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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1

From Maxwell Garnett to Debye Model forElectromagnetic Simulation of Composite

Dielectrics—Part II: RandomCylindrical 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 intendedto describe an inhomogeneous composite medium containing in-clusions of one or several types in a host matrix as an equivalenthomogeneous medium. The Maxwell Garnett mixing rule is widelyused to describe effective electromagnetic properties (permittivityand permeability) of composites, in particular, biphasic materials,containing inclusions of canonical shapes (spherical, cylindrical, orellipsoidal). This paper presents a procedure for deriving an equiv-alent Debye model that approximates the geometry-based MaxwellGarnett model for randomly distributed cylindrical inclusions. Thederived Debye model makes the equivalent dielectric material suit-able for any time-domain electromagnetic simulations.

Index Terms—Composite material, cylindrical inclusions, Debyemodel, frequency-dependent material.

I. INTRODUCTION

COMPOSITE materials are widely used in various elec-tromagnetic applications from dc to optical frequencies.

Engineering new composite materials with desirable propertiesand advanced characteristics for different RF and microwaveapplications, including electromagnetic compatibility (EMC)/electromagnetic interference (EMI) problems, may need intensenumerical simulations. Wideband time-domain electromagneticsimulations, e.g., based on the finite-difference time-domain(FDTD) numerical method, require representing material fre-quency responses as rational-fractional analytical functions. Thesimplest form of such an analytical function with a pole of thefirst order is the Debye dependence.

The semianalytical Debye representation of the Maxwell Gar-nett (MG) mixing formula for biphasic mixtures containing

Manuscript received August 23, 2010; revised January 28, 2011 and April20, 2011; accepted July 17, 2011.

M. H. Nisanci, F. de Paulis, and A. Orlandi are with the UAq Electromag-netic Compatibility Laboratory, Department of Electrical Engineering, Uni-versity of L’Aquila, L’Aquila 67100, Italy (e-mail: [email protected];[email protected]; [email protected]).

M. Y. Koledintseva and J. L. Drewniak are with the Electromagnetic Compat-ibility Laboratory, Missouri University of Science and Technology, Rolla, MO65401 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2011.2162845

spherical inclusions in a host matrix is proposed in Part I [1].The limit for the MG model is the so-called quasi-static approx-imation. This means that the size of inhomogeneities must bemuch less than the wavelength in the medium. The MG for-mulation is applicable to dielectric–dielectric mixtures, as wellas dielectric-conducting inclusion mixtures at volume concen-trations of conducting inclusions below the percolation thresh-old [2]. Inclusion shape affects the percolation threshold, andfor elongated cylindrical inclusions the latter is inverse propor-tional to the aspect ratio a, which is defined as the ratio of thelength to the diameter of inclusions. The percolation thresholdfor a mixture containing conducting sticks can be evaluated asapproximately (1/a, . . . , 5/a) [2]. As soon as the mixture is de-scribed by the MG formalism (within the limits of its validity),its frequency characteristics of the complex effective permit-tivity can be represented as a single-term or multiterm Debyedependence.

However, consideration of composites containing elongatedcylindrical inclusions, such as conducting fibers, is of a greatpractical importance, for example, for a design of absorberswith desired frequency characteristics of absorption and reflec-tion of electromagnetic waves, or suppression of surface cur-rents. Carbon-fiber filled polymer materials are widely used forthese purposes, for example, to design light-weight shieldingenclosures (see [3] and [4] and references therein).

The objective of this paper is to derive an analytical proce-dure for defining parameters of a Debye model equivalent tothe MG model for randomly oriented and spatially distributedconducting or dielectric cylindrical inclusions in a dielectrichost matrix. The equivalent Debye model depends upon theoriginal electromagnetic characteristics of the host and inclu-sions. Once the analytical expressions are developed, they canbe easily implemented in the time-domain numerical electro-magnetic solvers. There is no need in an intermediate stageof curve-fitting frequency characteristics of mixtures to De-bye terms for implementation of composite materials in time-domain numerical simulations. There is a direct transition frommaterial parameters of mixtures to their numerical modelingin electromagnetic structures. This will expedite the processof engineering electromagnetic materials with frequency de-pendences desirable for practical applications, including EMCpurposes.

