systematic analysis and engineering of absorbing materials...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011 317

Systematic Analysis and Engineering of Absorbing Materials ContainingMagnetic Inclusions for EMC Applications

Marina Y. Koledintseva�, Jianfeng Xu�, Soumya De�, James L. Drewniak�, Fellow, IEEE, Yongxue He�, andRichard Johnson�

Missouri University of Science and Technology, Rolla, MO 65401 USALaird Technologies, San Jose, CA 95131 USA

A methodology to efficiently design novel products based on magneto-dielectric materials containing ferrite or magnetic alloy inclu-sions is presented. The engineered materials should provide desirable frequency responses to satisfy requirements of electromagneticcompatibility/immunity over RF and microwave bands. The methodology uses an analytical model of a composite magneto-dielectricmaterial with both frequency-dependent permittivity and permeability. The Bruggeman asymmetric rule for effective permeability of acomposite is modified to take into account demagnetization factors of inclusions, and is shown to be applicable to platelet magnetic in-clusions. Complex permittivity and permeability are extracted from the transmission-line measurements. A novel accurate and efficientcurve-fitting procedure has been developed for approximating frequency dependencies of both permittivity and permeability of mag-neto-dielectric materials by series of Debye-like frequency terms, which is important for wideband full-wave numerical time-domainsimulations. Results of numerical simulations for a few structures containing magneto-dielectric sheet materials and their experimentalvalidation are presented.

Index Terms—Absorbing media, causality, composite material, electromagnetic compatibility, ferrites, frequency response, magneticmaterials, microstrip line, transmission line measurements.

I. INTRODUCTION

A design of wideband nonconductive absorbing shieldingenclosures, protecting screens, wallpaper, coatings with

specific filtering properties, and gaskets is important for solvingnumerous problems of electromagnetic compatibility (EMC)and improving immunity of electronic equipment [1], [2].Composite electromagnetic wave absorbers (EMWA) andnoise-suppressor sheets (NSS) protect susceptible devices,components, and circuits by absorbing undesirable radiation,by eliminating possible surface currents and cavity resonances,and by diverting or terminating unwanted coupling paths.Combining dielectric or conducting inclusions with ferrite ormagnetic alloy inclusions in a composite may substantiallyincrease the absorption level in the frequency range of interest[3].

To engineer EMWA and NSS composite materials, includingnanocomposites, it is important to adequately predict widebandfrequency responses of constitutive electromagnetic parameters(permittivity and permeability), as well as concentration depen-dences of these composites. There are many different mixingrules available in the present-day literature (see, e.g., [4] andreferences therein), but every rule has its own limitations. Thus,for an important case of composites filled with ferromagneticmetal powders, currently there is no standard and unified exper-imentally validated mixing rule to calculate dependences bothon frequency and concentration, especially if inclusions are non-spherical [5]. One of the objectives of this paper is to present amodel for effective permeability of mixtures containing mag-netic (ferrite or ferromagnetic alloy) platelets.

Manuscript received July 01, 2010; revised September 15, 2010; acceptedSeptember 23, 2010. Date of current version January 26, 2011. Correspondingauthor: M. Y. Koledintseva (e-mail: [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/TMAG.2010.2084991

Another objective is to analyze a few scenarios of applyingdifferent magneto-dielectric absorbing sheet materials usingfull-wave numerical finite-difference time-domain (FDTD)technique, and verify modeling results by experiments. Anadvantage of time-domain numerical techniques is the pos-sibility of having broadband responses. To effectively modelmagneto-dielectric materials in time domain, it is importantto represent frequency characteristics of both complex per-mittivity and permeability of these materials as analyticalrational-fractional functions that would satisfy Kramers-Krönigcausality relations (KKR) [6], for example, sums of the Debyeterms with the poles of the first order [7]. If a material exhibitsnarrowband resonances, then Lorentzian terms with poles ofthe second order should be used [8]. However, the present studyis limited to the Debye terms only, since the majority of mi-crowave absorbing materials can be described in terms of Debyedependencies only. The materials whose both permittivity andpermeability frequency functions can be represented throughDebye terms are called double-Debye materials (DDM) [9].

