (b) Imaginary part. Analytical derivations of the Debye parameters for cylindrical inclusions will be much more cumbersome than for spherical. This is because of the more complex MG formulation (1) for cylindrical inclusions than for spherical ([1], (1)). Therefore, an alternative approach is proposed. It uses the evaluation of the limits of the real part of (1), ε eff-mg-r, for the evaluation of the static and optic limit values of permittivity ε sd = lim ε eff-mg-r (4a) ω 0 ε D = lim ε eff-mg-r. (4b) ω To find the relaxation time τ D, the MG model in (1) is equated to the sought equivalent Debye model in (3a) ε eff-mg = ε D + (ε sd ε D ) (5) 1+jωτ D and solved for τ D as τ D (ω) = (ε sd ε D /ε eff-mg ε D ) 1. (6) jω The dc limit of (6), given in (7), allows for canceling out the frequency dependence, and the resultant τ D corresponds to the A. Case 2c Case 2c is characterized by a constant ε e for the host material, and a one-term Debye model for ε i (with parameters ε is, ε i, τ i ) representing the inclusions. The solutions of the limits in (4a) for ω 0 and in (4b) for ω provide two expressions of the same form. A generalized relation is provided in (8), shown at the bottom of this page, based on the two general parameters ε 1 and ε 2 ; it combines the analytical derivation based on (4a) and (4b), i.e., (8) can be used in Case 2c, instead of (4a), for calculating ε sd by substituting ε 1 = ε e and ε 2 = ε is ; (8) is representative of (4b) if ε D needs to be computed, substituting ε 1 = ε e and ε 2 = ε i. This procedure is summarized in (9). Then, (9a) and (9b) are substituted into (7) to obtain τ D. For the sake of brevity, the closed-form expression is not provided here, but it can be found in [14]. Alternatively, the value of τ D can be evaluated numerically from (7) ε sd = ε LIM (ε e,ε is ). (9a) ε D is obtained by substituting ε 1 = ε e and ε 2 = ε i into (8) ε D = ε LIM (ε e,ε i ). (9b) B. Case 3c Case 3c is dual to Case 2c. Host matrix is frequencydependent ε e, and inclusions are taken with constant ε i.for this case, the parameters of the equivalent Debye model are obtained using (8) and the same method as explained in Case 2c ε sd = ε LIM (ε es,ε i ) ε D = ε LIM (ε e,ε i ). (10a) (10b) The value of τ D is obtained from (7) using its closed-form expression provided in [14]. C. Case 4c Both the host ε e (ε es, ε e, τ e ) and the inclusion ε i (ε is, ε i, τ i ) in Case 4c are dielectric materials characterized by the Debye dependences. The effective permittivity of this composite calculated using the MG mixing rule leads to have two steps in the real part and two peaks in the imaginary part, as shown in Fig. 1. This behavior could be fitted by a two-term Debye model. It is initially done approximating ε eff-mg with the sum of two ε LIM (ε 1,ε 2 )=3ε 1 (ε 1 ε 2 ) 2 (f 1)N 2 z ε 2 (ε 1 ε 2 )N z +(1 f)ε 2 1 +(1+(2/3)f)ε 1 ε 2 +(1/3)fε 2 2 3(ε 1 ε 2 ) 2 (f 1)N 2 z 3(fε 1 (f 1)ε 2 )(ε 1 ε 2 )N z +2ε 1 (((3/2) + f)ε 1 (f (3/2))ε 2 ). (8)

4 4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY terms, ε Part1-MG and ε Part2-MG. The first term ε Part1-MG is an MG model computed by (1) with constant ε e = ε e, whereas ε i has a Debye dependence ε i (ε is, ε i, τ i ). The second term ε Part2-MG is an MG model computed by (1) with ε e Debyedependent ε e (ε es, ε e, τ e ) and constant ε i = ε i. These two terms include the Debye behavior of ε e and ε i separately. However, the sum of these two terms introduces an offset in the obtained equivalent model, since the combination of ε Part1-MG and ε Part2-MG takes into account twice the level of the high frequency permittivity of both materials (ε i, and ε e are included in both ε Part1-MG and ε Part2-MG ). This offset is not constant over the frequency. The accuracy of the approximation can be improved by introducing a correction term ε Part3-D. Thus, the more accurate representation of the two Debye dependences of Case 4c is summarized in ε eff-mg ε Part1-MG + ε Part2-MG ε Part3-D. (11) The correction factor ε Part3-D is constructed directly as a Debye model whose parameters (ε s-part3, ε -Part3, τ Part3 ) need to be computed. The three parameters associated with ε Part3-D are defined as follows. The left-hand side term in (11) and also the first two terms on the right-hand side of (11) are known; this assumption is employed and the limits