NATL INST. Of STAND S TECH R.I.C. N!ST

PUBLICATiONSi

AlllOM MbBlll

NisrUnited States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1371

Effective Medium Theory for

Ferrite-Loaded Materials

Richard G. Geyer

Joseph Mantese

James Baker-Jarvis

)0

1753

1371

194

Effective Medium Theory for

Ferrite-Loaded Materials

Richard G. Geyer

Joseph Mantese

James Baker-Jarvis

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

October 1994

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, AratI Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1371, 24 pages (October 1994)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1994

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Contents

1 Introduction 2

2 Theory 2

2.1 Modified Maxwell- Garnett Formulation 2

2.2 Lorentz-Lorenz, Maxwell-Garnett, and Bruggeman Formulations 8

3 Measurements 10

4 Summary 17

5 References 17

ni

EFFECTIVE MEDIUM THEORY FOR

FERRITE-LOADED MATERIALS

Richard G. Geyer^, Joseph Mantese^, and James Baker- Jarvis^

A ferrite-loaded composite medium is modeled by spherical inclusions spaced

equally on a cubic lattice within a host matrix. Both the inclusions and host matrix

may be magnetically permeable and possess dielectric and magnetic loss. The ferriteloaded medium may be considered to consist of excited Hertzian electric and magnetic

dipole sources. Effective medium rules of a modified Maxwell-Garnett form can be

derived by analysis of plane- wave propagation through the composite. These rules do

not yield symmetric characterization of two-phase media. They are compared with

other effective medium theories (Lorentz-Lorenz, Maxwell-Garnett, and Bruggeman)

and broadband coaxial transmission line data measured on ferrite-loaded titanates of

known composition. The modified Maxwell-Garnett rules give both lower and upper bounds for the effective permittivities and permeabilities of the composite and

yield accurate estimates of bulk electric and magnetic properties for low volumetric

inclusion loading. The Bruggeman formahsm yields the best predictive permittivity

and permeabihty values when volumetric percentages of the inclusions and host matrix are approximately equal. Generally, maximal magnetic loss factors occur at a frequency where the static initial permeability decreases by a factor of one-half, and

the relaxation frequency for ferrite composites increases with decreasing static initial

permeability.

Key words: composites; effective medium; ferrites; microwave; mixing rules; permeabihty; permittivity

^Electromagnetic Fields Division

^General Motors Research Laboratories, 30500 Mound Road, Warren, MI 48090-9055
1 Introduction

Dielectric and magnetic composite materials have complex permittivity and permeability properties that are determined by their constituents. Frequently, spectral

characteristics for the dielectric and magnetic properties that differ from those of individual constituent materials are needed. Hence it is necessary to produce a material

whose electromagnetic properties follow some specified behavior. This may be accom- phshed by either loading a material or by making multiphase mixtures. The loading

constituent may take the form of small spheres, ellipsoids, platelets, rods, or other

shapes. The bulk properties of the composite will depend on the alignment of the

loading particles and therefore may be dielectrically or magnetically anisotropic. The

loading particles may also interact with each other. Thus, the size, shape, and alignment of the loading material particles enter into mathematical formalisms describing

the electromagnetic behavior.

Effective medium theory has been studied for many years [1-6]. In this paper

we evaluate the apphcability of coupled electric- and magnetic-field integral equation

solutions for the effective propagation constant and impedance of a two-phase mixture that is modelled by a uniform lattice of spheres embedded in a matrix [4]. In

the formalism the spherical loading particles and matrix may have any dielectric or magnetic properties. The inclusions may be electrically and magnetically penetrable

or nonpenetrable. For example, porosity in a polycrystalline ceramic or expanded

polymer may be modeled by lossless inclusions whose relative permittivity is unity.

Similarly, the microscopic structure of various metal glasses, colloidal solutions, and

metallic films may be modeled by nonpenetrable metallic inclusions. Calculated dielectric and magnetic properties corresponding to certain assumed microstructures

may be compared with those observed in laboratory measurements. In the limiting case where the sphere radius is much less than the wavelength in the matrix, the

mathematical formalism simplifies markedly.