A new accurate and efficient technique to approximate ex-perimental or modeled (using corresponding mixing rules) fre-quency dependencies for permeability and permittivity of DDMmaterials is also described in detail in this paper. This curve-fit-ting technique is based on application of Legendre polynomialsand least-square regression analysis, and the results of curve-fit-ting guarantee satisfying KKR, which is extremely important fornumerical modeling correctness.

The presented methodology of designing absorbing materialsincludes an FDTD numerical code that allows for effective mod-eling of complex geometries containing frequency-dispersivematerials and evaluating absorbing properties of the engineeredmaterials, as well as filtering or shielding properties of struc-tures that contain these materials. Such a code (EZ-FDTD) hasbeen developed in the EMC Laboratory of Missouri Universityof Science & Technology [9]–[11]. The EZ-FDTD uses aux-iliary differential equations for incorporating DDM. Currently,this code allows for taking into account up to five Debye terms

0018-9464/$26.00 © 2010 IEEE

318 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011

both in permittivity and in permeability frequency responses,but this is not a fundamental limitation of the curve-fitting orFDTD technique: as many terms as required by reasonable ac-curacy can be used in the code.

II. EFFECTIVE MAGNETIC AND DIELECTRIC

PROPERTIES OF MIXTURES

Suppose that and are relative permittivities of theinclusions and base (background) materials, respectively,and they both can be complex functions of frequency:

. Effective permittivity of thismixture can be modeled with the most commonly usedMaxwell Garnett (MG) mixing rule [4], if the mixture satisfiesquasi-static conditions, and the volume fraction of inclusionsis below the percolation threshold, if inclusions are conducting.Also, it is known that the MG mixing rule gives good agreementwith experiment only when the contrast between the materialparameter of a host matrix and inclusions is comparatively low[5]. In the case of randomly oriented ellipsoidal inclusions withdepolarization factors , where correspond to ,

, and directions, the effective permittivity of the mixture canbe calculated through the MG rule as

(1)

If inclusions are all aligned, the corresponding MG formulagives components of the diagonal permittivity tensor

(2)

with index corresponding to , , and directions.The formulas analogous to (1) and (2), obtained by simple

replacing , can be used to predict behavior ofeffective permeability in nonaligned case, or in thecase of aligned inclusions.

(3)

and(4)

In the case of a nonmagnetic base material, . Permit-tivities and permeabilities

can be complex functions of frequency. However, itis important to mention that the intrinsic permeability of inclu-sions in the general case is different from the permeability

of the bulk magnetic material (ferrite or alloy) these in-clusions are made of. These values and can be relatedthrough the crush parameter , which is associated with demag-netization along the magnetic grain boundaries, and determineslocal magnetic interaction between neighboring grains, whenthe bulk material is crushed into a powder [12], [13]. There aresome other reasons for the difference between the permeabilityof bulk material and powders. Thus, the magnetic material pro-

cessing for making a powder (such as milling, etc.) may affectintrinsic magnetic fields and, therefore, the permeability. Also,in single-domain particles, the parameters of the ferromagneticresonance depend on the shape of the particle [14].

Magnetic platelets are of special interest due to their high in-ternal field of shape anisotropy. Their form (demagnetization)factors are related to the magnetic anisotropy field incide inclu-sions, even if there is no crystallographic anisotropy in the ma-terial. Internal magnetic moment lies in the plane of the platelet,while demagnetization field is normal to this plane [15].

This section presents an attempt to take into account the pro-nounced shape anisotropy of disk-like inclusions. It is knownthat the MG prediction is comparatively accurate for sphericalor stone-shaped magnetic inclusions [16]. Demagnetization fac-tors are present in (3) and (4), but these equations more accu-rately take into account shape of inclusions only at lower levelsof permeability. The MG mixing formula (1) substantially over-estimates effective permeability, when considering simultane-ously high ( 1) permeabilities and aspect ratios of inclusions.