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2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY

II. LIMIT-BASED ANALYTICAL APPROACH

Herein, dielectric–dielectric and conductor–dielectric mix-tures containing elongated cylindrical inclusions are considered.The MG rule in form (1) as given in [5]–[8] can be applied tomixtures with inclusions of nonspherical shape, since it containsdepolarization, or form, factors of inclusions Nik

εeff -MG

= εe +(1/3)f(εi − εe)

∑3k=1 (εe/εe + Nik (εi − εe))

1 − (1/3)f(εi − εe)∑3

k=1 (Nik/εe + Nik (εi − εe)).

(1)

In (1), εe and εi are the host and inclusion relative permittivities,respectively, and f is the volume fraction of the inclusions. TheMG rule is known to be applicable to conducting inclusionsat concentrations below percolation [1]. Depolarization factorsNik of elongated cylindrical inclusions, in the assumption thattheir diameter d is much less than its length l, are calculated [6]through their aspect ratio a = l/d as

Ni3 =12

ln(a +

√a2 − 1/a −

√a2 − 1

)a − 2

√a2 − 1

(√a2 − 1

)3

(2a)

Ni1 = Ni2 =1 − Ni3

2(2b)

where indices k = 1, 2, and 3 correspond to the Cartesiancoordinates associated with each individual inclusion.

The Debye model for a dispersive dielectric material is

εD = ε∞ +(εs − ε∞)1 + jωτ

(3a)

and a conductive inclusion material can be described as

εD = ε∞ +σ

jωε0(3b)

where εs and ε∞ are the static and “optic” limit permittivityvalues, τ is the relaxation time, ω is the angular frequency, σis the material conductivity, and ε0 is the permittivity in freespace.

The objective of this study is to obtain analytical expres-sions for the Debye parameters from the MG formulation (1)and to reproduce the frequency-dependent behavior of the ef-fective permittivity (1). A well-known approach for extractingthis equivalent Debye model relies on various optimization pro-cedures. Two examples of the optimization procedures are agenetic algorithm as described in [9]–[12], and the curve-fittingprocedure based on regression analysis and Legendre polyno-mials [13]. Any curve-fitting procedure needs programming ofoptimization procedures and setting initial search parametersand criteria specific for each particular case. This might requiresubstantial computer resources, and the user must have an ad-vanced expertise in the particular optimization procedure, sincedetermining ranges of initial parameter pool and optimizationsettings is a kind of an art that might affect not only computa-tional time and memory consumption, but also accuracy of thefit.

TABLE ICASES OVERVIEW

At the same time, derivation and implementation of analyti-cal expressions for the direct evaluation of the equivalent Debyemodel is fast and does not require particular programming re-sources. Herein, six different cases are considered taking into ac-count several combinations of host/inclusion permittivity types;this is done because the derivation of the Debye parametersfrom the original MG model may differ depending on the hostand inclusion types. These cases are the same as those used inPart I [1] for the spherical inclusions, but they are related toelongated cylindrical inclusions:

1) Case 1c: εe = constant, εi = constant;2) Case 2c: εe = constant, εi = Debye;3) Case 3c: εe = Debye, εi = constant;4) Case 4c: εe = Debye, εi = Debye;5) Case 5c: εe = constant, εi = lossy;6) Case 6c: εe = Debye, εi = lossy.The “c” stands for cylindrical inclusions, to the contrast of “s”

for spherical inclusions in Part I [1]. The cases are summarizedin Table I. The values in the table are given just as particularcases, for which computations have been run. For example, εs =2.2 corresponds to Teflon; inclusions in Cases 2c-A and 4c-Aare Barium Titanate (they provide high dielectric contrast withthe Teflon host material); εs = 2.5 corresponds to chloroprenerubber inclusions, whose dielectric contrast with Teflon is com-paratively low. As for conducting inclusions, the conductivityvalues are chosen in the range for carbon.