of (11) for ω 0 and ω are computed. This leads to approximate the static (ε s-part3 ) and high-frequency (ε -Part3 ) permittivity of the correction Debye term as in (12a) and (12b), respectively. The τ Part3 parameter is simply approximated by averaging the relaxation time for the host τ e and inclusion τ i materials (12c): ε s-part3 = lim ω 0 ε Part1-MG + lim ω 0 ε Part2-MG lim ω 0 ε eff-mg (12a) ε -Part3 = lim ε Part1-MG + lim ε Part2-MG lim ε eff-mg ω ω ω (12b) τ Part3 = τ e + τ i. (12c) 2 At this stage, (12a) (12c) are evaluated in terms of the original MG model parameters. The first two terms have an MG form, whereas the third one has a Debye form. In order to have a fully Debye description of (11), the first two terms should be converted in the Debye form. This can be done similarly to Case 2c for the first term ε Part1-MG, and to Case 3c for the second term ε Part2-MG. Then ε eff-mg = ε Part1-D + ε Part2-D ε Part3-D. (13) D. Case 5c Case 5c considers conductive inclusions ε i (ε i, σ i ), as in (3b), embedded in a dielectric host with constant ε e. When (4a) is applied, the following expression can be obtained: ε sd = 1 ( ε e 3fN 2 z 3Nz 2 +3N z + f ). (14a) 3 N z (N z 1) (f 1) When (4b) is applied. one obtains from (8) ε D = ε xlim (ε e,ε i ). (14b) Fig. 2. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 2c-A with a = 5. (a) Real part. (b) Imaginary part. Average error AE = 0.0% for the real part comparison; AE = 0.015% for the imaginary part comparison. The expression for τ D is obtained from (7) and is provided, for the sake of brevity, in [14]. E. Case 6c Conductive inclusions ε i (ε i, σ i ) are embedded in a Debyedependent host material ε e (ε es, ε e, τ e ) in this case. The behavior of real and imaginary parts of the effective MG permittivity in this case is very similar to a two-term Debye model, as in Case 4c. This can be observed from the dashed curves in Fig. 1. Therefore, the same three terms in (11) are used to derive the equivalent Debye model that approximates the results from the MG rule. The first two terms in (11) are computed similar to Case 4c. The third term is different from the one used for Case 4c. This last term is set to a constant value ε Part3, since the parameters ε -Part3 and ε s-part3 are very close to each other (their difference is around 1%). The analytical relationship of the approximated model is then given by (15), which reduces to a sum of pure Debye terms if the approach developed for Cases 2c and 3c is applied to the first two terms of (14), obtaining (16): ε eff-mg = ε Part1-MG + ε Part2-MG ε Part3 (15) ε eff-mg = ε Part1-D + ε Part2-D ε Part3. (16)

5 NISANCI et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION OF COMPOSITE DIELECTRICS 5 Fig. 3. GDM results for the pair of curve in Fig. 2 for f = 20.1% and a = 5. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Fig. 5. GDM results for the pair of curve in Fig. 4 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Fig. 4. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 2c-A with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 0.0% for the real part comparison; AE = 0.02% for the imaginary part comparison. III. RESULTS AND DISCUSSION The five aforementioned formulations are applied to the cases in Table I. The real and imaginary parts of permittivity ε eff-mg computed by the MG model (1) are compared with their counterparts coming from the equivalent Debye model ε eq-debye whose parameters (ε sd, ε D, τ D ) are evaluated by applying the proposed formulation. The quality of the agreement is quantified according to the IEEE Standard P1597 [15] by using the feature selective validation technique (FSV) [16]. The global difference measure (GDM), which is the figure of merit of FSV indicating the quality of the global agreement between two different datasets, is the combination of other two parameters that take into account both the amplitude and feature differences. Fig. 6. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 3c-A with a = 5. (a) Real part. (b) Imaginary part. Average error AE = 0.0% for the real part comparison; AE = 0.03% for the imaginary part comparison. The GDM is computed only for the largest volume fraction value considered, f = 20.1%. The differences between the curves are also quantified computing the percentage average error AE, as in (17); this procedure is applied to the same sets of data sets for which the GDM is evaluated. In the following examples, dielectric dielectric mixtures (Cases 2c 4c) and dielectric conductive mixtures (Cases 5c and 6c) are considered up to the volume fraction of 20.1%. This limit was set because this concentration ( 20%) get close to the percolation threshold, where the MG model may become inapplicable E (f) = Re (ε eff-mg (f)) Re (ε eff-debye (f)) (17a) Re (ε eff-mg (f)) N f =1 AE = E (f) 100. (17b) N