2 Theory

2.1 Modified Maxwell- Garnett Formulation

Consider the case (figure 1) in which the matrix material, characterized by e^,//^, is loaded by spherical particles of radius a that have complex permittivity and permeability e*fi* and that are equally spaced in a cubic lattice with distance d between

particle centers. Now allow a plane wave of unit amplitude, polarized in the x- direction and propagating in the z- direction, to be incident upon the matrix loaded with
T

d

1

O

O

o

o o

o o

o o

o o

o

o

o

Ep'^ip

^m' M^m

-^y

H,

rrr m

Figure 1. Uniform lattice of spheres in dielectric permeable matrix.

the spherical particles. Then, for e-''^' time dependence,

and

llii

E.

1 dE.

-ImZ

"1 ^o^~7m2

J^/^; d:

(1)

(2)

where 7^ = jk^ = ju)y/JP^, 11*^ = /io/^;^ - Mf^'r,m - Jf^'U)^ C = ^oK,m — ^o{(-'r,m~J^r,m)i ^^^ M*,m ' Km dcnote the relative complex permeability and complex

permittivity of the matrix. The permittivity and permeability of free space are Cq =

8.854 X lO"-'^ F/m and fiQ = Att x 10"^ H/m. At 2 = 0, the reflection coefficient F is given by

r =

(/^:///<//)' - if^uo'

(3)

where the reflected wave is assumed to be a plane wave and where fields propagate

in the loaded material with a bulk or effective propagation factor kg/j = Ljyje'jjfiljj.
The field at a given particle inclusion location (u, v, w) is the sum of the incident

field and a scattered field composed of all fields reradiated by the rest of the particles.

We may assume that the total exciting field is a function of only the coordinate w,

since all particles in the plane z = w are affected in the same way; that is E^ = E{w)

and Hy — H{w). At the center of a loading particle, all components of the scattered

field may be considered negligible. If we further allow the particle radius to be

sufficiently small, the field can be considered constant over its volume. Hence, the

effect of a loading particle can be found from a consideration of the scattering due to a sphere in a medium characterized by (e^,/ij^) that is excited by E[w) and H{w).

In general, we may consider the ferrite- loaded medium to consist of excited electric

and magnetic dipole sources [5], which may be described in terms of Hertz vectors,

n^ and H"^. In the matrix both Hertz vectors are multiples of e"-' '"''/r, where r is the distance from the source point {u^v,w) to the field point {x,y,z). The fields in

the matrix may be expressed as ^ = V(V • H^) + klW - ju^iiy X H-, (4)

and

H = V(V • H"^) ^klW" + jueiy x ^^ (5)

The electric Hertz vector has only an a;-component and the magnetic Hertz vector

only a y-component, so that

and

where C and D are determined by continuity of tangential field components at the

surface of a sphere in the matrix. Equations (4) and (5) must be written in spherical coordinates before application of continuity conditions. The contribution to the

scattered electric field from H^ may be written in spherical coordinates as

, , ,(d'^U ,2, A cos9cos(l> f d^U dU\

V{V-U^) + kl^W = ^Jsm^cos0U-^ + A:lf/ +--^fr^-^-—

+

sin(f> ( dU

r^ sin^ \d(j)
+ -[rcos<^(^sm^^ + cos^ —

sm p „ . „ . — cost^-

d(l>de

+ fc^r (/cos 6^ COS + ^- r COS0 - sm 0-^rd(f>

csc^fsm^— + cos<^—

(8)

where iir^ue^u^ are unit vectors in spherical coordinates, 11^ = u^f/, and [/ is a multiple of e~^'^'"'"/r. The contribution to the electric field from 11'^ is given by

