IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011 317
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Missouri University of Science and Technology, Rolla, MO 65401 USA
Laird Technologies, San Jose, CA 95131 USA
A methodology to efficiently design novel products based on magneto-dielectric materials containing ferrite or magnetic alloy inclusions is presented. The engineered materials should provide desirable frequency responses to satisfy requirements of electromagnetic compatibility/immunity over RF and microwave bands. The methodology uses an analytical model of a composite magneto-dielectric material with both frequency-dependent permittivity and permeability. The Bruggeman asymmetric rule for effective permeability of a composite is modified to take into account demagnetization factors of inclusions, and is shown to be applicable to platelet magnetic inclusions. Complex permittivity and permeability are extracted from the transmission-line measurements. A novel accurate and efficient curve-fitting procedure has been developed for approximating frequency dependencies of both permittivity and permeability of magneto-dielectric materials by series of Debye-like frequency terms, which is important for wideband full-wave numerical time-domain simulations. Results of numerical simulations for a few structures containing magneto-dielectric sheet materials and their experimental validation are presented.
Index Terms— Absorbing media, causality, composite material, electromagnetic compatibility, ferrites, frequency response, magnetic materials, microstrip line, transmission line measurements.
A
I. I NTRODUCTION design of wideband nonconductive absorbing shielding enclosures, protecting screens, wallpaper, coatings with specific filtering properties, and gaskets is important for solving numerous problems of electromagnetic compatibility (EMC) and improving immunity of electronic equipment [1], [2].
Composite electromagnetic wave absorbers (EMWA) and noise-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 or magnetic alloy inclusions in a composite may substantially increase the absorption level in the frequency range of interest
[3].
To engineer EMWA and NSS composite materials, including nanocomposites, it is important to adequately predict wideband frequency responses of constitutive electromagnetic parameters
(permittivity and permeability), as well as concentration dependences of these composites. There are many different mixing rules available in the present-day literature (see, e.g., [4] and references therein), but every rule has its own limitations. Thus, for an important case of composites filled with ferromagnetic metal powders, currently there is no standard and unified experimentally validated mixing rule to calculate dependences both on frequency and concentration, especially if inclusions are nonspherical [5]. One of the objectives of this paper is to present a model for effective permeability of mixtures containing magnetic (ferrite or ferromagnetic alloy) platelets.
Manuscript received July 01, 2010; revised September 15, 2010; accepted
September 23, 2010. Date of current version January 26, 2011. Corresponding author: M. Y. Koledintseva (e-mail: marinak@mst.edu).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2010.2084991
Another objective is to analyze a few scenarios of applying different magneto-dielectric absorbing sheet materials using full-wave numerical finite-difference time-domain (FDTD) technique, and verify modeling results by experiments. An advantage of time-domain numerical techniques is the possibility of having broadband responses. To effectively model magneto-dielectric materials in time domain, it is important to represent frequency characteristics of both complex permittivity and permeability of these materials as analytical rational-fractional functions that would satisfy Kramers-Krönig causality relations (KKR) [6], for example, sums of the Debye terms with the poles of the first order [7]. If a material exhibits narrowband resonances, then Lorentzian terms with poles of the second order should be used [8]. However, the present study is limited to the Debye terms only, since the majority of microwave absorbing materials can be described in terms of Debye dependencies only. The materials whose both permittivity and permeability frequency functions can be represented through
Debye terms are called double-Debye materials (DDM) [9].
A new accurate and efficient technique to approximate experimental or modeled (using corresponding mixing rules) frequency dependencies for permeability and permittivity of DDM materials is also described in detail in this paper. This curve-fitting technique is based on application of Legendre polynomials and least-square regression analysis, and the results of curve-fitting guarantee satisfying KKR, which is extremely important for numerical modeling correctness.