The proposed new analytical model for effective permeabilityof a composite containing magentic inclusions is based on acombination of the Bruggeman asymmetric rule (BAR) [17] andBruggeman symmetric rule (BSR) [4].

The BAR for effective permeability is written as [17]

(5)

For the mixtures containing arbitrary-shaped magnetic crumbsor spherical inclusions, the BAR gives predictions which agreewell with experimental data [9]. However, in (5) the demagne-tization factors are missing.

At the same time, it is known that the BSR, which is a formof the Bruggeman effective medium theory (EMT), accounts forshape factors of inclusions [18]

(6)

For disk-shaped inclusions with an aspect ratio(ratio of their diameter to thickness), the axial demagnetizationfactor is approximately [19]

(7)

while the other two depolarization factors. Indeed, disk-like inclusions could be approximated as

oblate spheroids.It was noticed that for high inclusion-host permeability con-

trast and volume fractions of magnetic inclusions over 30%, theresultant obtained through the BSR (6) is much higher thanthe values predicted by BAR (5). This happens even if the as-pect ratio of disk-like inclusions is small, , and thecorresponding demagnetization factors are close to those in thespherical case .

The objective is to modify the BAR in such a way that it wouldbe possible to apply it to disk-shaped inclusions. It is appealingto introduce the correction factor which would depend both onconcentration and aspect ratio of inclusions, and it wouldbe possible to calculate effective permeability for disk-shapedinclusions , knowing the corresponding effective

KOLEDINTSEVA et al.: SYSTEMATIC ANALYSIS AND ENGINEERING OF ABSORBING MATERIALS 319

Fig. 1. Effective permeability calculated using BSR (EMT) and correspondingcurve-fitting to retrieve the function � �� .

relative permeability in spherical case

(8)

Let us assume that this factor would be the same forthe BSR (6), too, so that

(9)

Fig. 1 presents curves as functions of the aspect ratiofor different inclusion concentrations p calculated using BSR

(6). These curves are calculated for the magnetic material withintrinsic static permeability of inclusions , which cor-responds to the bulk permeability and the crushparameter [9].

A fitting dependence for the factor is obtained byanalyzing numerous dependencies as those in Fig. 1. It can bewell approximated by a simple analytical expression

(10)

where corresponds to the spherical case calculatedusing BSR (5), and and are fitting parameters. This is animportant result for extending the BAR model.

Fig. 2(a) shows the effective permeability of the nonalignedmixture calculated using the modified BAR (5) and (8). Fig. 2(b)shows the permeability curves obtained using MG rule for non-aligned inclusions. In these calculations, the volume fraction ofinclusions is 25%, and the initial data is the same as in Fig. 1:

and MHz. Fig. 2(a) and (b) show that asaspect ratio increases, static permeability increases, and the losspeak slightly shifts to the lower frequencies.

It is seen that the MG rule predicts higher permeability thanthe modified BAR. As is mentioned above, the dependence ofMG on shape is known to be more accurate at lower perme-ability levels than at higher. For this reason, it is believed thatthe modified BAR predicts effective permeability for nonspher-ical inclusions more realistically than MG. For carbonyl iron(CI) flakes with , G, and aspectratio , the calculated data agrees quite well with experi-ments [20], as is shown in Fig. 3.

Fig. 2. Effective permeability calculated using: (a) Modified Bruggeman asym-metric rule (MBAR); (b) Maxwell Garnett (MG) rule.

