IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 …web.mst.edu/~marinak/files/My_publications/Papers/MG_Debye_I.pdf · high-frequency (optic limit) relative permittivity, respectively,

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1

From Maxwell Garnett to Debye Model forElectromagnetic Simulation of Composite Dielectrics

Part I: Random Spherical InclusionsFrancesco de Paulis, Student Member, IEEE, Muhammet Hilmi Nisanci,

Marina Y. Koledintseva, Senior Member, IEEE, James L. Drewniak, Fellow, IEEE, and Antonio Orlandi, Fellow, IEEE

Abstract—A semianalytical approach to obtain an equivalentDebye frequency dependence of effective permittivity for biphasicmaterials with random spherical inclusions from the well-knownMaxwell Garnett (MG) mixing rule is proposed. Different combi-nations of frequency characteristics of mixture phases (host andinclusions) are considered: when at least one of the phases is fre-quency independent; lossy (with dc conductivity); or with a knownsingle-term Debye frequency dependence. The equivalent Debyemodels approximate very well the frequency characteristics ob-tained directly from MG mixing rule. In some cases, there is anexact match between the two models, and a good approximation isachieved in the other cases and is quantified by the feature selec-tive validation technique. The parameters of the derived equivalentDebye model can be employed in full-wave time-domain numericalelectromagnetic codes and tools. This will allow for efficient wide-band modeling of complex electromagnetic structures containingcomposite materials with effective dielectric parameters obtainedthrough MG mixing rule.

Index Terms—Composite material, Debye model, frequency-dependent material, spherical inclusions.

I. INTRODUCTION

THE efficient modeling of wideband performance of com-plex structures containing composite materials is an im-

portant issue in computational electromagnetics. This kind ofmodeling is crucial for solving various problems related to elec-tromagnetic compatibility (EMC), electromagnetic immunity(EMI), and signal integrity. A composite material typically con-tains 3-D random or aligned spatial distribution of inclusionsof one or a few different types in a host (matrix) material.Since composite materials are typically complex structures, itis conventional to homogenize their properties, including fre-quency dependences of their electromagnetic properties (per-mittivity and permeability) using various mixing rules, includ-

Manuscript received August 23, 2010; revised January 28, 2011 and April14, 2011; accepted May 18, 2011.

F. De Paulis, M. H. Nisanci and A. Orlandi are with the UAq Electromag-netic Compatibility Laboratory, Deptartment 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.2011.2158217

ing Bruggeman’s effective medium theory [1], or generalizedBergman–Milton spectral function approach [2]. Homogeniza-tion is convenient for using the effective permittivity and per-meability in numerical simulations. Frequency-domain codesrequire knowledge of the effective homogenized electromag-netic parameters at every frequency point. This could be doneif the data of intrinsic electromagnetic parameters of the com-posite phases (ingredients) are available for every frequencypoint of interest. This is not always the case, since in practicematerial parameters might be known only at a few selected fre-quency points, and for matrix and inclusions these frequencypoints could be significantly different. This makes it difficult toobtain correct data to use in frequency-domain numerical sim-ulations. Besides, wideband simulations of complex structureswith composite materials in frequency domain may be veryinefficient.

The wideband behavior analysis of these equivalent mate-rials with effective electromagnetic parameters could be donemore efficiently using time-domain numerical techniques, e.g.,the finite difference time-domain technique [3], or finite inte-gration technique [4]. To model frequency dispersive materialsusing time-domain codes, it is important to represent materialparameters as rational-fractional functions. These are the sumsof Debye terms [5] with poles of the first order in nonresonancecases, and if frequency characteristics contain resonances, theLorentzian terms with the poles of the second order should beadded.

Even if the electromagnetic parameters of the composite’singredients are known only in a few separate frequency points,it is possible to restore their full Debye and Lorentzian databy an accurate curve fitting, using an appropriate optimizationtechnique, e.g., based on a genetic algorithm or Legendre poly-nomial and regression analysis algorithm [6]. Then, as soon asthe Debye and/or Lorentzian terms for intrinsic parameters ofeach composite phase are known, they can be used in an appro-priate homogenization procedure to obtain effective parametersof the composite. The resultant frequency dependences for ef-fective permittivity (or permeability) then could be representedas a series of new Debye and/or Lorentzian terms.