6 6 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 7. GDM results for the pair of curve in Fig. 6 for f = 20.1% and a = 5. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Spread = 1). Fig. 9. GDM results for the pair of curve in Fig. 8 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Fig. 8. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 3c-A with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 0.0% for the real part comparison; AE = 0.02% for the imaginary part comparison. A. Case 2c Case 2c-A is selected due to the relevant contrast between the electric permittivity of the host and inclusion materials. The comparisons between the original ε eff-mg and the equivalent ε eq-debye whose parameters (ε sd, ε D, τ D ) are evaluated by applying the proposed formulation are provided in Figs. 2 and 4 for aspect ratios a = 5 and 500, and volume fractions of inclusions f = 2.5%, 8.4%, and 20.1%. At higher volume fraction, the particles with an aspect ratio of a = 5 will definitely form a conducting path, since percolation threshold should be somewhere less than 1/a = 20% if the inclusions are conducting. Figs. 3 and 5 provide the evaluation of the GDM parameter associated to the data shown in Figs. 2 and 4, respectively. Fig. 10. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 4c-A with a = 5. (a) Real part.(b) Imaginary part. Average error AE = 0.17% for the real part comparison; AE = 18% for the imaginary part comparison. B. Case 3c The values of the parameters for Case 3c-A in Table I are considered to validate the method proposed in Section II-B. Three volume fractions f = 2.5%, 8.4%, and 20.1%, and two aspect ratios a = 5 and 500 are used for computing the effective permittivity (1) and its equivalent Debye model. Fig. 6 shows the comparison between the original MG model and its equivalent Debye model for a = 5. The GDM figure of merit related to the data in Fig. 6 is shown in Fig. 7. The analogous comparison and FSV assessment are repeated for a = 500 and shown in Figs. 8 and 9.

7 NISANCI et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION OF COMPOSITE DIELECTRICS 7 Fig. 11. GDM results for the pair of curve in Fig. 10 for f = 20.1% and a = 5. (a) Real part (Grade = 3, Spread = 2). (b) Imaginary part (Grade = 4, Spread = 4). Fig. 13. GDM results for the pair of curve in Fig. 12 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Fig. 12. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 4c-A with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 0.12% for the real part comparison; AE = 20.1% for the imaginary part comparison. Fig. 14. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 5c-A with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 3% for the real part comparison; AE = 22% for the imaginary part comparisons. The comparison of the curves in Figs. 6 9 demonstrates the accuracy of the proposed expressions for the evaluation of the Debye model equivalent to the original Maxwell Garnett model. C. Case 4c The fourth case considers the host and the inclusion materials described by a Debye model. Case 4c-A is selected for running this comparison due to the high contrast between the materials properties. The comparisons of the real and imaginary parts of ε eff-mg and ε eq-debye are given in Figs. 10 and 12 for different values of f and a = 5, 500, respectively. The three-term model provides an equivalent permittivity that follows the trend of the original Maxwell Garnett model. However, the agreement is not as good as in the previous cases. Fig. 15. GDM results for the pair of curve in Fig. 14 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Larger differences are found when higher volume fraction is considered (f = 20.1%) and for smaller values of the aspect ratio (a = 5). The FSV results in Figs. 11 and 13 help to quantify these differences.