—jupi*^V X n"^, where

V X n"^ = —Ur

r

+

r

r

. ,dV dV

sm 0^— + r cos 0^-

sm r cos U— sm t^-^rr- \ or od

(9)

w ,V, and y is a multiple of e •''"'"''/r. The dominant contributions to the electric field are from the induced electric

dipoles, particularly when the radii of the spherical loading particles become small

relative to the matrix propagation constant or under the condition \kmO'\ <C 1. Therefore one simplification that can be made is to consider only the Hertzian vector of

the electric type for the electric field. This condition further simplifies the formulation by reducing the angular derivatives of the Hertz vectors to zero. The spherical

components of the electric field exterior to a single loading sphere may then be written

Er = sin 9 cos

Eo = cos 6 cos

+ ki

1 d

r dr

+ kl

r^'

jkmr

a'-

r

-jkmr

(10)

(11)
and

E, sin (f) ld_ 2 r or

C-

-jkr, (12)

The total field external to a single particle is the sum of the total exciting field

E{w) plus the scattered field due to that particle. In order to match boundary conditions on continuity of the normal electric displacement field and tangential electric

field components at the surface of a particle, we must write the expressions for the

electric field internal to a sphere. In this case the Hertz vectors (or Green's function)

have the form of the zero-order spherical Bessel function or sin(A;pr)/r in order that

the fields be finite at the center of the sphere. Therefore, with the same condition

invoked that simphfied eqs (8) and (9) to eqs (10) through (12), the components of

the internal electric field of any sphere are

Er = sin 9 cos </>

Eg = COS 6 cos (f)

+ ki

"1 d

r dr

+ K

c

iSm[kpr)

C

,sin(A:pr)

and

E^ = — sin (f) -— e

r dr ^ C

,sm{kpr)

(13)

(14)

(15)

Application of continuity conditions on the radial component of the dielectric displacement field and on the tangential electric field gives, for Ik^a] <C 1,

C =

a^E{w)[ak,{e':^ + 2e;) cos{k,a) -f {kye*^ - {e*^ + 2e;)) sm{k,a)]

2fca(em - e;) cos{kpa) -f {kla^e*^ + e; - e*^) sin(A;pa)]

(16)

In an exactly analogous fashion, we may write the contribution to the scattered

magnetic field from a single sphere under the condition Ik^al *C 1 and, by applying

continuity conditions on the radial magnetic induction field and tangential magnetic

field, obtain

D

a^H{w)[kpa{fi*^ + 2fi;) cos(fcpa) + (fc^aV;, - (//;; + 2/^;)) sm{kj,a)]

'^[kp^il^m - f^*p) cos{kj,a) -I- (A;2aV;^ - {l^*m - f^p)) sin(^pa)]

(17)

The scattered external electric and magnetic fields of any sphere located at (w, v, w)

may now be written as
^scat ^ ^. 7-1/ \ a,p ^m (

'-' ^kl)^jujlil^H{w) f^*a,P fJ'm d

f^lp + '^f^m dz

-jkmr

(18)

and

TTScat ^2 Hy -a H{w) f^a,p - f^r K,p + '^t^*m dy (l^ + e) + ;^4^H

e — e d

<,p + 2e;, 5z

y-jkmT, (19)

where, in general, /i* ^ = /jLpG{kpa), e* ^ = e*G{kpa) are the apparent complex permeability and complex permittivity modified from the true complex permeability and

complex permittivity by the particle inclusion penetration factor,

G{kpa) = 2[sin(A;pa) — kpacos{kpa)]/[kpacos{kpa) + {k^a^ — l)sin(^pa)]. (20)

Note that E{w),H{w) are the total electric and magnetic fields, which consist

of the incident field (eq (1)) and the scattered fields from all other spheres. The

total field external to any single loading spherical particle is the sum of the exciting

field, which is the incident plane-wave field plus the re-radiated fields from other

spheres in the matrix, and the scattered field from that particle. If the scattered

fields from all the spheres on a uniform lattice are evaluated, we obtain, after some

derivation, the following coupled integral equations for the total electric and magnetic

field components, subject to the condition that | k^a |<C 1,

El{z) = e-^'-^^uEl{z) "'^

'0,,p

a,p ' m

/•oo 'UfP e* „ + 2e^ a,p ' m

vKH

fi + (?)^^;H f^*m d

K,p + 2/^;, dw

)g-Jfc^k-zl^^^

(21)

