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Modeling and Simulation of Dielectric Mixtures
using Finite Elements Method
C. Fărcaş1, R. Creţ2, D. Petreuş1 and N. Palaghiţă1
1
2
Faculty of Electrical Engineering,
Technical University of Cluj-Napoca,
Cluj-Napoca, Romania
Faculty of Electronics, Telecommunications and
Information Technology,
Technical University of Cluj-Napoca,
Cluj-Napoca, Romania,
Email: [email protected]
Abstract— We study the electromagnetic behavior and the
characteristics of dielectric mixtures. The effective permittivity of
non-homogenous mixtures is very important in order to design
composites materials with specific electrical characteristics.
Simulation results show the distribution of the electric field E
and the electric displacement D for ordered mixtures and for
those with random distribution of the inclusions. Knowing the
distribution of the electric field in the non-homogenous
compounds and determining its effective permittivity is
important in the design of composite materials with specific
electrical characteristics. The paper presents the simulation
results (realized in Maxwell 2D software). We are determined the
effect of shape, size and concentration of inclusions in host
material.
Keywords – dielectric permittivity, numerical modeling,
mixtures, finite elements method
I.
INTRODUCTION
Composite materials are widely used for electrical
applications. This fact led to an increased interest concerning
the study of the electromagnetic behavior and the
characterization of dielectric mixtures.
The main advantage of composites materials consists of the
fact that they are designed for special purposes [1], [2], [3].
The classical method, based on experimental trials, requires a
large amount of time and money, using the computer in the
early stage of the design represents an important advantage,
allowing us to estimate and correct the required properties.
Usually, composite dielectrics are represented by a
heterogeneous system and commonly two-phase composite
dielectrics consist of a host material with an inclusion of
another material. A composite dielectric is made of a polymer
that constitutes the host media, representing the matrix (with
relative permittivity 1), where organic or non-organic
inclusions (with relative permittivity 2) are distributed (in an
ordered or random way).
One can expressed the concentration of the inclusions in the
host materials as a volume fraction q, (where q is the ratio
between the volume of the inclusions and the total volume of
the mixture). For a low concentration of the inclusions (low
values of q), there are some analytical formulas available to
estimate the effective permittivity of the mixture ef, but for a
high concentration of the inclusions in the mixture (high values
of q, as in the case of composites materials used in electrical
insulators) the numerical simulations are required to study the
behavior of the dielectric material. For this purpose, one can
used software based on the finite element method, like
Maxwell software from ANSOFT Corporation. The electric
field E and the electric displacement D can be computed in
every point of the computation domain using the finite element
method. Based on average values of E and D, one can find the
effective permittivity of the composite materials.
The effective permittivity ef, of the mixture is dependent
on some factors:

the shape of inclusions;

the size of inclusions;

the distance between inclusions;

the concentration of inclusions;

the ratio between the relative permittivity of
inclusions (2) and the relative permittivity of host
materials, etc.
II.
THEORETICAL BACKGROUND
There are two ways to calculate the effective relative
permittivity: an analytical one and a numerical one.
A. Analytical Calculation Methods for the Permittivity of
Mixtures
The calculation of the permittivity of mixture on the basis
of the permittivity of the components and their concentration is
done by using various models. The calculation of effective
permittivity of the non-homogeneous dielectric is done by
either using equivalent schemes (in simple treatments,
available for the simple distributions of the phases) or theories
of the effective electric field or theories similar to these with
molecular approaches or considering a regular ordering of the
inclusions. It is considered a physical mixture, in which the
components do not chemically interact.
In the literature [1], [4], [5], there are several relations
available for computing the effective permittivity of dielectric
mixtures, based on different models of the non-homogeneous
dielectric mixtures. The most famous formulas are:

Maxwell – Garnett formula:
 eff  1

 eff  (d  1)1
 2  1
q
 2  d  11
0

B. Numerical Calculation Methods
In order to determine the effective permittivity, it is
important to determine the electric field distribution E  x, y 
within a domain. From the Gauss’ law [4] we can obtain the
electric field distribution:
where