Therefore, to numerically model a frequency-dispersive com-posite material in time domain one should 1) obtain its ho-mogenized effective electromagnetic responses, and 2) expressthese properties as a sum of the Debye (and/or Lorentzian)terms whose parameters should be computed. In majority ofEMC applications in RF and microwave frequency ranges, when

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2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY

resonances in material parameters are not noticeable, it is suf-ficient to approximate frequency characteristics of materials assums of the Debye terms only. For this reason, in the furtherconsideration we will omit Lorentzian terms.

It is attractive to get the Debye terms from the calculatedeffective electromagnetic parameters of the given mixture di-rectly, not applying any curve-fitting procedure. This means ananalytical derivation of the Debye parameters from the homog-enization formula. Obtaining such formulas for fast computa-tion of the Debye parameters from, for example, the MaxwellGarnett (MG) mixing rule is the objective of this paper. Thenthese Debye parameters could be directly used in any time-domain code through either recursive convolution, or auxiliarydifferential equation procedure, or any other algorithm that em-ploys time-domain responses of materials [7].

It is known that the MG mixing rule [8] is the most widespreadand convenient formula for predicting electromagnetic behaviorof mixtures, if electromagnetic parameters, volume fractions,and shapes of ingredients are known, assuming that these are3-D random mixtures of inclusions in a host matrix.

The MG model is the so called quasi-static approximation.It is valid, when the size of inhomogeneities is much less thanthe wavelength in the medium. The MG model is applicableto dielectric–dielectric mixtures, as well as mixtures containingconducting inclusions in a dielectric host (matrix) at volumeconcentrations of conducting inclusions below the percolationthreshold [9].

In the case of conducting inclusions, their volume fractionshould be less than the percolation threshold. Theoretically,spherical inclusions may start touching each other in a 3-Dperiodic lattice at the volume fraction above 47%. Practically,in a random mixture, the limit is not more than 20%–30%, sinceinclusions may interact and build conductive chains.

Intrinsic dielectric parameters of the host and inclusions in theMG formulation can be functions of frequency in the generalcase. In this paper, only effective permittivity will be consid-ered, though the analogous analysis can be applied to effectivepermeability as well.

The MG formula expresses the permittivity of an equivalenthomogeneous material as a function of the intrinsic permittivi-ties of the host and inclusions, as well as the inclusions volumefraction. The generalized MG formula contains depolarization,or form factors of inclusions as well [1, p. 268]. However, theexpression of the MG mixing rule is not in the form of the Debyelaw for permittivity.

Several approaches have been proposed to transform the MGformula into a sum of Debye terms, including different curve-fitting techniques [10]–[12]. However, any kind of curve-fittingalgorithm requires writing special codes for optimization, and auser’s experience in setting proper initial parameters and criteriafor search the optimal parameters. Besides, curve fitting to amultiterm Debye series may result in an increased complexityof the optimization procedures. Though running of optimizationcodes may be very fast (on the order of a few seconds), thepreparation may be difficult and time consuming.

This paper is devoted to the development of asemianalytical approach to retrieve the Debye terms from the

MG model of biphasic frequency-dispersive dielectric materialswith randomly placed spherical inclusions. Once the analyticalexpressions are developed, they can be easily implemented andsolved.

As soon as the mixture is described by the MG formalism(within the limits of its validity), its frequency characteristicsof the complex effective permittivity can be represented as asingle-term or multiterm Debye dependence.

By means of the proposed formulation the parameters ofthe Debye terms are computed based on the electromagneticcharacterization of the host and of the inclusions. Part II focusedon the development of the same approach for biphasic dielectricmaterials with a random distribution of cylindrical inclusions.

II. SEMIANALYTICAL APPROACH

The analytical relationship between the electrical and geo-metrical parameters of a biphasic dielectric material is given bythe MG equation [1], [13], [14]

εeff−MG = εe +3fεe(εi − εe)

εi + 2εe − f(εi − εe)(1)

where εe is the electric permittivity of the host material, εi isthe electric permittivity of the inclusion material, and f is thevolume fraction (volume percentage of the inclusions in theoverall volume).

An N-term Debye model for dispersive dielectric material isdefined as

εD = ε∞ +N∑

n=1

εsn − ε∞1 + jωτn

(2)

where εsn and ε∞ are the nth static dielectric constant and thehigh-frequency (optic limit) relative permittivity, respectively,and τ is the Debye constant, or relaxation time.