^.(.) . (^)le--..^:(.)|^-^./;(,,^|^^J(.) e

el .1

K.va,p + 2/.; C d

(22)

'afP 'm

where v = 47ra^/(3c^^) is the volumetric loading. The three terms on the right side

of eqs (21) and (22) may be identified respectively as the incident field at 2, the
scattered field from a spherical particle at the point z, and the scattered field from

all the other particles. These equations may be solved for the effective propagation

factor and impedance to give e*yy,/i*yy; that is,

<n = 4 f

1 + . n'^'T^flL ^ ' (23)

and

/ Su(u*-ur) \

At least formally, eqs (23) and (24), which have a modified Maxwell- Garnett form,

are consistent and correct in the dilute and dense extremes. In other words, when

there is no loading {v = 0), e*yy = e^, and //*yy = //^. When there is vanishing host

matrix (i/ = 1, \kpa\ > 1), e*jj = e; and //*yy = //;.

2.2 Lorentz-Lorenz, Maxwell-Garnett, and Bruggeman

Formulations

The effective dielectric and magnetic properties given by eqs (23) and (24) incorporate

composite microstructure into the formalism and do not yield symmetric characterization of two-phase media. In this respect, they differ from other effective medium

theories (Lorentz-Lorenz, Maxwell- Garnett, and Bruggeman) used for representing

heterogeneous mixtures [7-11]. Generally, all these theories yield a mixture rule of

the form

where ^*yy,^^ represent either the complex permittivity or permeability for the ef- fective medium and the host medium, and 0^ , i/jj, . .

. , i/'n represent either the complex

permittivity or permeability for the inclusions of types 1, 2, . .

. , n. z/j, 1/2, . .. , i^n denote the volume fractions of inclusions 1,2, ...,n. The underlying assumptions of

the Lorentz- Lorenz (LL), Maxwell-Garnett (MG), and Bruggeman (B) models are

also spherical inclusion geometry and dipole interactions. Spherical inclusions produce the smallest dipole moment, given the amount of particle polarizable material.

The primary difference in the LL, MG, and B models is in the choice of the "host"

medium. The LL approximation was essentially developed to describe point polarizable particles embedded in vacuum. In this case 0^ = 1 so that eq (25) simply

becomes

8
The LL model usually is an approximation for the roughness involved in island formation film growth up to a percentage volume fraction of about 20.

The normal MG approximation corresponds to inclusions in a matrix background

other than vacuum. For a two-phase medium, eq (25) then becomes

reff + '2^:n Vr + 2^;;' ^

^

which works fairly well if the inclusions make up a small fraction of the total volume.

For both the modified MG and MG formulations, different effective permittivities

and permeabilities result when the roles of the matrix host and particle inclusions are interchanged, even if the respective volume fractions remain the same. Bruggeman proposed making the properties of the niatrix or host the same as the

effective medium itself, yielding the following two- phase composite effective medium

symmetric formulation,

Hp^ ril^^y^ll

^= 0. (28)

The Bruggeman formulation for a two-phase composite leads to the solution of the

quadratic,

r,r2 + (^i*(2^i - ^2) + V'2*(2^2 - ^1)) V'e*// - 2(C//)' = 0. (29)

Equations (23) and (24) also differ from the LL, MG, and B rules in that the

loading particle permeabihty and permittivity properties are modified by a complex

amplitude and phase wave number function that quantifies how well the field penetrates the particle. For example, the penetration factor modifies the dielectric and

magnetic properties of the loading particles in accordance to the ratio of size and

electrical wavelength in the spheres and is given in figure 2 as a function of \kj,a\.