Bruggeman formula:
1  q 1   eff 
 2   eff
q
1  d  1 eff
 2  d  1 eff


log  ef 
 q log 
i
i
 x, y 
0




~ 1
E
Ed 




~ 1
D
Dd 


Lorenz – Lorenz formula:
 eff  1
 1
 1
 1  q  1
q 2
0
 eff  2
1  2
2  2

q represents the concentration,
d – the dimension of space (2 for the bi-dimensional
problems, or 3 in case of three-dimensional media).
The concentration q can be expressed as a volume fraction:
q
V2

V1

where
V2 represents the volume of the inclusions and
V1 represents the total volume of the material.
In order to verify the results, the computed effective
permittivity of mixture can be compared with the upper and
lower limits established by Wiener:



~
Knowing the average electric field E and average electric
displacement, the effective permittivity is determined with:
 eff

~
D
 ~
E

2) Method 2:
The method 2 supposes the calculation of the electrostatic
energy over the computational domain. The effective
permittivity is obtained from:

1
1
1 1
 eff
E 2 d 
EDd 
2

2



the lower limit:
 eff min 



where

represents the charge density on the domain.
1) Method 1:
This method supposes the calculation of the average
~
~
electric field E and average electric displacement D over the
entire computational domain, with:
i


The effective permittivity of a composite can be computed
using one of the methods [4], [6]:
Lichtenecker – Rother formula (or logarithmical law of
mixtures):

   r  0 E ( x, y )    x, y  

1 2
III.


 eff max  1  q 1  q 2 

q1  1  q  2
the upper limit:
THE PRINCIPLE OF THE METHOD
The principle of the method of finite elements is based on
the integral formulation of the problems with partial derivates.
This formulation can be of a variation type when an
equivalence between the solution of the differential problem
and the function that optimizes a functional, usually a Lagrange
function of the system, is done. The domain places itself in
small-sized but finite elements, on which the known function is
approximated with the help of a function of linear attempt (for
elements of first rank), polynomial (elements of second rank)
or more complex (macroelements).
The effective permittivity of the dielectric mixture depends
on the shape and size of the inclusions. So, we have studied the
dielectric mixtures with various shapes of the inclusions:
cylindrical, rectangular, squared, triangular, horizontal ellipsoid
(with the long axis parallel to the electric field) and vertical
oriented ellipsoid (with the axis perpendicular to the electric
field), as in figure 1.
`
ε2
ε2
ε2
ε1
ε1
ε1
a)
b)
c)
a)
Figure 3. Matrix structured dielectric mixture: a) ordered way (matrix);
b)disordered way (statistical).
`
ε2
ε2
We performed simulations in Maxwell 2D software for the
structures from figure 3, with inclusion’s cross-sections like in
figure 1. For the simulations one used two types of mixtures:
ε2
ε1
ε1
ε1
d)
e)
f)
Figure 1. Binary dielectric structures with different shapes of the inclusion’s
cross-section: a) circle; b) rectangle; c) square; d) triangle; e) horizontal
ellipsoid; f) vertical ellipsoid)
The modeling was done in 2D and we study the influence
of the inclusion’s geometry upon the effective permittivity εef.
We modeled a simple binary structure with different shapes of
the inclusion’s cross-section (considered to be cylinders or
infinitely long prisms, for 2D modeling). The computation
domain was considered to be a square with the side of 10 m
(figure 2).