The conductive material with conductivity σ over the mi-crowave band can be modeled as

εD = ε∞ +σ

jωε0(3)

ε0 in (3) is the permittivity of vacuum.The derivation of an equivalent Debye model from the

geometry-based MG model requires the identification of severalcases. Each case is based on the electric properties of both thehost and the inclusion materials. If the host matrix is relativelynondispersive, it can be characterized by a constant dielectricpermittivity. However, if the host matrix is substantially lossyand dispersive, it should be described in terms of at least thefirst-order Debye model. As for inclusions, in some cases theycan be approximated as a dielectric with constant permittivity,or with the Debye frequency dependence, or a lossy materialthat requires a nonzero conductivity. Therefore, in this paper,six different cases will be considered.Case 1s: εe = constant, εi = constant.Case 2s: εe = constant, εi = Debye.Case 3s: εe = Debye, εi = constant.Case 4s: εe = Debye, εi = Debye.Case 5s: εe = constant, εi = conductive.Case 6s: εe = Debye, εi = conductive.


DE PAULIS et al.: FROM MAXWELL GARNETT TO DEBYE MODEL FOR ELECTROMAGNETIC SIMULATION 3

TABLE ICASE OVERVIEW

The cases are identified by a progressive number followedby an “s” standing for spherical inclusions. This is done toseparate them from the cylindrical inclusions considered inPart II. The case of conductive host material is not considered,since in the majority of cases matrix is dielectric, while inclu-sions are conducting. All the cases are characterized by differenttypes and values of the host and inclusions material parameters.These cases are summarized in Table I. The values in the tableare given just as particular cases, for which computations arerun. For example, εs = 2.2 corresponds to Teflon; inclusionsin the Cases 2c-A and 4c-A are Barium Titanate (they providehigh dielectric contrast with the Teflon host material); εs = 2.5corresponds to chloroprene rubber inclusions, whose dielectriccontrast with Teflon is comparatively low). As for conductinginclusions, the conductivity values are chosen in the range forcarbon.

Case 1s is characterized by constant dielectric properties forboth host and inclusion materials. Thus, the constant value ofεeff−MG (1) is ready to be used inside any time-domain nu-merical simulation. The different properties and values of εe

and εi impact on the frequency dependence of εeff−MG . Fig. 1show the real and imaginary parts of εeff−MG , computed by(1), for the cases from 2s A to 6s A of Table I. The corre-sponding B cases are not shown in these figures for sake ofbrevity.

The real and imaginary parts of εeff−MG in Fig. 1 show atrend similar to that of an equivalent Debye model, which isvery convenient to use in numerical simulations. Cases 2s, 3s,and 5s can be related to a one-term Debye model. Cases 4sand 6s have frequency-dependent dielectric properties for bothhost and inclusion materials, and can be approximated by atwo-term Debye model. Thus, the key issue is to find the an-alytical expressions for the parameters (εsD , ε∞D , τD ) of theequivalent Debye model as function of the parameters (εe , εi , f)of the original MG model. This will be done in the Sectionslater, separately for each of the five cases from Case 2s toCase 6s.

A. Case 2s

Case 2s is characterized by a constant εe for the host material,and the one-term Debye model for εi (with parameters εis ,εi∞, τ i) representing the inclusions. After inserting the Debyeexpression of εi in (1) and some algebraic manipulation, one

Fig. 1. εeff −M G for Cases 2s A – 6s A with f = 46.8%: (a) real part and(b) imaginary part.

can obtain

εeff−MG =NUM (εeff−MG)DEN (εeff−MG)

(4a)

NUM (εeff−MG) =εe (εis + 2εe + 2fεis − 2fεe)

εis + 2εe − fεis + fεe

+ jεeωτi(εi∞ + 2εe + fεi∞ − 2fεe)

εis + 2εe − fεis + fεe

(4b)

DEN(εeff−MG) = 1 + jωτiεi∞ + 2εe − fεi∞ + fεe

εis + 2εe − fεis + fεe.

(4c)

A one-term equivalent Debye model derived from (2) forN = 1 can be rewritten in the same form as (4a)–(4c) as

εeq−Debye = ε∞D +εsD − ε∞D

1 + jωτD=

NUM (εeq−Debye)DEN (εeq−Debye)

(5a)

NUM (εeq−Debye) = εsD + jωε∞D τD (5b)

DEN (εeq−Debye) = 1 + jωτD . (5c)



The parameters (εsD , ε∞D , τD ) of the equivalent Debye modelof εeff−MG are computed by equating (4b) to (5b), and (4c) to(5c) giving as result

εsD =εe (εis + 2εe + 2fεis − 2fεe)

εis + 2εe − fεis + fεe(6a)

ε∞D =εe (εi∞ + 2εe + fεi∞ − 2fεe)

εi∞ + 2εe − fεi∞ + fεe(6b)

τD =τi (εi∞ + 2εe − fεi∞ + fεe)

εis + 2εe − fεis + fεe. (6c)

This derivation implies that the original MG model for Cases2s is rigorously equivalent to a one-term Debye model.