When the size of the loading spheres becomes small relative to wavelength in the

sphere, G(|A;pa|) « 1. This corresponds to the case where the loading spherical inclusions are completely penetrable and /i*p = //*,e*_p = Cp- The effective values e*y^ and

/i*yy can be found throughout the complex plane for high particle permittivity and

permeability.
25

20

15

10 — 5

Q. 3

O

-10

-15 -

_l I I I

I

I I I I I

I I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I II I I I

I

I I I I

I

I I M_

-20 'i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 10

Ikpal

Figure 2. Penetration function modifying properties of loading particles.

3 Measurements

Over the years many methods have been developed for permittivity and permeability

determination [12,13]. These methods include free-space, open-ended coaxial probe,

cavity-resonators, and transmission-hne techniques. Each technique has its own inherent limitations. Transmission-line techniques, usually made in rectangular or coaxial

waveguides, are the simplest of the relatively accurate ways of measuring permeability and permittivity. Coaxial hues are broadband in the TEM dominant mode and

therefore are attractive for spectral characterization of lossy magnetic materials, despite the problems of measurement uncertainty in complex permittivity determination

introduced by potential air gaps between the sample and the coaxial line center con- ductor. Details of two-port, reference-plane invariant scattering parameter relations

that can be used for determining permittivity and permeability are given elsewhere

[13]. One set of equations for single sample dielectric and magnetic measurements

from two-port scattering parameters is

5*11522 - ^21512 = exp(-27o(LatV - L))

R'

I- B?T'^'

(30)

10
and

where

(5,2 + S2,)/2 = exp(-7„(i.., - i))Ill^, (31)

ji^tllZ^^ (32)

T = exp(-7l), (33)

V ^c Clab

Ci,ac a-nd c/oft are speed of light in vacuum and laboratory, uj is angular frequency,

Ac is cutoff transmission-line wavelength, e*,//* are the effective relative complex

permittivity and permeability relative to vacuum, and Lair,L are air-line and sample

lengths. Equations (30) and (31) may be solved either explicitly or implicitly as a system of nonlinear scattering equations at each frequency or by using a nonlinear

regression model over the entire frequency range [14]. The respective advantages of

either approach are discussed in [13]. The total attenuation loss a of a sample under

test for TEM mode structure in a coaxial transmission line is given by

a = Re{'y)

^vac = ^^\jt',ii', [(1 + tan2 b,Yl\\ + tan2 <5^)V2 4. tan b, tan (5^-1], (36)

vac

where tan^g — 4'/^r ^i^^ tan ^^ = /ir//^r- Room-temperature two-port coaxial

transmission-line measurements were made with a 14 mm line. Dominant TEM mode

scattering- parameter data were taken with an automatic network analyzer (ANA).

The ANA was calibrated such that phase measurement uncertainties were no greater

than 20 millidegrees over the measurement band (1 MHz to 2 GHz); this is within

the noise level of the network analyzer. Differential uncertainty analyses, which in- cluded sources of measurement error due to uncertainties in the magnitude and phase

11
of the scattering parameters, sample holder and sample dimensions were also performed. The influence of gaps between the sample and sample holder was mitigated

by conductive paste painted on the external surfaces of the sample that are in contact with the sample holder. Use of conductive paste minimizes uncertainty due

to gap problems, particularly with respect to permittivity characterization. At 100

MHz, uncertainties in both permeability and magnetic loss factor determination are approximately 2 percent. The "static" or low-frequency initial permeability is wellcharacterized. However, low-frequency (less than 10 MHz) phase instabihty that is more prominent in measurements on lower-loss samples is observed in the data. For

low-volumetric ferrite loading, we also see spurious mode generation at frequencies

greater than 1 GHz. This is due to sample imperfections and becomes more evident

on samples having lower dielectric loss. Examples of calculated effective permittivity and permeability of ferrite-titanate

mixtures whose constituents have measured room-temperature dielectric and magnetic spectra shown in figures 3, 4, and 5 are given in figures 6 and 7. These examples

compare predicted effective permittivity and permeabihty results from the modified