Case 1c is included in the list even though it provides aconstant effective permittivity εeff -MG that is already suitable fortime-domain numerical simulations. The other cases are dealingwith frequency-dependent parameters of the host matrix and/orinclusions, resulting in the frequency-dependent εeff -MG . Fig. 1shows the real and imaginary parts of εeff -MG as functions offrequency for the subcases A of the cases 2c–5c in Table I. Inthis figure, the curves are calculated for the inclusion volumefraction 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 tospherical inclusions, as in [1]. A one-term equivalent Debyemodel is associated with Cases 2c, 3c, and 5c. The two cases,4c and 6c, may be described by a two-term equivalent Debyemodel. The goal herein is to find analytical expressions forthe Debye parameters (εsD , ε∞D , τD ) as the functions of theoriginal parameters (εe , εi , f , Nik ) used in the MG mixing rule.


NISANCI et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION OF COMPOSITE DIELECTRICS 3

Fig. 1. εeff -M G for Cases 2c-A–6c-A with f = 20.1. (a) Real part. (b) Imag-inary part.

Analytical derivations of the Debye parameters for cylindricalinclusions will be much more cumbersome than for spherical.This is because of the more complex MG formulation (1) forcylindrical inclusions than for spherical ([1], (1)). Therefore, analternative approach is proposed. It uses the evaluation of thelimits of the real part of (1), εeff -MG-R , for the evaluation of thestatic and “optic” limit values of permittivity

εsD = limω→0

εeff -MG-R (4a)

ε∞D = limω→∞

εeff -MG-R . (4b)

To find the relaxation time τD , the MG model in (1) is equatedto the sought equivalent Debye model in (3a)

εeff -MG = ε∞D +(εsD − ε∞D )

1 + jωτD(5)

and solved for τD as

τD (ω) =(εsD − ε∞D /εeff -MG − ε∞D ) − 1

jω. (6)

The dc limit of (6), given in (7), allows for canceling out thefrequency dependence, and the resultant τD corresponds to the

position of the relaxation peak

τD = limω→0

Re (τD (ω))

= limω→0

Re(

(εsD − ε∞D /εeff -MG − ε∞D ) − 1jω

)

. (7)

This approach is employed for each case listed in Table I toget analytical expressions of the equivalent Debye model.

A. Case 2c

Case 2c is characterized by a constant εe for the host material,and a one-term Debye model for εi (with parameters εis , εi∞,τ i) representing the inclusions.

The solutions of the limits in (4a) for ω → 0 and in (4b) forω→∞ provide two expressions of the same form. A generalizedrelation is provided in (8), shown at the bottom of this page,based on the two general parameters ε1 and ε2 ; it combines theanalytical derivation based on (4a) and (4b), i.e., (8) can be usedin Case 2c, instead of (4a), for calculating εsD by substitutingε1 = εe and ε2 = εis ; (8) is representative of (4b) if ε∞D needs tobe computed, substituting ε1 = εe and ε2 = εi∞. This procedureis summarized in (9). Then, (9a) and (9b) are substituted into (7)to obtain τD . For the sake of brevity, the closed-form expressionis not provided here, but it can be found in [14].

Alternatively, the value of τD can be evaluated numericallyfrom (7)

εsD = εLIM (εe , εis) . (9a)

ε∞D is obtained by substituting ε1 = εe and ε2 = εi∞ into (8)

ε∞D = εLIM (εe , εi∞) . (9b)

B. Case 3c

Case 3c is dual to Case 2c. Host matrix is frequency-dependent εe , and inclusions are taken with constant εi . Forthis case, the parameters of the equivalent Debye model areobtained using (8) and the same method as explained in Case 2c

εsD = εLIM (εes , εi) (10a)

ε∞D = εLIM (εe∞, εi) . (10b)

The value of τD is obtained from (7) using its closed-formexpression provided in [14].