III. EXTRACTION OF MATERIAL FREQUENCY DEPENDENCES

This section describes the new proposed curve-fitting pro-cedure to approximate frequency dependencies of materials assums of the Debye-like terms. This curve-fitting procedure canbe applied both to experimentally obtained (for example, using7/3 mm coaxial airline technique and a vector network analyzer)and to modeled through mixing rules and of compos-ites [8]

(11)

and

(12)

Such representation of frequency dependencies of dielectric andmagnetic properties as sums of Debye-like terms is conve-nient for using them in the FDTD simulations. Previously, acurve-fitting technique based on the genetic algorithm (GA)optimization was used for extracting parameters of the Debyeterms from experimentally available data [7], [8]. The GA flow-chart is shown Fig. 4(a). Though the GA yields a global op-timum, it is a very tedious task to obtain proper results using


Fig. 3. Experimental [20] and obtained using modified Bruggeman asymmetricrule (MBAR) permeability (a) real part, and (b) imaginary part.

Fig. 4. Flowcharts for curve-fitting: (a) genetic algorithm, and (b) Legendrepolynomial approximation and regression analysis.

the GA method. To make the GA converge fast to the optimalsolution, a user must make a good preliminary guess and test nu-merous initial values at the very start, which is time-consumingand requires the user’s knowledge on using the GA and experi-ence working with the code.

A powerful technique used in numerical methods for curve-fitting the measured data is an estimation of parameters by theprinciple of least squares. This method can reduce random and

systematic errors that may be due to fluctuations in the measureddata [21]. In addition, sometimes in polynomial-fitting the fol-lowing artifice is used: the given set of data points is transformedto another form, which is more easily represented by polyno-mials. For example, the given set of points may be multipliedby some function resulting in a linearized or a more simplifiedpolynomial representation of the points. This requires decidingwhich function would be suitable for this transformation, and asstated in [22], there is no particular simple rule to follow.

In the proposed approach, the flowchart of which is shown inFig. 4(b), an orthogonal-polynomial fitting for the Debye curvesby using the discrete Legendre polynomials was chosen. Thistechnique allows for modeling permittivity and permeability asmulti-term Debye curves with as many terms as required by theaccuracy of curve-fitting and the ability of the numerical electro-magnetic code to handle this number of terms. Legendre poly-nomials are used as the basis functions to model the measureddata. In general, the -degree discrete Legendre polynomialcan be expressed as [23]

(13)

for , . Herein, is the parameter of theLegendre polynomial; and are the backward factorialfunctions of the order defined by

(14)

and the binomial coefficient is .In terms of multiple regression of Legendre polynomials up

to the degree, any smooth curve can be expressed as[24]

(15)

whereifif

is analogous to Kronecker

symbol in the sampling frequency points. Then the sum of thesquares of the differences between the experimental data andthe smooth curve obtained as in (15) is given by

(16)

Applying the least-squares criterion and performing somecalculations, one could obtain the following expression for theregression coefficients

(17)

where . The fitted curve estimated by the leastsquares method is obtained by substituting (17) to (15).

The final step in this approach is using the Matlab lsqcurvefitfunction [25] to approximate the smooth curve in (15) by theDebye dependencies (11) and (12) for complex permittivity and


Fig. 5. Measured and curve-fitted permittivity of an absorbing material: (a) realpart; and (b) imaginary part.

permeability, respectively. The lsqcurvefit function finds the co-efficients that best fit data using the nonlinear least squares re-gression method as

(18)

where is the data obtained using the Legendre polynomials(15), and is the Debye data. The vector

is the set of Debye parameters for permittivity(11), or is for permeability (12). The index

is the order of the corresponding Debye termsin (11) and (12), and is the order of the frequency samplingpoint. These vectors as arguments of are the Debye pa-rameters of interest. The measured and curve-fitted data for per-mittivity and permeability of an absorbing magneto-dielectricmaterial are shown in Figs. 5 and 6. The GA curve-fitting al-gorithm was realized for two Debye terms in permittivity, andthree Debye terms in permeability. Realization of GA with moreDebye terms is cumbersome and time-consuming. The resultsof curve-fitting using regression analysis with Legendre poly-nomials shown in these figures were obtained with five Debye

Fig. 6. Measured, curve-fitted, and modeled using MBAR permeability of anabsorbing material.

terms for permittivity, and five Debye terms for permeability.As is mentioned above, there is no fundamental limitation forthe number of terms. It can be seen that the Legendre polyno-mials provide a very efficient and robust way to perform a curvefit of the measured (or modeled) microwave permittivity andpermeability by the Debye curves. As is shown by Kirkpatrickand Heckman [26], and as is implemented in [27], Legendrepolynomials have several favorable properties for curve-fitting.These properties are the following: the functions are orthogonal;there is flexibility to fit sparse data; higher orders are estimablefor high levels of curve complexity; and computations convergefast.