mixtures 1 was polyethylene with glass inclusions
and

mixtures 2 was polyethylene with barium titanate
(BaTiO3) inclusions.
The relative permittivities of the components are: 1=2.25
for the host material (polyethylene) and 2=3.78 for the glass
inclusions, respectively 2 =1000 for the BaTiO3 inclusions.
The results of the numerical computation of the effective
permittivity are given in table 1.
TABLE I.
VARIATION OF EFFECTIVE PERMITTIVITY OF TWO MIXTURES
WITH DIFFERENT SHAPES OF THE INCLUSION’S CROSS SECTION.
ε
Shape
Circle
Rectangle
Square
Triangle
Ellipsoid || to E
Ellipsoid  on E
V
0
n
V1=0V
b)
V2=10V
εeff
V
0
n
Figure 2. The computation domain with the boundary conditions assigned.
εeff1 for mixture 1
εeff2 for mixture 2
2.611893226
2.626417563
2.615193870
2.619990037
2.339934803
2.328272277
4.066342398
4.589550387
4.278775943
5.254818519
2.723327579
2.519362426
In figure 4 it is presented the variation of electric potential
when the inclusion has a triangular shape for the mixture 1
(polyethylene with glass inclusions) and shape for the mixture
2 (polyethylene with BaTiO3 inclusions).
The concentration of the mixture was considered q=0.3,
and the surface of the inclusion was computed as:

Sinclusion  q  S host square 

The study regarding the effect of the inclusion’s position
inside the dielectric mixture on effective permitivity, we
performed some simulations on some structures with 25
inclusions (with 5x5 squares of 1m side). Each of these
inclusions are positioned inside a square, in an ordered (matrix)
or disordered way (statistical), figures 3. Two types of mixtures
were modeled for different concentrations.
Figure 4. Variation of electric potential when the inclusion has a triangular
shape.
The distribution of the electric field E and electrical
displacement D are presented in figures 5a and 5b for the
mixture 2 (polyethylene with barium titanate). One can see that
the distribution of the electric field is uniform inside the
inclusions. Larger values of the field can also be observed
between inclusions, or between inclusions and the plates of the
plane capacitor.
2,440
2,430
2,420
εeff2
2,410
2,400
2,390
2,380
2,370
2,360
2,350
0
4
8
12
16
20
24
Distance [um]
b)
a)
Figure 6. Variation with distance between inclusions of effective
permittivity for a)mixture 1 and b)mixture 2.
b)
Figure 5. Variation of electric field (a) and electric desplacement (b).
The effective permittivities of mixtures vary with the
distance between inclusions. The radius of an inclusion was
considered to be r=2 microns. We studied the variation of the
effective permittivity by varying the distance between the two
inclusions, from the minimum of d=0 to the maximum of
d=12r (24 microns). In the table II it is presented the variation
of effective permittivity of mixtures with distances between
inclusions.
TABLE II.
VARIATION OF EFFECTIVE PERMITTIVITY OF TWO MIXTURES
WITH VARIATION OF THE DISTANCE BETWEEN INCLUSIONS.
εeff
Distance
d=0
d=2μm
d=4μm
d=6μm
d=8μm
d=10μm
d=12μm
d=14μm
d=16μm
d=18μm
d=20μm
d=22μm
d=24μm
εeff1 for mixture 1
εeff2 for mixture 2
2.278093619
2.277027228
2.276761658
2.276762259
2.276782985
2.276878173
2.277005645
2.277203074
2.277402008
2.277670073
2.277932512
2.278259099
2.278593454
2.431498328
2.365913617
2.359967690
2.357939643
2.357184891
2.357129213
2.357362583
2.357950836
2.358622804
2.359595494
2.360597417
2.361886507
2.363191827
IV.
The paper presents a study of two types of mixtures. The
first mixture has the relative permittivity of inclusions closed to
the permittivity of the host material and the second mixture has
the relative permittivity of inclusions very different to the
permittivity of the host material. The results of 2D simulations
are presented in the paper. Studying the results of simulations
one can say that the shape, size, and space distribution of the
inclusions is significant only for mixtures with high values of
the ratio k between the permittivities of the components. The
distribution of the electric field inside the inclusions is uniform,
but larger values of the field can be observed between
inclusions, or between inclusions and the plates of the plane
capacitor considered as a computational domain. For larger
values of the voltage applied, we must check if E is below the
high voltage breakdown value.
REFERENCES
[1]
[2]
[3]
Figure 6 show the variation of effective permittivity with
distance between inclusions for mixture 1 and 2.
[4]
2,279
εeff1
2,279
[5]
2,278
[6]
2,278
2,277
2,277
0
4
8
12
Distance [um]
a)
16
20
CONCLUSIONS
24
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