B. Case 3s

Case 3s contains a host with a frequency-dependent εe andinclusions of constant εi . This case is dual to Case 2s.

However, (1) is not symmetric with respect to εe and εi .This means that substituting εe and εi in (1) does not resultin an expression similar to (4b) and (4c) with εe and εi inter-changed. Because of this the procedure of Case 2s can not befollowed.

Fig. 1 shows a one-term Debye-like behavior for εeff−MG forCase 3s (subcase A in particular). The real part of εeff−MG−R ischaracterized by the constant values at low (∼105 Hz) and “opticlimit” frequencies (>1011 Hz) frequencies, and by a monotonictransition between these two values at around 1010 Hz. Theimaginary part εeff−MG−I is characterized by a peak value atthe same frequency of the transition of the real part, and goes tozero at low and high frequencies.

The mathematical limits of the real part εeff−MG−R at fre-quency going to zero and to infinity are as in (7a) and (7b).These are the static, εsD , and “optic limit” permittivity, ε∞D , ofthe equivalent Debye model of εeff−MG . The value of τD is asin (7c)

εsD = limω→0

εeff−MG−R =ε2es (2 − 2f) + εesεi (1 + 2f)

εes (2 + f) + εi (1 − f)

(7a)

ε∞D = limω→∞

εeff−MG−R =ε2e∞ (2 − 2f) + εe∞εi (1 + 2f)

εe∞ (2 + f) + εi (1 − f)

(7b)

τD =1ω∗ (7c)

where ω∗ is the frequency at which the imaginary part ofεeff−MG−I (1) reaches maximum. The derivative of εeff−MG−I

with respect to ω at this frequency ω∗ goes to zero

ω∗ :dεeff−MG−I (ω)

dω

∣∣∣∣ω∗

= 0. (7d)

The closed-form expression for τD is quite cumbersome andit is omitted herein; however, it is available in [15]. The value ofω∗ can be alternatively evaluated by solving (7d) numerically.The set of expressions (7a)–(7d) define an analytical procedurefor deriving the parameters of a one-term equivalent Debye

model associated with the geometry-based MG formulation re-lated to Case 3s.

C. Case 4s

Case 4s is characterized by the Debye dependence of both thehost εe (εes ,εe∞,τe ) and the inclusions εi(εis ,εi∞,τ i) dielectricmaterials. This material configuration results in an effectivepermittivity εeff−MG of the MG model with two peaks in theimaginary part, as shown by the dot-dashed curves Fig. 1(a)and (b). This behavior cannot be approximated by a simple one-term Debye model, as done for Cases 2s and 3s. The shapeof the curves in Fig. 1 shows that there is a superposition ofat least two Debye terms due to the combined effect of twomaterials (host and inclusions) with different Debye dispersionparameters.

The corresponding mathematical model that approximates theMG formulation (1), when εe and εi are described by single-termDebye dependencies (2), consists of the sum of three differentterms

εeff−MG ≈ εPart1−MG + εPart2−MG − εPart3−D . (8)

The first term in (8) εPart1−MG is the MG model (1) computedfor the base material with constant high-frequency permittivityεe = εe∞ and Debye inclusion material εi(εis ,εi∞,τ i) as in (2).The second term in (8), εPart2−MG , is the MG model (1) com-puted with the Debye host εe (εes ,εe∞,τe ) as in (2) and constantparameters of inclusions, εi = εi∞. These first two terms takeinto account the Debye behavior of εe and εi separately. How-ever, the sum of these two terms takes into account twice thelevel of the high-frequency permittivity for both materials, thusa correction term is introduced, εPart3−D , as correction factor.It is constructed directly as a Debye model whose parameters(εs−Part3 , ε∞−Part3 , τPart3) need to be computed.