The presented methodology of designing absorbing materials includes an FDTD numerical code that allows for effective modeling of complex geometries containing frequency-dispersive materials and evaluating absorbing properties of the engineered materials, as well as filtering or shielding properties of structures that contain these materials. Such a code (EZ-FDTD) has been developed in the EMC Laboratory of Missouri University of Science & Technology [9]–[11]. The EZ-FDTD uses auxiliary 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 or
FDTD technique: as many terms as required by reasonable accuracy can be used in the code.
II. E FFECTIVE
P
M AGNETIC AND
ROPERTIES OF M
D IELECTRIC
IXTURES
Suppose that and are relative permittivities of the inclusions and base (background) materials, respectively, and they both can be complex functions of frequency:
. Effective permittivity of this mixture can be modeled with the most commonly used
Maxwell Garnett (MG) mixing rule [4], if the mixture satisfies quasi-static conditions, and the volume fraction of inclusions is below the percolation threshold, if inclusions are conducting.
Also, it is known that the MG mixing rule gives good agreement with experiment only when the contrast between the material parameter of a host matrix and inclusions is comparatively low
[5]. In the case of randomly oriented ellipsoidal inclusions with depolarization factors , where correspond to ,
, and directions, the effective permittivity of the mixture can be calculated through the MG rule as cessing for making a powder (such as milling, etc.) may affect intrinsic magnetic fields and, therefore, the permeability. Also, in single-domain particles, the parameters of the ferromagnetic resonance depend on the shape of the particle [14].
Magnetic platelets are of special interest due to their high internal field of shape anisotropy. Their form (demagnetization) factors are related to the magnetic anisotropy field incide inclusions, even if there is no crystallographic anisotropy in the material. 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 pronounced shape anisotropy of disk-like inclusions. It is known that the MG prediction is comparatively accurate for spherical or stone-shaped magnetic inclusions [16]. Demagnetization factors are present in (3) and (4), but these equations more accurately take into account shape of inclusions only at lower levels of permeability. The MG mixing formula (1) substantially overestimates effective permeability, when considering simultaneously high ( 1) permeabilities and aspect ratios of inclusions.
The proposed new analytical model for effective permeability of a composite containing magentic inclusions is based on a combination of the Bruggeman asymmetric rule (BAR) [17] and
Bruggeman symmetric rule (BSR) [4].
The BAR for effective permeability is written as [17]
(1)
(5)
If inclusions are all aligned, the corresponding MG formula gives components of the diagonal permittivity tensor
(2)
For the mixtures containing arbitrary-shaped magnetic crumbs or spherical inclusions, the BAR gives predictions which agree well with experimental data [9]. However, in (5) the demagnetization factors are missing.
At the same time, it is known that the BSR, which is a form of the Bruggeman effective medium theory (EMT), accounts for shape factors of inclusions [18] with index corresponding to , , and directions.
The formulas analogous to (1) and (2), obtained by simple replacing , can be used to predict behavior of effective permeability case of aligned inclusions.
in nonaligned case, or in the
(6)
(3)
For disk-shaped inclusions with an aspect ratio
(ratio of their diameter to thickness), the axial demagnetization factor is approximately [19]
(7) and
(4)
In the case of a nonmagnetic base material, tivities and permeabilities
. Permitcan be complex functions of frequency. However, it is important to mention that the intrinsic permeability of inclusions in the general case is different from the permeability of the bulk magnetic material (ferrite or alloy) these inclusions are made of. These values and can be related through the crush parameter , which is associated with demagnetization along the magnetic grain boundaries, and determines local magnetic interaction between neighboring grains, when the bulk material is crushed into a powder [12], [13]. There are some other reasons for the difference between the permeability of bulk material and powders. Thus, the magnetic material prowhile the other two depolarization factors
. Indeed, disk-like inclusions could be approximated as oblate spheroids.
It was noticed that for high inclusion-host permeability contrast and volume fractions of magnetic inclusions over 30%, the resultant obtained through the BSR (6) is much higher than the values predicted by BAR (5). This happens even if the aspect ratio of disk-like inclusions is small, , and the corresponding demagnetization factors are close to those in the spherical case .