An important validation for causality involving theKramers-Krönig relations (KKR) could also be providedusing this method. The real and imaginary parts of the par-ticular material parameter (permittivity or permeability),according to the causality principle, are not independent. KKRare the integral relations which express this interdependence.For example, for permittivity they are [6]

(19)


Fig. 7. Microstrip line geometry coated with absorbing material: (a) experi-mental test structure, and (b) FDTD model setup.

where the principal parts of integrals are taken. The analogousrelations are valid for real and imaginary parts of complexpermeability.

Approximating measured permittivity and permeability byseries of the Debye terms assures that the resultant curve-fittedfrequency dependencies will be causal, while the initial mea-sured data might violate causality in some frequency point orregions due to measurement errors. Causality of data which isthen used in numerical codes is extremely important for the nu-merical stability and physical meaningfulness of the modelingresults. To assure causality when curve-fitting measured datausing the proposed approach, one could either first curve fit theimaginary data, and then apply the KKR to restore the real part,or, vice versa, first curve fit the real data, and then restore theimaginary part using the KKR. To correctly restore data usingthe KKR, some measured points for the curve to be restored areneeded, since the KKR are valid up to some constant offset.

The regression analysis-Legendre polynomial curve-fittingprocedure and KKR check may be applied several times inturn to real and imaginary parts to minimize the total errorof curve-fitting both real and imaginary parts, while assuringcausality of the curve-fitted response. The weight coefficientsof curve-fitting accuracy are assigned to each frequency rangeof interest, as well as to real and imaginary parts of permittivityand permeability.

The curve-fitted data obtained using the Legendre polyno-mial-least squares regression method was successfully used inthe FDTD simulations, and the results are discussed in the nextsection.

IV. FDTD SIMULATIONS AND MEASUREMENTS

The effect of various absorbing sheet materials on frequencycharacteristics of a specially designed microstrip line shown inFig. 7(a) has been studied. The length of the board is 14.7 cmwith a 3.5 mm wide trace. The height of the dielectric is 1 mmand the relative permittivity is 3.53 with a tangent loss of 0.001to provide the 50- characteristic impedance of the board. Themicrostrip line was operating in two regimes, short- and open-circuited. Measurements are carried out in the range from 0.9 to6 GHz using Agilent vector network analyzer E-5071C.

The same structure was modeled using the FDTD codes withthe bulk discretization cells incorporating Debye dielectricand magnetic material (DDM). The modeling setup is shownin Fig. 7(b). In the simulations, the length was tuned to takeinto account the effect of two connectors. The cell size alongdirection is 0.1 mm, while the cell sizes along and directionsare 0.5 mm.

Fig. 8. FDTD modeled and measured input impedance of the bare board in theopen-circuit case: (a) real part, and (b) imaginary part.

Fig. 8 shows the simulation results of the bare board togetherwith the measured data for the open-circuit termination. As isseen from this figure, the agreement between the simulated andmeasured results in the short-circuit case is excellent for bothreal and imaginary parts of the input impedance through thewhole frequency range of interest. The difference of the reso-nance magnitudes may be explained by the perfect electric con-ductor assumption for ground plane and trace in the simulationand not sufficient loss in the board dielectric.