8 8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 16. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 5c-B with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 3% for the real part comparison; AE = 19% for the imaginary part comparisons. Fig. 18. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 6c-A with a = 5. (a) Real part. (b) Imaginary part. Average error AE = 0.18% for the real part comparison; AE = 50.2% for the imaginary part comparisons. Fig. 17. GDM results for the pair of curve in Fig. 16 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, Fig. 19. GDM results for the pair of curve in Fig. 18 for f = 20.1% and a = 5. (a) Real part (Grade = 3, Spread = 2). (b) Imaginary part (Grade = 5, Spread = 5). D. Case 5c Case 5c is related to conductive inclusions embedded in a constant permittivity host. The input parameters are taken from Case 5c-A and 5c-B from Table I. Fig. 14 compares ε eff-mg, computed using the parameters of Case 5c-A, with ε eq-debye whose parameters are computed as in Section II-D. The computations are done for different values of volume fraction f = 2.5%, 8.4%, and 20.1%, and a = 500. Fig. 15 is the corresponding FSV chart to show an acceptable agreement between the two models. Fig. 16 compares ε eff-mg, computed using the parameters of Case 5c-B, with ε eq-debye. The FSV results in Fig. 17 make more clear that the proposed approach increases its accuracy as the values of f and a are large. E. Case 6c Case 6c-A from Table I is considered here varying the aspect ratio a from 5 to 500. The trend demonstrated by the previous cases is confirmed; larger aspect ratios make the proposed equivalent Debye model more accurate. Fig. 18 compares the original MG curves from Case 6c-A with the equivalent Debye model for a small value of a = 5. The differences between the two models, both in the real and in the imaginary parts, are evident. They are quantified by the GDM charts. Different conclusions can be carried out looking at Figs. 20 and 21, in which the comparison is done using the larger value of the aspect ratio, a = 500.

9 NISANCI et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION OF COMPOSITE DIELECTRICS 9 Fig. 20. Comparisons of the original Maxwell Garnett model (solid curve) and the computed equivalent Debye model (dashed curve) for Case 6c-A with a = 500. (a) Real part. (b) Imaginary part. Average error AE = 3.2% for the real part comparison; AE = 22.5% for the imaginary part comparisons. between the MG mixing formulation and its equivalent Debye model is obtained when only one ingredient in a biphasic mixture has a frequency dispersive behavior, while the other is nondispersive. This conclusion holds for cases 2c and 3c; very good agreement between the original Maxwell Garnett model and the derived Debye model is confirmed by the FSV GDM parameter; it provides always an excellent agreement, and by the computed average error, it is always less than 0.03%. The case 5c related to conductive inclusions in a constant permittivity host material provides good comparisons, as indicated by the excellent GDM parameter. However, both the real and imaginary parts of the two models include values very close to zero; thus, small differences between the two models could lead to very large values of the computed average error (i.e., around 3% for the real part, and around 20% for the imaginary part). These considerations hold also for the imaginary part of the other two cases 4c and 6c, in which the average error increases also due to the less accurate approximation of the two-term Debye dependence. The comparisons of the real parts of cases 4c and 6c, instead, provide good agreement from the GDM and average error parameters. Thus, these parameters can be easily computed and included in a time-domain electromagnetic solver for taking into account the dispersive properties of the equivalent permittivity, without the need of modeling the complex geometry related to the randomly distributed cylindrical inclusions. Thus, an intermediate stage of curve-fitting frequency characteristics of mixtures to Debye terms for implementation of composite materials in time-domain numerical simulations is not needed. There is a direct transition from material parameters of mixtures to their numerical modeling in electromagnetic structures, which will expedite the process of engineering electromagnetic materials with desirable frequency dependences. REFERENCES Fig. 21. GDM results for the pair of curve in Fig. 20 for f = 20.1% and a = 500. (a) Real part (Grade = 1, (b) Imaginary part (Grade = 1, The larger aspect ratio (a = 500 versus a = 5) makes the model more accurate, as can be easily seen from the