C. Case 4c

Both the host εe (εes , εe∞, τe ) and the inclusion εi(εis , εi∞,τ i) in Case 4c are dielectric materials characterized by the De-bye dependences. The effective permittivity of this compositecalculated using the MG mixing rule leads to have two steps inthe real part and two peaks in the imaginary part, as shown inFig. 1. This behavior could be fitted by a two-term Debye model.It is initially done approximating εeff -MG with the sum of two

εLIM(ε1 , ε2) = 3ε1(ε1 − ε2)2(f − 1)N 2

z − ε2(ε1 − ε2)Nz + (1 − f)ε21 + (1 + (2/3)f)ε1ε2 + (1/3)fε2

2

3(ε1 − ε2)2(f − 1)N 2z − 3(fε1 − (f − 1)ε2)(ε1 − ε2)Nz + 2ε1(((3/2) + f)ε1 − (f − (3/2))ε2)

. (8)



terms, εPart1-MG and εPart2-MG . The first term εPart1-MG is anMG model computed by (1) with constant εe = εe∞, whereasεi has a Debye dependence εi(εis , εi∞, τ i). The second termεPart2-MG is an MG model computed by (1) with εe Debye-dependent εe (εes , εe∞, τe ) and constant εi = εi∞. These twoterms include the Debye behavior of εe and εi separately. How-ever, the sum of these two terms introduces an offset in theobtained equivalent model, since the combination of εPart1-MGand εPart2-MG takes into account twice the level of the high fre-quency permittivity of both materials (εi∞, and εe∞ are includedin both εPart1-MG and εPart2-MG ). This offset is not constantover the frequency. The accuracy of the approximation can beimproved by introducing a correction term εPart3-D . Thus, themore accurate representation of the two Debye dependences ofCase 4c is summarized in

εeff -MG ≈ εPart1-MG + εPart2-MG − εPart3-D . (11)

The correction factor εPart3-D is constructed directly as a Debyemodel whose parameters (εs-Part3 , ε∞-Part3 , τPart3) need to becomputed. The three parameters associated with εPart3-D aredefined as follows. The left-hand side term in (11) and also thefirst two terms on the right-hand side of (11) are known; this as-sumption is employed and the limits of (11) for ω→0 and ω→∞are computed. This leads to approximate the static (εs-Part3) andhigh-frequency (ε∞-Part3) permittivity of the correction Debyeterm as in (12a) and (12b), respectively. The τPart3 parameteris simply approximated by averaging the relaxation time for thehost τe and inclusion τ i materials (12c):

εs-Part3 = limω→0

εPart1-MG + limω→0

εPart2-MG − limω→0

εeff -MG

(12a)

ε∞-Part3 = limω→∞

εPart1-MG + limω→∞

εPart2-MG − limω→∞

εeff -MG

(12b)

τPart3 =τe + τi

2. (12c)

At this stage, (12a)–(12c) are evaluated in terms of the originalMG model parameters. The first two terms have an MG form,whereas the third one has a Debye form. In order to have afully Debye description of (11), the first two terms should beconverted in the Debye form. This can be done similarly to Case2c for the first term εPart1-MG , and to Case 3c for the secondterm εPart2-MG . Then

εeff -MG = εPart1-D + εPart2-D − εPart3-D . (13)

D. Case 5c

Case 5c considers conductive inclusions εi(εi∞, σi), as in(3b), embedded in a dielectric host with constant εe . When (4a)is applied, the following expression can be obtained:

εsD =13

εe

(3fN 2

z − 3N 2z + 3Nz + f

)

Nz (Nz − 1) (f − 1). (14a)

When (4b) is applied. one obtains from (8)

ε∞D = εxLIM (εe∞, εi∞) . (14b)

Fig. 2. Comparisons of the original Maxwell Garnett model (solid curve) andthe 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 partcomparison; AE = 0.015% for the imaginary part comparison.

The expression for τD is obtained from (7) and is provided, forthe sake of brevity, in [14].

