S.A. Korchagin
Composite materials modeling using equivalent electrical circuits.
Article tutor:PhD D.V.Terin
Language advisor:A.V. Vozdvizhenskaya
Abstract
The models of the Maxwell - Garnett and Bruggeman were considered for
dielectric permittivity of composite materials. Composite consisting of conductor
particles in an insulating matrix was analyzed by the electrical equivalent circuit
method. This method has certain advantages including in the possibility of
effective methods of attracting electric circuit theory.
Keywords: composite materials, the effective-medium model, the equivalent
circuit diagram method, determination of the dielectric constant.
Composite materials find wide application in many fields of human
activity, including electronics and electrical engineering to create capacitors, power
resistors, sensors, memory and other devices.
Acquisition of new knowledge about the object of study is closely related
to the definition of its characteristics, which in many cases is difficult, time
consuming, costly, and often totally inaccessible for direct measurement. An
example of this is the problem of determining the physical characteristics of an
inhomogeneous material inside the body. In this case, the most effective (and
sometimes the only) way to research it is mathematical and computer modeling.
The purpose of the work is to describe a mathematical model of a
composite medium consisting of a dielectric matrix with conductive particles, to
develop an equivalent circuit diagram of the medium. Our primary task isto
compare the results obtained on the basis of the relations Bruggeman, Maxwell Garnett and the method of equivalent circuits for the complex permittivity of the
mixture component parameters in the mixture and external influences.[3]
Effective medium theory is applied to determine the dielectric constant
composite. The essence of the model is that the set of clusters that make up the
composite is considered as a kind of new medium with a so-called effective
dielectric constant. Thus, knowing the settings for each of the composite
components, their geometrical form and concentration can be obtained to
determine characteristics of the composite medium as a whole. Models of Maxwell
– Garnet and Bruggemanare most widely used as models of the effective medium
theory. [1]
Maxwell - Garnet model is valid and can be used only in the case when one
material is matrix and the other constitutes an isolated inclusion therein, the
proportion of which is small[4].
𝜀𝑒𝑓𝑓 − 𝜀2
𝜀1 − 𝜀2
= 𝑓1
,
𝜀𝑒𝑓𝑓 + 2𝜀2
𝜀1 + 𝜀2
(1)
Bruggeman model is used in the statistical environment, ie if in the
composite matrix and cannot be isolated inclusions. There is a medium formed by
spherical particles out of the two materials. In this model, it is assumed that each of
the particles of the materials placed in the so-called effective medium with an
effective dielectric constant 𝜀𝑒𝑓𝑓 , different from the dielectric constant materials
with a dielectric constant and 𝜀1 , 𝜀2 , within the material.[2]
𝑓1
𝜀1 − 𝜀𝑒𝑓𝑓
𝜀2 − 𝜀𝑒𝑓𝑓
+ 𝑓1
= 0,
2𝜀𝑒𝑓𝑓 + 𝜀1
2𝜀𝑒𝑓𝑓 + 𝜀2
(2)
The picture shows the equivalent circuit of the composite dielectric matrix
with conducting inclusions. It also shows the formulas for calculating the complex
permittivity of the material.
Pict. 1
Pict. 2
Graphs of dependence of the material on the frequency of an external
electric field are plotted.
Pict.3
The use of equivalent circuits is more convenient as electrical, mechanical
and magnetic components of the composite are presented in the form of electrical
equivalents. The method of equivalent circuit has an advantage of applying electric
circuit theory effective methods, and also in that the problems is partly solved
already during production. Application of method of equivalent circuits can give us
the same accurate results as a direct solution of the wave equation.
References
[1] Buchelnikov V.D.,.Louzguine-Luzgin D.V, Xie G., Li S., Yoshikawa N., Sato
М., AnzulevichА.Р., Bychkov I.V., Inoue А. Heating of metallic powders bу
microwaves: Experiment and theory / // Journal of Applied Physics. 2008. Vol.
104. № 1. P. 113505–10
[2] But’ko L.N., Buchel’nikov V.D., Bychkov I.V. Absorption of Electromagnetic
Waves in a Nonmagnetic Conductor–Ferromagnet Structure // Physics of the Solid
State. 2010. Vol. 52. №10. P. 2154–2163
[3] Lutich A A et al. Appl. Phys. B 84, 327. 2006
[4] Maxwell-Garnett J C Philos. Trans. R. Soc. London 203, 385. 1904