The three parameters associated with εPart3−D are defined asfollows. The left side term in (8) and also the first two termson the right side of (8) are known; this assumption is employedand the limits of (8) for ω going to zero and infinity are com-puted. This leads to approximate the static (εs−Part3) and high-frequency (ε∞−Part3) permittivity of the correction Debye termas in (9a) and (9b), respectively. The τPart3 parameter is simplyapproximated by averaging the relaxation time for the host τe

and inclusion τ i materials (9c)

εs−Part3 = limω→0

εPart1−MG + limω→0

εPart2−MG − limω→0

εeff−MG

(9a)

ε∞−Part3 = limω→∞

εPart1−MG + limω→∞

εPart2−MG

− limω→∞

εeff−MG (9b)

τPart3 =τe + τi

2. (9c)

At this stage, all the three terms in (8) are evaluated in terms ofthe original MG model. The first two terms have the MG form,whereas the third one has a Debye form. In order to have a fullyDebye description of (8), the first two terms should be convertedin the Debye form. This can be done similarly to procedures of



getting the first term εPart1−MG in Case 2s, and second termεPart2−MG in Case 3s. After this algebraic manipulation (8)becomes

εeff−MG = εPart1−D + εPart2−D − εPart3−D . (10)

D. Case 5s

Case 5s considers conductive inclusions εi(εi∞,σi) as in (3),embedded in a dielectric host with constant εe .

A perfect equivalent Debye model can be derived in this casefollowing the same steps as done for Case 2s. The relationshipsthat provide the parameters of the equivalent Debye model arethe following:

εsD =εe (1 + 2f)

1 − f(11a)

ε∞D =εe (εi∞ + 2fεi∞ + 2εe − 2fεe)

(εi∞ − fεi∞ + 2εe + fεe)(11b)

τD =ε0 (εi∞ − fεi∞ + 2εe + fεe)

σi (1 − f). (11c)

E. Case 6s

The conductive inclusions εi(εi∞,σi) in this case are em-bedded in a Debye host material εe (εes ,εe∞,τe ). The resultantbehavior of real and imaginary parts of the effective MG per-mittivity is very similar to the two-term Debye model, as inCase 4s. This can be stated by observing the dashed curves inFig. 1. Therefore, the same approach as for Case 4s is appliedfor deriving an equivalent Debye model that approximates thegeometry-based MG model, and there are three terms as in (8)defined as follows.

The first term εPart1−MG is a MG model computed by (1)with constant εe = εe∞ and conductive inclusions εi(εi∞,σi) asin (3). The second term εPart2−MG is a MG model computedby (1) with the Debye εe (εes ,εe∞,τe ) as in (2) and constantεi = εi∞ taken at the “optic limit.” The third term εPart3−D is thecorrection factor. It is described by the Debye model (εs−Part3 ,ε∞−Part3 , τPart3), which can be computed using (9a)–(9c).

For the values of the parameters considered in Case 6s it canbe obtained that ε∞−Part3 ≈ εs−Part3 with the difference lessthan 1%. Therefore, the Debye model for εPart3−D reduces to aconstant value εPart3 , and (8) becomes

εeff−MG = εPart1−MG + εPart2−MG − εPart3 . (12)

In order to have a fully Debye description of (12), the first twoterms should be converted in the Debye form. This can be doneby using the same procedures as for the first term εPart1−MG inCase 5s, and for the second term εPart2−MG as in Case 3s. Thethird term εPart3 is computed as (9a). Then

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

III. RESULTS AND DISCUSSION

Five approaches developed earlier are applied to the casespresented in Table I. The real and imaginary parts of the MG

Fig. 2. Comparison of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 2s A: (a) real part and(b) imaginary part. Average error AE = 0.0% both for the real and imaginarypart comparisons.

Fig. 3. GDM results for the pair of curve in Fig. 2 for f = 46.8%: (a) real part(Grade = 1, Spread = 1), and (b) imaginary part (Grade = 1, Spread = 1).

permittivity εeff−MG (1) are compared with the permittivity dataεeq−Debye obtained using the equivalent Debye model. The pa-rameters of the latter are evaluated by applying the proposedformulation. The agreement between the sets of curves that arecompared is quantified by applying the feature selective valida-tion (FSV) technique [16]–[19], as required by the recent IEEEStandard P1597 [20]. The calculated figure of merit is the globaldifference measure (GDM). GDM shows the global, i.e., gen-eral trend and details or features, agreement between differentdatasets. The differences between the curves are also quanti-fied computing the percentage average error AE, as in (14); thisprocedure is applied to the same sets of datasets for which the



Fig. 4. Comparisons of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 3s A: (a) real part and(b) imaginary part. Average error AE = 0.0% for the real part comparison;AE = 0.07% for the imaginary part comparisons.