The objective is to modify the BAR in such a way that it would be possible to apply it to disk-shaped inclusions. It is appealing to introduce the correction factor which would depend both on concentration and aspect ratio of inclusions, and it would be possible to calculate effective permeability for disk-shaped inclusions , knowing the corresponding effective

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: SYSTEMATIC ANALYSIS AND ENGINEERING OF ABSORBING MATERIALS 319
Fig. 1. Effective permeability calculated using BSR (EMT) and corresponding curve-fitting to retrieve the function
F (p; u)
.
relative permeability
Let us assume that this factor the BSR (6), too, so that in spherical case
(8) would be the same for
(9)
Fig. 1 presents curves as functions of the aspect ratio for different inclusion concentrations p calculated using BSR
(6). These curves are calculated for the magnetic material with intrinsic static permeability of inclusions responds to the bulk permeability
, which corand the crush parameter [9].
A fitting dependence for the factor is obtained by analyzing numerous dependencies as those in Fig. 1. It can be well approximated by a simple analytical expression
(10)
Fig. 2. Effective permeability calculated using: (a) Modified Bruggeman asymmetric rule (MBAR); (b) Maxwell Garnett (MG) rule.
where corresponds to the spherical case calculated using BSR (5), and and are fitting parameters. This is an important result for extending the BAR model.
Fig. 2(a) shows the effective permeability of the nonaligned mixture calculated using the modified BAR (5) and (8). Fig. 2(b) shows the permeability curves obtained using MG rule for nonaligned inclusions. In these calculations, the volume fraction of inclusions is 25%, and the initial data is the same as in Fig. 1: and MHz. Fig. 2(a) and (b) show that as aspect ratio increases, static permeability increases, and the loss peak slightly shifts to the lower frequencies.
It is seen that the MG rule predicts higher permeability than the modified BAR. As is mentioned above, the dependence of
MG on shape is known to be more accurate at lower permeability levels than at higher. For this reason, it is believed that the modified BAR predicts effective permeability for nonspherical inclusions more realistically than MG. For carbonyl iron
(CI) flakes with , G, and aspect ratio , the calculated data agrees quite well with experiments [20], as is shown in Fig. 3.
III. E XTRACTION OF M ATERIAL F REQUENCY D EPENDENCES
This section describes the new proposed curve-fitting procedure to approximate frequency dependencies of materials as sums of the Debye-like terms. This curve-fitting procedure can be applied both to experimentally obtained (for example, using
7/3 mm coaxial airline technique and a vector network analyzer) and to modeled through mixing rules ites [8] and and of compos-
(11)
(12)
Such representation of frequency dependencies of dielectric and magnetic properties as sums of Debye-like terms is convenient for using them in the FDTD simulations. Previously, a curve-fitting technique based on the genetic algorithm (GA) optimization was used for extracting parameters of the Debye terms from experimentally available data [7], [8]. The GA flowchart is shown Fig. 4(a). Though the GA yields a global optimum, it is a very tedious task to obtain proper results using

320 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011 systematic errors that may be due to fluctuations in the measured data [21]. In addition, sometimes in polynomial-fitting the following artifice is used: the given set of data points is transformed to another form, which is more easily represented by polynomials. For example, the given set of points may be multiplied by some function resulting in a linearized or a more simplified polynomial representation of the points. This requires deciding which function would be suitable for this transformation, and as stated in [22], there is no particular simple rule to follow.
In the proposed approach, the flowchart of which is shown in
Fig. 4(b), an orthogonal-polynomial fitting for the Debye curves by using the discrete Legendre polynomials was chosen. This technique allows for modeling permittivity and permeability as multi-term Debye curves with as many terms as required by the accuracy of curve-fitting and the ability of the numerical electromagnetic code to handle this number of terms. Legendre polynomials are used as the basis functions to model the measured data. In general, the -degree discrete Legendre polynomial can be expressed as [23]
(13) for ,
Legendre polynomial;
. Herein, and functions of the order defined by is the parameter of the are the backward factorial
Fig. 3. Experimental [20] and obtained using modified Bruggeman asymmetric rule (MBAR) permeability (a) real part, and (b) imaginary part.