E. Case 6c

Conductive inclusions εi(εi∞, σi) are embedded in a Debye-dependent host material εe (εes , εe∞, τe ) in this case. The behav-ior of real and imaginary parts of the effective MG permittivityin this case is very similar to a two-term Debye model, as inCase 4c. This can be observed from the dashed curves in Fig. 1.Therefore, the same three terms in (11) are used to derive theequivalent Debye model that approximates the results from theMG rule. The first two terms in (11) are computed similar toCase 4c. The third term is different from the one used for Case4c. This last term is set to a constant value εPart3 , since theparameters ε∞-Part3 and εs-Part3 are very close to each other(their difference is around 1%).

The analytical relationship of the approximated model is thengiven by (15), which reduces to a sum of pure Debye terms ifthe approach developed for Cases 2c and 3c is applied to thefirst two terms of (14), obtaining (16):

εeff -MG = εPart1-MG + εPart2-MG − εPart3 (15)

εeff -MG = εPart1-D + εPart2-D − εPart3 . (16)



Fig. 3. GDM results for the pair of curve in Fig. 2 for f = 20.1% and a =5. (a) Real part (Grade = 1, Spread = 1). (b) Imaginary part (Grade = 1,Spread = 1).

Fig. 4. Comparisons of the original Maxwell Garnett model (solid curve) andthe 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 realpart comparison; AE = 0.02% for the imaginary part comparison.

III. RESULTS AND DISCUSSION

The five aforementioned formulations are applied to the casesin Table I. The real and imaginary parts of permittivity εeff -MGcomputed by the MG model (1) are compared with their counter-parts coming from the equivalent Debye model εeq-Debye whoseparameters (εsD , ε∞D , τD ) are evaluated by applying the pro-posed formulation. The quality of the agreement is quantifiedaccording to the IEEE Standard P1597 [15] by using the fea-ture selective validation technique (FSV) [16]. The global dif-ference measure (GDM), which is the figure of merit of FSVindicating the quality of the global agreement between two dif-ferent datasets, is the combination of other two parameters thattake into account both the amplitude and feature differences.


Fig. 6. Comparisons of the original Maxwell Garnett model (solid curve) andthe 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 partcomparison; AE = 0.03% for the imaginary part comparison.

The GDM is computed only for the largest volume fractionvalue considered, f = 20.1%. The differences between thecurves are also quantified computing the percentage averageerror AE, as in (17); this procedure is applied to the same setsof data sets for which the GDM is evaluated. In the follow-ing examples, dielectric–dielectric mixtures (Cases 2c–4c) anddielectric–conductive mixtures (Cases 5c and 6c) are consid-ered up to the volume fraction of 20.1%. This limit was setbecause this concentration (∼20%) get close to the percolationthreshold, where the MG model may become inapplicable

E (f) =Re (εeff -MG (f)) − Re (εeff -Debye (f))

Re (εeff -MG (f))(17a)

AE =

∑Nf =1 E (f)

N× 100. (17b)



Fig. 7. GDM results for the pair of curve in Fig. 6 for f = 20.1% and a = 5.(a) Real part (Grade = 1, Spread = 1). (b) Imaginary part (Grade = 1, Spread= 1).

Fig. 8. Comparisons of the original Maxwell Garnett model (solid curve) andthe 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 realpart comparison; AE = 0.02% for the imaginary part comparison.

A. Case 2c

Case 2c-A is selected due to the relevant contrast betweenthe electric permittivity of the host and inclusion materials. Thecomparisons between the original εeff -MG and the equivalentεeq-Debye whose parameters (εsD , ε∞D , τD ) are evaluated byapplying the proposed formulation are provided in Figs. 2 and 4for aspect ratios a = 5 and 500, and volume fractions of inclu-sions f = 2.5%, 8.4%, and 20.1%. At higher volume fraction,the particles with an aspect ratio of a = 5 will definitely forma conducting path, since percolation threshold should be some-where less than ∼1/a = 20% if the inclusions are conducting.Figs. 3 and 5 provide the evaluation of the GDM parameterassociated 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 witha = 5. (a) Real part.(b) Imaginary part. Average error AE = 0.17% for the realpart comparison; AE = 18% for the imaginary part comparison.