GDM is evaluated

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

Re(εeff−MG(f))(14a)

AE =

∑Nf =1 E (f)

N· 100. (14b)

The concentration of spherical dielectric inclusions in anotherdielectric, the matrix, may be theoretically up to 46.8%, i.e.,when inclusions almost start touching each other. This limitis also theoretically applicable to conductive inclusions in adielectric, until inclusions touch each other in a dense 3-Dpackage. However, practically, it is known that the MG forspherical conducting inclusions starts significantly deviate fromexperiment at volume fractions exceeding about 20%–30% (thisis because of the arising more complex inclusion–inclusionand inclusion–matrix interactions). In the following examples,dielectric–dielectric mixtures (Cases 2s–4s) are “safely” consid-ered up to the concentration of 46.8%. In the Cases 5s and 6s,conducting inclusions in a dielectric matrix are considered alsoat the concentrations up to the 46.8% limit. It is seen, though,that in Case 6, when the matrix is the Debye dielectric, the repre-sentation of the MG formula by the Debye parameters becomesless accurate near the relaxation frequencies at the concentrationof conducting inclusions greater than ∼ 20%.

Fig. 5. Comparison of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 2s B: (a) real part, and(b) imaginary part. Average error AE = 0.03% for the real part comparison; AE= 0.2% for the imaginary part comparisons.

Fig. 6. GDM results for the pair of curve in Fig. 4 for f = 46.8%: (a) real part(Grade = 1, Spread = 1), and (b) imaginary part (Grade = 1, Spread = 1).

A. Case 2s

The first case considered is Case 2s A in Table I. This is thecase with the high contrast between the constant host permittiv-ity εe = 2 and both Debye parameters of the inclusions, εis =1900 and εi∞ = 280. This makes the effect of inclusions uponthe equivalent Debye model dominant.

Five different values of the volume fraction f = 2.5%, 8.4%,20.1%, 39.3%, and 46.8% are considered; these cases allowstudying the formulation robustness with respect to the variationof the inclusion volume fraction f. Fig. 2 shows the comparisonbetween the MG and the equivalent Debye models. There is aperfect agreement between the pair of curves due to the fact thatthe parameters in (6) are the exact solution of the set of equationsobtained equating (4b), (4c) and (5b), (5c). Fig. 3(a) and (b) showthe GDM parameter for the comparison of the corresponding



Fig. 7. Comparisons of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 4s A: (a) real part, and(b) imaginary part. Average error AE = 0.05% for the real part comparison; AE= 17.9% for the imaginary part comparisons.

Fig. 8. GDM results for the pair of curve in Fig. 7 for f = 46.8. (a) Real part(Grade = 1, Spread = 2), and (b) imaginary part (Grade = 3, Spread = 3).

real and imaginary parts as in Fig. 2. These calculations aredone for the volume fraction of f = 46.8%.

B. Case 3s

The values of the parameters associated with Case 3s A andCase 3s B in Table I are considered for validating the methodproposed in Section II-B. Three values of volume fraction f =2.5%, 20.1%, and 46.8% are used for computing the effectivepermittivity (1) and its equivalent Debye model. The results arepresented in Figs. 4 and 5. They agree very well, even though theequivalent Debye model does not stem from an exact solution,but represents an approximation of the original model. Fig. 6provides a quantification of the comparison through the FSVtechnique. The FSV was applied to the datasets of Case 3s A

Fig. 9. Comparisons of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 5s B. (a) Real part.(b) Imaginary part. Average error AE = 0.0% both for the real and imaginarypart comparisons.

Fig. 10. GDM results for the pair of curve in Fig. 9 for f = 46.8%: (a) Realpart (Grade = 1, Spread = 2) and (b) imaginary part (Grade = 3, Spread = 3).

for the volume fraction of inclusions of f = 46.8%. From Figs. 3and 6, it is seen that the approximated formulation in (7) and (8)provides an excellent match with the exact one. A same FSVresponse is achieved for Case 3s B.

C. Case 4s

The fourth case considers the host and the inclusions mate-rials described by a Debye model. Case 4s A is selected forrunning this comparison, since the Debye parameters associ-ated with the inclusions are much higher (εis = 1900, εi∞ =280) than those related to the host (εes = 2.5, εe∞ = 2.2). Withthis choice, the differences between the two materials are morenoticeable. The comparison of the real and imaginary part of



Fig. 11. Comparisons of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 6s A: (a) real part and(b) imaginary part. Average error AE = 1.5% for the real part comparison;AE = 51% for the imaginary part comparisons.