(14) and the binomial coefficient is .
In terms of multiple regression of Legendre polynomials up to the
[24] degree, any smooth curve can be expressed as
(15) where if if is analogous to Kronecker symbol in the sampling frequency points. Then the sum of the squares of the differences between the experimental data and the smooth curve obtained as in (15) is given by
(16)
Fig. 4. Flowcharts for curve-fitting: (a) genetic algorithm, and (b) Legendre polynomial approximation and regression analysis.
Applying the least-squares criterion and performing some calculations, one could obtain the following expression for the regression coefficients
(17) the GA method. To make the GA converge fast to the optimal solution, a user must make a good preliminary guess and test numerous initial values at the very start, which is time-consuming and requires the user’s knowledge on using the GA and experience working with the code.
A powerful technique used in numerical methods for curvefitting the measured data is an estimation of parameters by the principle of least squares. This method can reduce random and where . The fitted curve estimated by the least squares method is obtained by substituting (17) to (15).
The final step in this approach is using the Matlab lsqcurvefit function [25] to approximate the smooth curve in (15) by the
Debye dependencies (11) and (12) for complex permittivity and

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Fig. 6. Measured, curve-fitted, and modeled using MBAR permeability of an absorbing material.
Fig. 5. Measured and curve-fitted permittivity of an absorbing material: (a) real part; and (b) imaginary part.
permeability, respectively. The lsqcurvefit function finds the coefficients that best fit data using the nonlinear least squares regression method as
(18) where is the data obtained using the Legendre polynomials
(15), and is the Debye data. The vector
(11), or is the set of Debye parameters for permittivity is for permeability (12). The index is the order of the corresponding Debye terms in (11) and (12), and is the order of the frequency sampling point. These vectors as arguments of are the Debye parameters of interest. The measured and curve-fitted data for permittivity and permeability of an absorbing magneto-dielectric material are shown in Figs. 5 and 6. The GA curve-fitting algorithm was realized for two Debye terms in permittivity, and three Debye terms in permeability. Realization of GA with more
Debye terms is cumbersome and time-consuming. The results of curve-fitting using regression analysis with Legendre polynomials shown in these figures were obtained with five Debye terms for permittivity, and five Debye terms for permeability.
As is mentioned above, there is no fundamental limitation for the number of terms. It can be seen that the Legendre polynomials provide a very efficient and robust way to perform a curve fit of the measured (or modeled) microwave permittivity and permeability by the Debye curves. As is shown by Kirkpatrick and Heckman [26], and as is implemented in [27], Legendre polynomials 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 estimable for high levels of curve complexity; and computations converge fast.
An important validation for causality involving the
Kramers-Krönig relations (KKR) could also be provided using this method. The real and imaginary parts of the particular material parameter (permittivity or permeability), according to the causality principle, are not independent. KKR are the integral relations which express this interdependence.
For example, for permittivity they are [6]
(19)

322 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011
Fig. 7. Microstrip line geometry coated with absorbing material: (a) experimental test structure, and (b) FDTD model setup.
where the principal parts of integrals are taken. The analogous relations are valid for real and imaginary parts of complex permeability.
Approximating measured permittivity and permeability by series of the Debye terms assures that the resultant curve-fitted frequency dependencies will be causal, while the initial measured data might violate causality in some frequency point or regions due to measurement errors. Causality of data which is then used in numerical codes is extremely important for the numerical stability and physical meaningfulness of the modeling results. To assure causality when curve-fitting measured data using the proposed approach, one could either first curve fit the imaginary data, and then apply the KKR to restore the real part, or, vice versa, first curve fit the real data, and then restore the imaginary part using the KKR. To correctly restore data using the KKR, some measured points for the curve to be restored are needed, since the KKR are valid up to some constant offset.