MG and B formalisms with measured broadband transmission line data. The predicted results used only the measured constituent data of the two-phase composite

given in figures 3, 4, and 5. Generally, the measured composite data compared well

with both the B and modified MG mixing rules, provided the proper matrix was

chosen in the MG case. Examples of B, modified MG ferrite matrix, modified MG

titanate matrix, and experimental data for permittivity and permeability at 100 MHz

are shown as a function of volume percent titanate in figures 8 and 9. Generally, for

low volumetric percentages of titanate, the modified MG rule worked well with the

ferrite as the host matrix and the titanate particles as the guest inclusions. Similarly,

for high volumetric percentages of titanate, the modified MG rule compared well with

measured data with the titanate chosen as the host matrix and the ferrite particles as the guest inclusions. The B formalism yielded the best predictive permittivity

and permeability values when volumetric percentages of the titanate and ferrite were

approximately equal. In general, the modified MG and inverse (interchange of matrix

and inclusion roles) MG rules give both lower and upper bounds for the effective

permittivities and permeabihties of the composite. Overall measured magnetic dispersion for several titanate-ferrite composite mixtures is shown in figures 10 and 11.

Maximal magnetic loss factors were observed at a frequency where the static initial

permeability decreased by a factor of approximately one-half. In addition, the relaxation frequency increases as the volume percent titanate increases, or as the static

initial permeability decreases.

12
100 "1 I I I I I I I

I

I I I I I I I I

I

I I I M I I I

I

I I I I I r I — Permittivity

--- Dielectric Loss Factor

10

0.1

0.01

I 11 • I ft M l< t

' I' I I'll II

S'i'i

llil^fe-.^.'i^.ii _]. .I [ I 1 1 1 1

1 ' I I

I I I 1

1 I 0.001 0.01 0.1 10

Frequency (GHz)

Figure 3. Measured dielectric spectra of magnesium-copper-zinc ferrite.

uuu 1 1 1 1 r 11 1| 1 1 1 1 1 1 M{ 1 1 1 1 1 1 1 l| 1 1 1 M Jl L

: — Permeability

-

- - - Magnetic Loss Factor .

100

10 r /

N. X

1 - \ ;

n 1 1 1 1 1 1 ] 1 1 1 1 r 1 t 1 1 1 il 1 1 1 1 1 t 1 1 1 0.001 0.01 0.1 10 Frequency (GHz)

Figure 4. Measured magnetic spectra of magnesium-copper-zinc ferrite.

13
10000

1000

100

10

1 0.001

-1 1—I—n—r-| 1 1 1 1—I

n—rq — Permittivity

--- Dielectric Loss Factor

_1_ I I i 1 I I L_ _I 1 I I 1 I L_ 0.01 Frequency (GHz)

0.1

Figure 5. Measured dielectric spectra of barium titanate.

10000

1000

n 1—I I I 1 1 I I M I Mr r Bruggeman

Modified Maxwell-Gamett/Titanafe Matrix Experimental Data: 87.5% BaTiOs

'^ 100

10

1 I 1 1

I r I I I I I I 1 I 0.001 0.01 0.1 1 10 Frequency (GHz)

Figure 6. Spectral comparisons of ferrite- titanate composite relative permittivity with

modified Maxwell-Garnett and Bruggeman predictions.

14
1000

100

i 10

~' '

I I I 11 i| I I I M M i{ 1

1 I I I 1 1 i |

1

1 I I I 1 1 q Bruggeman — - Modified Maxwell-Garnett/Ferrite Matrix Experimental Data: 25% BaTiOs

0.1 I 1—I

I I I II i l 1—I I I I II il I I

I 0.001 0.01 0.1 1 10 Frequency (GHz)

Figure 7. Spectral comparisons of ferrite-titanate composite relative permeability

with modified Maxwell- Garnett and Bruggeman predictions.