Fig. 20 compared to Fig. 18. This kind of conclusion is proven by the FSV results shown in Fig. 21 compared to those reported in Fig. 19. IV. CONCLUSION This paper extends the methodology developed in Part I to the randomly distributed cylindrical inclusions inside a dielectric host. The equivalent Debye model obtained starting from the Maxwell Garnett formulation provides the expression of a frequency-dependent equivalent permittivity. Some cases are considered covering real-world combinations of values for the host and inclusion materials. The best matching [1] F. de Paulis, M. H. Nisanci, M. Y. Koledintseva, and A. Orlandi, From Maxwell Garnett to Debye model for electromagnetic simulation of composite dielectrics Part I: Random spherical inclusions, IEEE Trans. Electromag. Compat., to be published. [2] A. N. Lagarkov and A. K. Sarychev, Electromagnetic properties of composites containing elongated conducting inclusions, Phys.Rev.B,vol.53, no. 9, pp , [3] M. Y. Koledintseva, J. Drewniak, R. DuBroff, K. Rozanov, and B. Archambeault, Modeling of shielding composite materials and structures for microwave frequencies, Progr. Electromagn. Res. B, vol. 15, pp , [4] I. M. De Rosa, R. Mancinelli, F. Sarasini, M. S. Sarto, and A. Tamburrano, Electromagnetic design and realization of innovative fiber-reinforced broad-band absorbing screens, IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pp , Aug [5] A. Sihvola and J. A. Kong, Effective permittivity of dielectric mixtures, IEEE Trans. Geosc. Remote Sens., vol.26,no.4,pp ,Jul [6] M. Y. Koledintseva, R. E. DuBroff, and R. W. Schwartz, A Maxwell Garnett model for dielectric mixtures containing conducting particles at optical frequencies, Progr. Electromagn. Res., vol. 63, pp , [7] M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio, Progr. Electromagn. Res., vol. 66, pp , [8] M. Y. Koledintseva, R. E. DuBroff, R. W. Schwartz, and J. L. Drewniak, Double statistical distribution of conductivity and aspect ratio

10 10 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY of inclusions in dielectric mixtures at microwave frequencies, Progr. Marina Y. Koledintseva (M 95 SM 03) received Electromagn. Res., vol. 77, pp , the M.S. and Ph.D. degrees from the Radio Engineering [9] H. S. Park, I. S. Choi, J. K. Bang, S. H. Suk, S. S. Lee, and H. T. Kim, Optimized design of radar absorbing materials for complex targets, J. Electromagn. Waves Appl., vol. 18, no. 8, pp , [10] M. Y. Koledintseva, J. Wu, J. Zhang, J. L. Drewniak, and K. N. Rozanov, Representation of permittivity for multi-phase dielectric mixtures in FDTD modeling, in Proc. IEEE Symp. Electromag. Compat., Santa Clara, CA, Aug. 9 13, 2004, vol. 1, pp [11] Z.-Q. Meng, Autonomous genetic algorithm for functional optimization, Progr. Electromagn. Res., vol. 72, pp , [12] M. Donelli, S. Caorsi, F. de Natale, D. Franeeschini, and A. Massa, A versatile enhanced genetic algorithm for planar array design, J. Electromagn. Waves Appl., vol. 18, no. 11, pp , [13] J. Xu, M. Y. Koledintseva, S. De, A. Radchenko, R. E. DuBroff, and J. L. Drewniak, FDTD modeling of absorbing materials for EMI applications, in Proc Asia-Pacific Symp. Electromagn. Compat., Beijing, China, Apr , pp Department, Moscow Power Engineering In- stitute (Technical University) [MPEI(TU)], Moscow, Russia, in 1984 and 1996, respectively. From 1983 to 1999, she worked as a Researcher with the Ferrite Laboratory, MPEI(TU), and from 1997 to 1999 she combined research with teaching as an Associate Professor in the same university. Since January 2000, she has been working as a Research Professor with the Electromagnetic Compatibility (EMC) Laboratory, Missouri University of Science and Technology (MS&T), formerly known as the University of Missouri-Rolla, Rolla. Her research interests include microwave engineering, analytical and numerical modeling of interaction of electromagnetic waves with complex geometries and materials, engineering composite materials with desirable electromagnetic properties, and their application for electromagnetic compatibility. She has published more than 150 papers in peer-reviewed journals and proceedings of international [14] (2011). [Online]. Available: conferences, and is the author of seven patents (Russian Federation). mg2d/equations_part_ii.pdf [15] Standard for Validation of Computational Electromagnetics Computer Modeling and Simulation Part 1, IEEE Standard P1597, Dr. Koledintseva is a member of the TC-9 (Computational Electromagnetics) and a Secretary of TC-11 (Nanotechnology) Committees of the IEEE EMC Society. [16] A. P. Duffy, A. J. M. Martin, A. Orlandi, G. Antonini, T. M. Benson, and M. S. Woolfson, Feature selective validation (FSV) for validation of computational electromagnetic (CEM) Part I: The FSV Method, IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp , Aug Muhammet Hilmi Nisanci (S 11) was born in Istanbul, Turkey, in He received the B.S. and M.S. degrees from Suleyman Demirel University, Isparta, Turkey, in 2006 and 2009, respectively, both in electronic and telecommunication engineering. He is currently working toward the Ph.D. degree in electrical engineering at the University of L Aquila, L Aquila, Italy. He was involved in the research activities at the UAq Electromagnetic Compatibility (EMC) Laboratory, L Aquila, from February 2007 to March His research interests include the numerical analysis of general electromagnetic problems, reverberation/anechoic chambers, interaction of electromagnetic field with dielectrics and composite media, their modeling and application for EMC. James L. Drewniak (S 85 M 90 SM 01 F 07) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana- Champaign, Champaign, in 1985, 1987, and 1991, respectively. He is currently with Electromagnetic Compatibility (EMC) Laboratory, Electrical Engineering Department, Missouri University of Science and Technology, Rolla. His research and teaching interests include electromagnetic compatibility in high-speed digital and mixed-signal designs, signal and power integrity, electronic packaging, EMC in power electronic based systems, electronics, and antenna design. Dr. Drewniak is an Associate Editor for the IEEE TRANSACTIONS ON ELEC- TROMAGNETIC COMPATIBILITY. Francesco de Paulis (S 08) was born in L Aquila, Italy, in He received the Laurea and Specialistic degree (summa cum laude) in electronic engineering both from University of L Aquila, L Aquila, Italy, in 2003 and 2006, respectively. In August 2006, he joined the Electromagnetic Compatibility (EMC) Laboratory, Missouri University of Science and Technology (formerly University of Missouri- Rolla) Rolla, where he received the M.S. degree in electrical engineering in May He is currently working toward the Ph.D. degree at the University of L Aquila. He was involved in the research activities at the UAq EMC Laboratory, L Aquila, from August 2004 to August 2006 and at the UMR EMC Laboratory, Rolla, from August 2006 to May From June 2004 to June 2005, he had an internship at Selex Communications, L Aquila, within the layout/si/pi design group. He is currently a Research Assistant at the UAq EMC Laboratory, University of L Aquila. His main research interests include in developing fast and efficient analysis tool for SI/PI and design of high-speed signal on PCB, RF interference in mixed-signal system, EMI problem investigation on PCBs, and composite material for shielding. Mr. de Paulis received the Past President s Memorial Award from the IEEE EMC Society in 2010 and He was the recipient of the Best Paper Award at the IEEE International Symposium on EMC in 2009 and 2010, and the IEC DesignCon Paper Award in 2010 and Antonio Orlandi (M 90 SM 97 F 07) was born in Milan, Italy, in He received the Laurea degree in electrical engineering from the University of Rome La Sapienza, Rome, Italy, in He was with the Department of Electrical Engineering, University of Rome La Sapienza, from 1988 to Since 1990, he has been with the Department of Electrical Engineering, University of L Aquila, L Aquila, where he is currently a Full Professor and Chair of the UAq Electromagnetic Compatibility (EMC) Laboratory. He is the author of more than 230 technical papers in the field of EMC in lightning protection systems and power drive systems. His current research interests include numerical methods and modeling techniques to approach signal/power integrity and EMC/EMI issues in high-speed digital systems. Dr. Orlandi is the recipient of the IEEE TRANSACTIONS ON ELECTROMAG- NETIC COMPATIBILITY Best Paper Award in 1997; the IEEE EMC Society Technical Achievement Award in 2003; the IBM Shared University Research Award in 2004, 2005, and 2006; the CST University Award in 2004, and the IEEE International Symposium on EMC Best Paper Award in 2009 and He is currently an Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, a member of the Education, TC-9 Computational Electromagnetics, and the Past Chairman of the TC-10 Signal Integrity Committees of the IEEE EMC Society. From 1996 to 2000, he was an Associate Editor of the IEEE TrANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY; from 2001 to 2006, he was an Associate Editor of the IEEE TRANSACTIONS ON MOBILE COMPUTING; and from 1999 to the end of the symposium, he was Chairman of the TC-5 Signal Integrity Technical Committee of the International Zurich Symposium and Technical Exhibition on EMC.