εeff−MG and εeq−Debye are given in Fig. 7(a) and (b), respec-tively. The approximation introduced by the sum of the threeterms in (10) does not give a perfect agreement between theoriginal MG and the equivalent Debye models. However, themain features introduced by the two Debye dependencies (dueto the host and inclusion materials) are reproduced. The good-ness of the equivalent Debye model is quantified in Fig. 8 bythe FSV response.

D. Case 5s

The Case 5s is characterized by conductive inclusions. Theequivalent Debye model is derived analytically in terms of theinput dielectric properties of host material and inclusions inthe equivalent Debye model, as detailed in Section II-D. TheCase 5s B in Table I is considered for validating the derivedequivalent Debye model. The inclusion conductivity is σi =4·104 S/m. The comparison of the results is presented in Fig. 9.The FSV quantification is shown in Fig. 10 for both the real andthe imaginary part. The fully analytical derivation employed inthis case leads to a perfect matching between the equivalentDebye and the original MG models. This is because the equiv-alent Debye model is obtained using the same approach as in

Fig. 12. Comparisons of the original MG model (solid curve) and the com-puted equivalent Debye model (dashed curve) for Case 6s B: (a) real part, and(b) imaginary part. Average error AE = 2.1% for the real part comparison;AE = 54% for the imaginary part comparisons.

Case 2s. It is worth to note that Case 5s A with σi = 4·103 S/mdemonstrates the similar model features.

E. Case 6s

Herein, the comparison of the original MG model εeff−MG(1) and its equivalent Debye model εeq−Debye for Case 6s A isprovided. This case is related to conductive inclusions embeddedin a Debye-dependent host material. Cases 6s A (σi = 4·103)and 6s B (σi = 4·104 ) are considered for validating the proposedapproach. εeff−MG and εeq−Debye for Cases 6s A and 6s B arecompared in Figs. 11 and 12, respectively. The FSV results inFigs. 13 and 14 are employed for quantifying the differencesbetween the two models. This combination of values providesthe worst approximation among all the cases.

The results of comparison for those two cases are very similar.The equivalent Debye model has the same trend as the originalMG model, even though the derived Debye model is not ableto catch the first step in the real part (and the first peak in theimaginary part) at around 1010 Hz for Case 6s A, and around1011 Hz for Case 6s B.



Fig. 13. GDM results for the pair of curve in Fig. 11 for f = 46.8%: (a)Real part (Grade = 3, Spread = 3), and (b) imaginary part (Grade = 4,Spread = 3).

Fig. 14. GDM results for the pair of curve in Fig. 12 for f = 46.8%: (a) Realpart (Grade = 2, Spread = 2), and (b) imaginary part (Grade = 4, Spread = 4).

IV. CONCLUSION

The method that we proposed is an efficient semianalyticalway for obtaining an equivalent Debye model from the MGmixing rule for the permittivity of a composite dielectric ma-terial. There is no need for special algorithms to curve-fit per-mittivity of an equivalent homogeneous material to a seriesof Debye terms. The computation time for running the devel-oped analytical formulas is negligible (the order of millisec-onds). Some cases are considered covering real-world combi-nations of values for the host and inclusions materials. Thebest matching between the MG mixing formulation and itsequivalent Debye model is obtained when only one ingredi-ent in a biphasic mixture has a frequency dispersive behav-ior, while the other is nondispersive. This conclusion holdsfor Cases 2s, 3s, and 5s; the very good agreement betweenthe original MG model and the derived Debye model is con-firmed by the FSV GDM parameter, it provides always an “ex-cellent” agreement, and by the computed average error, it isalways less than 0.2%. The worst approximation is obtainedwhen both ingredients have a frequency dispersive response, asfor Cases 4s and 6s; the comparisons of the real parts is stillgood; this is confirmed by both the FSV results (the “excel-lent” bar is always the largest), and by the average error, alwaysless than 2.1%. The FSV outputs for the imaginary part com-parisons include also the “fair” and “poor” bars, even thoughthe largest bar value is held by the “excellent” or the “verygood” category. The average error, in the imaginary part com-parisons, is less meaningful since it is sensitive to the very smallvalues of the data toward dc and infinity; thus, small differ-ences between the two models could lead to very large valuesof the computed average error (i.e., more than 50% for theCase 6s).

The practical outcome is to be able to incorporate bipha-sic Debye model in time-domain simulations. This is impor-

tant for finding out whether the particular mixture satisfies in-tended requirements. The practical scenarios can, for exam-ple, be application of composite materials to reduce common-mode currents, cavity resonances, absorb unwanted radiation,or terminate unwanted coupling paths in particular EMI/EMCproblems.