B. Case 3c

The values of the parameters for Case 3c-A in Table I areconsidered to validate the method proposed in Section II-B.Three volume fractions f = 2.5%, 8.4%, and 20.1%, and twoaspect ratios a = 5 and 500 are used for computing the effectivepermittivity (1) and its equivalent Debye model. Fig. 6 shows thecomparison between the original MG model and its equivalentDebye model for a = 5. The GDM figure of merit related to thedata in Fig. 6 is shown in Fig. 7. The analogous comparison andFSV assessment are repeated for a = 500 and shown in Figs. 8and 9.



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. 12. Comparisons of the original Maxwell Garnett model (solid curve)and the computed equivalent Debye model (dashed curve) for Case 4c-A witha = 500. (a) Real part. (b) Imaginary part. Average error AE = 0.12% for thereal part comparison; AE = 20.1% for the imaginary part comparison.

The comparison of the curves in Figs. 6–9 demonstrates theaccuracy of the proposed expressions for the evaluation of theDebye model equivalent to the original Maxwell Garnett model.

C. Case 4c

The fourth case considers the host and the inclusion materialsdescribed by a Debye model. Case 4c-A is selected for runningthis comparison due to the high contrast between the materialsproperties. The comparisons of the real and imaginary parts ofεeff -MG and εeq-Debye are given in Figs. 10 and 12 for differentvalues of f and a = 5, 500, respectively.

The three-term model provides an equivalent permittivity thatfollows the trend of the original Maxwell Garnett model. How-ever, the agreement is not as good as in the previous cases.


Fig. 14. Comparisons of the original Maxwell Garnett model (solid curve)and the computed equivalent Debye model (dashed curve) for Case 5c-A witha = 500. (a) Real part. (b) Imaginary part. Average error AE = 3% for the realpart comparison; AE = 22% for the imaginary part comparisons.


Larger differences are found when higher volume fraction isconsidered (f = 20.1%) and for smaller values of the aspect ra-tio (a = 5). The FSV results in Figs. 11 and 13 help to quantifythese differences.



Fig. 16. Comparisons of the original Maxwell Garnett model (solid curve)and the computed equivalent Debye model (dashed curve) for Case 5c-B witha = 500. (a) Real part. (b) Imaginary part. Average error AE = 3% for the realpart comparison; AE = 19% for the imaginary part comparisons.


D. Case 5c

Case 5c is related to conductive inclusions embedded in aconstant permittivity host. The input parameters are taken fromCase 5c-A and 5c-B from Table I. Fig. 14 compares εeff -MG ,computed using the parameters of Case 5c-A, with εeq-Debyewhose parameters are computed as in Section II-D. The com-putations are done for different values of volume fraction f =2.5%, 8.4%, and 20.1%, and a = 500. Fig. 15 is the corre-sponding FSV chart to show an acceptable agreement betweenthe two models. Fig. 16 compares εeff -MG , computed using theparameters of Case 5c-B, with εeq-Debye . The FSV results inFig. 17 make more clear that the proposed approach increasesits accuracy as the values of f and a are large.

Fig. 18. Comparisons of the original Maxwell Garnett model (solid curve)and the computed equivalent Debye model (dashed curve) for Case 6c-A witha = 5. (a) Real part. (b) Imaginary part. Average error AE = 0.18% for the realpart comparison; AE = 50.2% for the imaginary part comparisons.

Fig. 19. GDM results for the pair of curve in Fig. 18 for f = 20.1% anda = 5. (a) Real part (Grade = 3, Spread = 2). (b) Imaginary part (Grade = 5,Spread = 5).

E. Case 6c

Case 6c-A from Table I is considered here varying the aspectratio a from 5 to 500. The trend demonstrated by the previ-ous cases is confirmed; larger aspect ratios make the proposedequivalent Debye model more accurate. Fig. 18 compares theoriginal MG curves from Case 6c-A with the equivalent Debyemodel for a small value of a = 5. The differences between thetwo models, both in the real and in the imaginary parts, areevident. They are quantified by the GDM charts. Different con-clusions can be carried out looking at Figs. 20 and 21, in whichthe comparison is done using the larger value of the aspect ratio,a = 500.