Fig. 9 shows the simulation and measured results for the sameopen-circuited board covered with same absorbing sheet as dis-cussed in Section III. The FDTD model used the Debye termsobtained by the Legendre polynomial and regression curve fit(see the corresponding curves in Figs. 5 and 6). Thickness ofthe absorbing sheet is 0.5 mm, and its width and length are both10 mm. The sheet is put directly upon the trace. As is seen fromFigs. 8 and 9, in the loaded case, resonances damp and shiftto the lower frequencies, and this does not contradict physics.The positions of resonance peaks in the simulation and mea-surements agree well in the considered frequency range. Somediscrepancy in amplitudes of peaks is due to the difference inthe measured and modeled material parameters of the absorber,probable gap between the layer and the board, and underesti-mated loss on the bare board. The comparatively good agree-ment validates the correctness of the simulation method.

From the above analysis, it is clear that the new approachfor frequency dependent material parameter extraction is a very


Fig. 9. Simulated and measured input impedance of the microstrip line coveredwith the absorber: (a) real part; (b) imaginary part.

promising method and will be used for future studies for thisfield of research.

V. CONCLUSION

The methodology of designing new magneto-dielectric ab-sorbing materials and structures on their basis for different EMCapplications is presented. These applications may include ra-diation from heatsinks, parasitic resonances within enclosures,spurious radiation from chips and other active circuit compo-nents, etc. The proposed new mixing rule for predicting effectivemagnetic parameters of composites over wide frequency rangegives reasonable agreement with measured results. FDTD codewith accurate and satisfying causality relations curve-fitting ofcomplex-shaped frequency characteristics by series of Debyeterms is an efficient tool to evaluate whether a material couldbe a successful candidate for mitigating an electromagnetic in-terference. The optimization for choosing proper dielectric andmagnetic properties of materials and their ingredients in com-posites, as well as geometries (configuration and thickness oflayers) for particular practical problems can be done based onthe results of this work.

REFERENCES

[1] S. Celozzi, R. Araneo, and G. Lovat, Electromagnetic Shielding. NewYork: Wiley, 2008.

[2] P. S. Neelakanta, Handbook of Electromagnetic Materials: Monolithicand Composite Versions and Their Applications. Boca Raton, FL:CRC Press, 1995.

[3] M. Y. Koledintseva, P. C. Ravva, J. L. Drewniak, A. A. Kitaitsev, andA. A. Shinkov, “Engineering of ferrite-graphite composite media formicrowave shields,” in Proc. Int. IEEE Symp. Electromagn. Compat.,Portland, OR, Aug. 14–18, 2006, vol. 3, pp. 598–602.

[4] A. H. Sihvola, Electromagnetic Mixing Formulas and Applications.U.K.: IEE Publishing, 1999.

[5] A. N. Lagarkov and K. N. Rozanov, “High-frequency behavior ofmagnetic composites,” J. Magn. Magn. Mater., vol. 321, no. 14, pp.2082–2092, Jul. 2009.

[6] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics ofContinuous Media, 2nd ed. Oxford, NY: Pergamon, 1984, sec. 82.

[7] J. Xu, M. Y. Koledintseva, Y. He, R. E. DuBroff, J. L. Drewniak, andB. Matlin, “Measurement of electromagnetic parameters and FDTDmodelling of ferrite cores,” in Proc. IEEE Symp. Electromag. Compat.,Austin, TX, Aug. 17–21, 2009, pp. 83–88.

[8] M. Y. Koledintseva, J. Wu, J. Zhang, J. L. Drewniak, and K. N.Rozanov, “Representation of permittivity for multi-phase dielectricmixtures in FDTD modeling,” in Proc. IEEE Symp. Electromagn.Compat., Santa Clara, CA, Aug. 9–13, 2004, vol. 1, pp. 309–314.

[9] M. Koledintseva, J. Drewniak, Y. Zhang, J. Lenn, and M. Thoms,“Engineering of ferrite-based composite materials for shielding enclo-sures,” J. Magn. Magn. Mater., vol. 321, pp. 730–733, 2009.