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Francesco de Paulis (S’08) was born in L’Aquila,Italy in 1981. He received the Laurea degree andthe Specialistic degree (summa cum laude) in elec-tronic engineering both from University of L’Aquila,L’Aquila, Italy, in 2003 and 2006, respectively. InAugust 2006, he joined the Electromagnetic Compat-ibility (EMC) Laboratory at the Missouri Universityof Science and Technology (formerly University ofMissouri-Rolla) Rolla, where he received the M.S.degree in electrical engineering in May 2008. He iscurrently working toward in the Ph.D. degree at the

University of L’Aquila.He was involved in the research activities at the UAq EMC Laboratory from

August 2004 to August 2006, L’Aquila and at the UMR EMC Laboratory, Rolla,from August 2006 to May 2008. From June 2004 to June 2005, he had an intern-ship at Selex Communications, L’Aquila, within the layout/SI/PI design group.He is currently a Research Assistant at the UAq EMC Laboratory, Universityof L’Aquila. His main research interests include in developing fast and efficientanalysis tool for SI/PI and design of high speed signal on PCB, RF interferencein mixed-signal system, EMI problem investigation on PCBs, and compositematerial for shielding.

Mr. de Paulis received the Past President’s Memorial Award from the IEEEEMC Society in 2010 and in 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.

Muhammet Hilmi Nisanci was born in Istanbul,Turkey, in 1983. He received the B.S. and M.S. de-grees from Suleyman Demirel University, Isparta,Turkey, in 2006 and 2009, respectively, both inelectronic and telecommunication engineering. Heis currently working toward the Ph.D. degree inelectrical engineering at University of L’Aquila,L’Aquila, Italy.

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

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.

Marina Y. Koledintseva (M’95–SM’03) receivedthe M.S. and Ph.D. degrees in 1996 in 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 of MPEI (TU), and from1997 to 1999 combined research with teaching asan Associate Professor in the same University. SinceJanuary 2000, she has been working as a ResearchProfessor with the Electromagnetic Compatibility

(EMC) Laboratory of the Missouri University of Science and Technology(MS&T), formerly known as the University of Missouri-Rolla, Rolla. Her scien-tific interests include microwave engineering, analytical and numerical modelingof interaction of electromagnetic waves with complex geometries and materials,engineering composite materials with desirable electromagnetic properties, andtheir application for electromagnetic compatibility. She has published over 150papers in peer-reviewed journal and proceedings of international conferences,and is an 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 (F’09) received the B.S., M.S.,and Ph.D. degrees in electrical engineering from theUniversity of Illinois, Urbana-Champaign, in 1985,1987, and 1991, respectively.

He is currently with Electromagnetic Compati-bility (EMC) Laboratory in the Electrical Engineer-ing Department, Missouri University of Science andTechnology. His research and teaching interests in-clude electromagnetic compatibility in high-speeddigital and mixed-signal designs, signal and powerintegrity, electronic packaging, electromagnetic com-

patibility in power electronic based systems, electronics, and antenna design.Dr. Drewniak is an Associate Editor for the IEEE TRANSACTIONS ON EMC.

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 En-gineering, University of Rome “La Sapienza” from1988 to 1990. Since 1990, he has been with theDepartment of Electrical Engineering, University ofL’Aquila, where he is currently a Full Professorand Chair of the UAq Electromagnetic Compatibility(EMC) Laboratory. He is the author of more than 230

technical papers in the field of electromagnetic compatibility in lightning pro-tection systems and power drive systems. His current research interests includenumerical methods and modeling techniques to approach signal/power integrity,EMC/EMI issues in high speed digital systems.

Dr. Orlandi is the recipient of IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY Best Paper Award in 1997, the IEEE EMC Society TechnicalAchievement Award in 2003, the IBM Shared University Research Award in2004, 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 iscurrently Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY, member of the “Education,” TC-9 “Computational Electro-magnetics” and Past Chairman of the TC-10 “Signal Integrity” Committeesof the IEEE EMC Society. From 1996 to 2000 has been Associate Editor ofthe IEEE TrANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, from 2001 to2006 served as Associate Editor of the IEEE TRANSACTIONS ON MOBILE COM-PUTING and from 1999 to the end of the Symposium was Chairman of the TC-5“Signal Integrity” Technical Committee of the International Zurich Symposiumand Technical Exhibition on EMC.

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 …web.mst.edu/~marinak/files/My_publications/Papers/MG_Debye_I.pdf · high-frequency (optic limit) relative permittivity, respectively,

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