10000

1000

100

10

1

1

'

1

1

1 1 1 ' ' I1 1 1

1

1 1 1 1

1

1 1 1 1 11 1 1 1

1

1 1 1 1 1

1 1 1 1

1

I

I I I I

I

I I I I

I

I T I I

I

• Bruggeman

Modified Maxwell-Garnett/Ferrite Matrix

* Modified Maxwell-Garnett/Titanate Matrix

+ Experimental Data: 100 MHz

1 1 1 1 1 1 1

1 1 1 1 I J t I I I I I 1 11 1I I 1 1 1 1 1 1 10 20 30 40 50 60 70 80 90 100

Volume Percent BaTiOa

Figure 8. Permittivity of ferrite-titanate composite as a function of volume percent

titanate at 100 MHz.

15
10000

1000

100

lOr

I I I I II I I I [

I 1 I I

I

I I 1 I

I

I 1 I I

I

I 1 I I

I

I I I I

I

1 I I I

I

I 1 I I I

I I

• Bnjggeman

Modified Maxwell-Gamett/Ferrite Matrix * Modified Maxwell-Gamett/Titanate Matrix

+ Experimental Data: 100 MHz

1

I I 1 .1 1 I I

.

10 20 30 40 50 60 70 80 90 100

Volume Percent BaTiOa

Figure 9. Permeability of ferrite-titanate composite as a function of volume percent

titanate at 100 MHz.

1000

100

10

"T I I I I I M

I

1 I TTTTTT] I I I I I M 1

1

I I I I I I I Composite Permeability

vs. Volume % BaTiOa

1 I I I

I 1 1 1

1 ii I I

I 1 1 II ii I I 11 I I I 1 1 1 1

1

0.001 0.01 0.1 1 10

Frequency (GHz)

Figure 10. Measured relative permeability of ferrite-titanate composite as a function

of frequency.

16
1000

100

~i r—( I I 1 1 1] r—1—TTT I -I—I I I l[ u

1—I I I ri [] Composite Magnetic Loss

Factor vs. Volume % BaTi03

J 1

I 1 1 II ii 0.001 0.01 0.1 1 Frequency (GHz)

10

Figure 11. Measured relative magnetic loss factor of ferrite-titanate composite as a function of frequency.

4 Summary

Several approaches for predicting the spectral dielectric and magnetic behavior of

heterogeneous systems have been analyzed, subject to the simplifying constraint that

the loading particle radius is small relative to the wavelength in the enclosing host

matrix, but not necessarily small relative to the wavelength in the loading particle.

Examination of composite field behavior allows the engineering of a material's electromagnetic behavior over a broad frequency spectrum provided that the properties

of the constituents can be accurately measured. Structural effects not due to special

properties of the constituents may be imposed on the electromagnetic properties of

the composite.

5 References

[1] Maxwell, J.C., A treatise on electricity and magnetism. New York: Dover, 1954.

[2] Brown, W.F., "Dielectrics," in Handbuch der Physik, vol. 17, S-1, Springer Verlag,

Berhn, 1956.

17
[3] Bottcher, C.J.F. and Bordewijk, P., Theory of electric polarization, vols. 1, 2. New York: Elsevier, 1978.

[4] Kharadly, M.M.Z. and Jackson, W., "The properties of artificial dielectrics comprising arrays of conducting elements," Proc. lEE, vol. 66, part III, pp. 199-212,

1961.

[5] Lewin, L., "Electrical constants of spherical conducting particles in a dielectric,"

J. lEE (London), vol. 94, part III, pp. 65-68, 1947.

[6] Ding, K.H. and Tsang, L., "Effective propagation constants in media with densely

distributed dielectric particles of multiple sizes and permittivities, "in Progress in

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*U.S. GOVERNMENT PRINTING OFFICE: 1994-673-018/00053

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