[10] J. Xu, M. Y. Koledintseva, S. De, A. Radchenko, R. E. DuBroff, J.L. Drewniak, Y. He, and R. Johnson, “FDTD modeling of absorbingmaterials for EMI applications,” in Asia-Pacific Int. Symp. Elec-tromag. Compat.(AP-EMC’2010), Beijing, China, Apr. 12–16, 2010,pp. 173–176.

[11] J. Xu, M. Koledintseva, Y. Zhang, Y. He, B. Matlin, R. E. DuBroff,J. L. Drewniak, and J. Zhang, “Complex permittivity and permeabilitymeasurements and finite-difference time-domain simulation of ferritematerials,” IEEE Trans. Electromagn. Compat., to be published.

[12] M. Y. Koledintseva, J. L. Drewniak, Y. He, and B. Matlin, “Model-ling of absorbing material containing magnetic platelets,” in 2009 EMCSymp., Kyoto, Japan, Jul. 20–24, 2009, no. 24P3-3, pp. 809–812.

[13] T. Tsutaoka, “Frequency dispersion of complex permeability in Mn-Znand Ni-Zn spinel ferrites and their composite materials,” J. Appl. Phys.,vol. 93, no. 5, pp. 2789–2796, 2003.

[14] R. Ramprasad, P. Zurcher, M. Petras, M. Miller, and P. Renaud, “Mag-netic properties of metallic ferromagnetic nanoparticle composites,” J.Appl. Phys., vol. 96, no. 1, pp. 519–529, Jul. 2004.

[15] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations andWaves. Boca Raton, FL: CRC Press, 1996.

[16] K. N. Rozanov, A. V. Osipov, D. A. Petrov, S. N. Starostenko, andE. P. Yelsukov, “The effect of shape distribution of inclusions on thefrequency dependence of permeability in composites,” J. Magn. Magn.Mater., vol. 321, pp. 738–741, Apr. 2009.

[17] D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Kon-stanten von heterogenen Substanzen,” Annalen der Physik, vol. 5, no.24, pp. 636–679, 1936.

[18] D. Polder and J. H. van Santen, “The effective permeability of mixturesof solids,” Physica, vol. 12, no. 5, pp. 257–271, 1946.

[19] J. Helszajn, “A method of determining the effective axial demagne-tizing factor of an irregular hexagonal magnetic insulator,” IEEE Mi-crow. Wireless Comp. Lett., vol. 15, no. 5, pp. 392–394, May 2005.

[20] M. A. Abshinova, A. V. Lopatin, N. E. Kazantseva, J. Vilcakova, and P.Saha, “Correlation between the microstructure and the electromagneticproperties of carbonyl iron polymer composites,” Composites: Part A,vol. 38, pp. 2471–2485, 2007.

[21] H.-P. Chan and K. Doi, “Determination of radiographic screen—Filmsystem characteristic curve and its gradient by use of a curve-smoothing technique,” Med. Phys., vol. 5, pp. 443–447, 1978.

[22] D. A. Brandreth, “Some practical aspects of curve fitting,” J. Chem.Educ., vol. 45, p. 657, 1968.

[23] N. Morrison, Introduction to Sequential Smoothing and Prediction.New York: McGraw-Hill, 1969.

[24] J. E. Gentle, Numerical Linear Algebra for Applications in Statistics.Berlin, Germany: Springer-Verlag, 1998, pp. 95–97.

[25] C. Moler, Numerical Computing With MATLAB sec. 5, Society forIndustrial and Applied Mathematics, 2004.

[26] M. Kirkpatrick and N. Heckman, “A quantitative genetic model forgrowth, shape, reaction norms, and other infinite-dimensional charac-ters,” J. Math. Biol., vol. 27, pp. 429–450, 1989.

[27] M. Lin and R. L. Wu, “A joint model for nonparametric functional map-ping of longitudinal trajectories and time-to-events,” BMC Bioinform.,vol. 7, no. 138, pp. 1–12, 2006.

systematic analysis and engineering of absorbing materials...

Documents