Fig. 20. Comparisons of the original Maxwell Garnett model (solid curve)and the computed equivalent Debye model (dashed curve) for Case 6c-A witha = 500. (a) Real part. (b) Imaginary part. Average error AE = 3.2% for thereal part comparison; AE = 22.5% for the imaginary part comparisons.


The larger aspect ratio (a = 500 versus a = 5) makes themodel more accurate, as can be easily seen from the Fig. 20compared to Fig. 18. This kind of conclusion is proven by theFSV results shown in Fig. 21 compared to those reported inFig. 19.

IV. CONCLUSION

This paper extends the methodology developed in Part I tothe randomly distributed cylindrical inclusions inside a dielec-tric host. The equivalent Debye model obtained starting fromthe Maxwell Garnett formulation provides the expression of afrequency-dependent equivalent permittivity.

Some cases are considered covering real-world combinationsof values for the host and inclusion materials. The best matching

between the MG mixing formulation and its equivalent Debyemodel is obtained when only one ingredient in a biphasic mix-ture has a frequency dispersive behavior, while the other isnondispersive. This conclusion holds for cases 2c and 3c; verygood agreement between the original Maxwell Garnett modeland the derived Debye model is confirmed by the FSV GDMparameter; it provides always an “excellent” agreement, and bythe computed average error, it is always less than 0.03%. Thecase 5c related to conductive inclusions in a constant permittiv-ity host material provides good comparisons, as indicated by the“excellent” GDM parameter. However, both the real and imag-inary parts of the two models include values very close to zero;thus, small differences between the two models could lead tovery large values of the computed average error (i.e., around 3%for the real part, and around 20% for the imaginary part). Theseconsiderations hold also for the imaginary part of the other twocases 4c and 6c, in which the average error increases also dueto the less accurate approximation of the two-term Debye de-pendence. The comparisons of the real parts of cases 4c and 6c,instead, provide good agreement from the GDM and averageerror parameters.

Thus, these parameters can be easily computed and includedin a time-domain electromagnetic solver for taking into accountthe dispersive properties of the equivalent permittivity, with-out the need of modeling the complex geometry related to therandomly distributed cylindrical inclusions.

Thus, an intermediate stage of curve-fitting frequency char-acteristics of mixtures to Debye terms for implementation ofcomposite materials in time-domain numerical simulations isnot needed. There is a direct transition from material parame-ters of mixtures to their numerical modeling in electromagneticstructures, which will expedite the process of engineering elec-tromagnetic materials with desirable frequency dependences.

REFERENCES

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Muhammet Hilmi Nisanci (S’11) was born in Istan-bul, Turkey, in 1983. He received the B.S. and M.S.degrees from Suleyman Demirel University, Isparta,Turkey, in 2006 and 2009, respectively, both in elec-tronic and telecommunication engineering. He is cur-rently working toward the Ph.D. degree in electricalengineering at the University of L’Aquila, L’Aquila,Italy.

He was involved in the research activities at theUAq Electromagnetic Compatibility (EMC) Labora-tory, L’Aquila, from February 2007 to March 2009.

His research interests include the numerical analysis of general electromagneticproblems, reverberation/anechoic chambers, interaction of electromagnetic fieldwith dielectrics and composite media, their modeling and application for EMC.

Francesco de Paulis (S’08) was born in L’Aquila,Italy, in 1981. He received the Laurea and Special-istic degree (summa cum laude) in electronic engi-neering both from University of L’Aquila, L’Aquila,Italy, in 2003 and 2006, respectively. In August2006, he joined the Electromagnetic Compatibility(EMC) Laboratory, Missouri University of Scienceand Technology (formerly University of Missouri-Rolla) Rolla, where he received the M.S. degree inelectrical engineering in May 2008. He is currentlyworking 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 2008. From June 2004 to June 2005, he hadan internship at Selex Communications, L’Aquila, within the layout/SI/PI de-sign group. He is currently a Research Assistant at the UAq EMC Laboratory,University of L’Aquila. His main research interests include in developing fastand efficient analysis tool for SI/PI and design of high-speed signal on PCB, RFinterference in mixed-signal system, EMI problem investigation on PCBs, andcomposite material for shielding.