The regression analysis-Legendre polynomial curve-fitting procedure and KKR check may be applied several times in turn to real and imaginary parts to minimize the total error of curve-fitting both real and imaginary parts, while assuring causality of the curve-fitted response. The weight coefficients of curve-fitting accuracy are assigned to each frequency range of interest, as well as to real and imaginary parts of permittivity and permeability.
The curve-fitted data obtained using the Legendre polynomial-least squares regression method was successfully used in the FDTD simulations, and the results are discussed in the next section.
IV. FDTD S IMULATIONS AND M EASUREMENTS
The effect of various absorbing sheet materials on frequency characteristics of a specially designed microstrip line shown in
Fig. 7(a) has been studied. The length of the board is 14.7 cm with a 3.5 mm wide trace. The height of the dielectric is 1 mm and the relative permittivity is 3.53 with a tangent loss of 0.001
to provide the 50characteristic impedance of the board. The microstrip line was operating in two regimes, short- and opencircuited. Measurements are carried out in the range from 0.9 to
6 GHz using Agilent vector network analyzer E-5071C .
The same structure was modeled using the FDTD codes with the bulk discretization cells incorporating Debye dielectric and magnetic material (DDM). The modeling setup is shown in Fig. 7(b). In the simulations, the length was tuned to take into account the effect of two connectors. The cell size along direction is 0.1 mm, while the cell sizes along and directions are 0.5 mm.
Fig. 8. FDTD modeled and measured input impedance of the bare board in the open-circuit case: (a) real part, and (b) imaginary part.
Fig. 8 shows the simulation results of the bare board together with the measured data for the open-circuit termination. As is seen from this figure, the agreement between the simulated and measured results in the short-circuit case is excellent for both real and imaginary parts of the input impedance through the whole frequency range of interest. The difference of the resonance magnitudes may be explained by the perfect electric conductor assumption for ground plane and trace in the simulation and not sufficient loss in the board dielectric.
Fig. 9 shows the simulation and measured results for the same open-circuited board covered with same absorbing sheet as discussed in Section III. The FDTD model used the Debye terms obtained by the Legendre polynomial and regression curve fit
(see the corresponding curves in Figs. 5 and 6). Thickness of the absorbing sheet is 0.5 mm, and its width and length are both
10 mm. The sheet is put directly upon the trace. As is seen from
Figs. 8 and 9, in the loaded case, resonances damp and shift to the lower frequencies, and this does not contradict physics.
The positions of resonance peaks in the simulation and measurements agree well in the considered frequency range. Some discrepancy in amplitudes of peaks is due to the difference in the measured and modeled material parameters of the absorber, probable gap between the layer and the board, and underestimated loss on the bare board. The comparatively good agreement validates the correctness of the simulation method.
From the above analysis, it is clear that the new approach for frequency dependent material parameter extraction is a very

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: SYSTEMATIC ANALYSIS AND ENGINEERING OF ABSORBING MATERIALS 323
Fig. 9. Simulated and measured input impedance of the microstrip line covered with the absorber: (a) real part; (b) imaginary part.
promising method and will be used for future studies for this field of research.
V. C ONCLUSION
The methodology of designing new magneto-dielectric absorbing materials and structures on their basis for different EMC applications is presented. These applications may include radiation from heatsinks, parasitic resonances within enclosures, spurious radiation from chips and other active circuit components, etc. The proposed new mixing rule for predicting effective magnetic parameters of composites over wide frequency range gives reasonable agreement with measured results. FDTD code with accurate and satisfying causality relations curve-fitting of complex-shaped frequency characteristics by series of Debye terms is an efficient tool to evaluate whether a material could be a successful candidate for mitigating an electromagnetic interference. The optimization for choosing proper dielectric and magnetic properties of materials and their ingredients in composites, as well as geometries (configuration and thickness of layers) for particular practical problems can be done based on the results of this work.
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