Mr. de Paulis received the Past President’s Memorial Award from the IEEEEMC Society in 2010 and 2011. He was the recipient of the Best Paper Awardat the IEEE International Symposium on EMC in 2009 and 2010, and the IECDesignCon Paper Award in 2010 and 2011.

Marina Y. Koledintseva (M’95–SM’03) receivedthe M.S. and Ph.D. degrees from the Radio Engi-neering 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 Researcherwith the Ferrite Laboratory, MPEI(TU), and from1997 to 1999 she combined research with teach-ing as an Associate Professor in the same univer-sity. Since January 2000, she has been working as aResearch Professor with the Electromagnetic Com-

patibility (EMC) Laboratory, Missouri University of Science and Technology(MS&T), formerly known as the University of Missouri-Rolla, Rolla. Her re-search interests include microwave engineering, analytical and numerical mod-eling of interaction of electromagnetic waves with complex geometries andmaterials, engineering composite materials with desirable electromagnetic prop-erties, and their application for electromagnetic compatibility. She has publishedmore than 150 papers in peer-reviewed journals and proceedings of internationalconferences, and is the author of seven patents (Russian Federation).

Dr. Koledintseva is a member of the TC-9 (Computational Electromagnet-ics) and a Secretary of TC-11 (Nanotechnology) Committees of the IEEE EMCSociety.

James L. Drewniak (S’85–M’90–SM’01–F’07) re-ceived the B.S., M.S., and Ph.D. degrees in electricalengineering from the University of Illinois at Urbana-Champaign, Champaign, in 1985, 1987, and 1991,respectively.

He is currently with Electromagnetic Compati-bility (EMC) Laboratory, Electrical Engineering De-partment, Missouri University of Science and Tech-nology, Rolla. His research and teaching interestsinclude electromagnetic compatibility in high-speeddigital and mixed-signal designs, signal and power

integrity, electronic packaging, EMC in power electronic based systems, elec-tronics, and antenna design.

Dr. Drewniak is an Associate Editor for the IEEE TRANSACTIONS ON ELEC-TROMAGNETIC COMPATIBILITY.

Antonio Orlandi (M’90–SM’97–F’07) was born inMilan, Italy, in 1963. He received the Laurea degreein electrical engineering from the University of Rome“La Sapienza,” Rome, Italy, in 1988.

He was with the Department of Electrical Engi-neering, University of Rome “La Sapienza,” from1988 to 1990. Since 1990, he has been with theDepartment of Electrical Engineering, University ofL’Aquila, L’Aquila, where he is currently a Full Pro-fessor and Chair of the UAq Electromagnetic Com-patibility (EMC) Laboratory. He is the author of more

than 230 technical papers in the field of EMC in lightning protection systemsand power drive systems. His current research interests include numerical meth-ods and modeling techniques to approach signal/power integrity and EMC/EMIissues 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 Tech-nical Achievement Award in 2003; the IBM Shared University Research Awardin 2004, 2005, and 2006; the CST University Award in 2004, and the IEEE In-ternational Symposium on EMC Best Paper Award in 2009 and 2010. He is cur-rently an Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY, a member of the “Education,” TC-9 “Computational Electro-magnetics,” and the Past Chairman of the TC-10 “Signal Integrity” Committeesof the IEEE EMC Society. From 1996 to 2000, he was an Associate Editorof the IEEE TrANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY; from 2001to 2006, he was an Associate Editor of the IEEE TRANSACTIONS ON MOBILE

COMPUTING; and from 1999 to the end of the symposium, he was Chairmanof the TC-5 “Signal Integrity” Technical Committee of the International ZurichSymposium and Technical Exhibition on EMC.

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